[{"data":1,"prerenderedAt":10819},["ShallowReactive",2],{"post-\u002Fblog\u002F2024\u002F2024-09-12-box-muller-transformation":3},{"id":4,"title":5,"body":6,"description":9426,"draft":10807,"enableComment":9481,"extension":10808,"image":10809,"meta":10810,"navigation":9481,"onday":10811,"path":10812,"seo":10813,"stem":10814,"summary":10815,"tags":10816,"__hash__":10818},"blog\u002Fblog\u002F2024\u002F2024-09-12-box-muller-transformation.md","Box-Muller Transformation",{"type":7,"value":8,"toc":10801},"minimark",[9,14,18,269,548,788,848,852,1002,1682,1742,2079,2082,2233,2236,2697,2905,3612,4225,4317,4571,4574,4843,5235,5603,5606,5752,5812,5816,5986,5989,6082,6395,6455,6458,6787,7036,7039,7071,7251,7283,7625,7656,7907,7966,8264,8267,8371,8430,9356,9416,9420,9756,9767,9781,9784,9803,9806,10787,10797],[10,11,13],"h2",{"id":12},"content","Content",[15,16,17],"p",{},"The Box-Muller transform is a method used to generate normally distributed random variables from uniformly distributed random variables.",[15,19,20,21,130,131,202,203,206,207,130,238,268],{},"Specifically, it involves selecting two random variables, ",[22,23,27,57],"span",{"className":24,"translate":26},[25],"katex","no",[22,28,31],{"className":29},[30],"katex-mathml",[32,33,35],"math",{"xmlns":34},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[36,37,38,52],"semantics",{},[39,40,41],"mrow",{},[42,43,44,48],"msub",{},[45,46,47],"mi",{},"U",[49,50,51],"mn",{},"1",[53,54,56],"annotation",{"encoding":55},"application\u002Fx-tex","U_1",[22,58,62],{"className":59,"ariaHidden":61},[60],"katex-html","true",[22,63,66,71],{"className":64},[65],"base",[22,67],{"className":68,"style":70},[69],"strut","height:0.8333em;vertical-align:-0.15em;",[22,72,75,80],{"className":73},[74],"mord",[22,76,47],{"className":77,"style":79},[74,78],"mathnormal","margin-right:0.10903em;",[22,81,84],{"className":82},[83],"msupsub",[22,85,89,121],{"className":86},[87,88],"vlist-t","vlist-t2",[22,90,93,116],{"className":91},[92],"vlist-r",[22,94,98],{"className":95,"style":97},[96],"vlist","height:0.3011em;",[22,99,101,106],{"style":100},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[22,102],{"className":103,"style":105},[104],"pstrut","height:2.7em;",[22,107,113],{"className":108},[109,110,111,112],"sizing","reset-size6","size3","mtight",[22,114,51],{"className":115},[74,112],[22,117,120],{"className":118},[119],"vlist-s","​",[22,122,124],{"className":123},[92],[22,125,128],{"className":126,"style":127},[96],"height:0.15em;",[22,129],{}," and ",[22,132,134,153],{"className":133,"translate":26},[25],[22,135,137],{"className":136},[30],[32,138,139],{"xmlns":34},[36,140,141,150],{},[39,142,143],{},[42,144,145,147],{},[45,146,47],{},[49,148,149],{},"2",[53,151,152],{"encoding":55},"U_2",[22,154,156],{"className":155,"ariaHidden":61},[60],[22,157,159,162],{"className":158},[65],[22,160],{"className":161,"style":70},[69],[22,163,165,168],{"className":164},[74],[22,166,47],{"className":167,"style":79},[74,78],[22,169,171],{"className":170},[83],[22,172,174,194],{"className":173},[87,88],[22,175,177,191],{"className":176},[92],[22,178,180],{"className":179,"style":97},[96],[22,181,182,185],{"style":100},[22,183],{"className":184,"style":105},[104],[22,186,188],{"className":187},[109,110,111,112],[22,189,149],{"className":190},[74,112],[22,192,120],{"className":193},[119],[22,195,197],{"className":196},[92],[22,198,200],{"className":199,"style":127},[96],[22,201],{},", which are uniformly distributed over the interval (",[22,204,205],{},"0,1","). The random variables ",[22,208,210,224],{"className":209,"translate":26},[25],[22,211,213],{"className":212},[30],[32,214,215],{"xmlns":34},[36,216,217,222],{},[39,218,219],{},[45,220,221],{},"X",[53,223,221],{"encoding":55},[22,225,227],{"className":226,"ariaHidden":61},[60],[22,228,230,234],{"className":229},[65],[22,231],{"className":232,"style":233},[69],"height:0.6833em;",[22,235,221],{"className":236,"style":237},[74,78],"margin-right:0.07847em;",[22,239,241,255],{"className":240,"translate":26},[25],[22,242,244],{"className":243},[30],[32,245,246],{"xmlns":34},[36,247,248,253],{},[39,249,250],{},[45,251,252],{},"Y",[53,254,252],{"encoding":55},[22,256,258],{"className":257,"ariaHidden":61},[60],[22,259,261,264],{"className":260},[65],[22,262],{"className":263,"style":233},[69],[22,265,252],{"className":266,"style":267},[74,78],"margin-right:0.22222em;"," are constructed as 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",[22,792,794,807],{"className":793,"translate":26},[25],[22,795,797],{"className":796},[30],[32,798,799],{"xmlns":34},[36,800,801,805],{},[39,802,803],{},[45,804,221],{},[53,806,221],{"encoding":55},[22,808,810],{"className":809,"ariaHidden":61},[60],[22,811,813,816],{"className":812},[65],[22,814],{"className":815,"style":233},[69],[22,817,221],{"className":818,"style":237},[74,78],[22,820,822,835],{"className":821,"translate":26},[25],[22,823,825],{"className":824},[30],[32,826,827],{"xmlns":34},[36,828,829,833],{},[39,830,831],{},[45,832,252],{},[53,834,252],{"encoding":55},[22,836,838],{"className":837,"ariaHidden":61},[60],[22,839,841,844],{"className":840},[65],[22,842],{"className":843,"style":233},[69],[22,845,252],{"className":846,"style":267},[74,78]," follow a normal distribution with a mean of 0 and a variance of 1.",[10,849,851],{"id":850},"derivation","Derivation",[15,853,854,855,130,883,911,912,130,957,1001],{},"Assume that ",[22,856,858,871],{"className":857,"translate":26},[25],[22,859,861],{"className":860},[30],[32,862,863],{"xmlns":34},[36,864,865,869],{},[39,866,867],{},[45,868,221],{},[53,870,221],{"encoding":55},[22,872,874],{"className":873,"ariaHidden":61},[60],[22,875,877,880],{"className":876},[65],[22,878],{"className":879,"style":233},[69],[22,881,221],{"className":882,"style":237},[74,78],[22,884,886,899],{"className":885,"translate":26},[25],[22,887,889],{"className":888},[30],[32,890,891],{"xmlns":34},[36,892,893,897],{},[39,894,895],{},[45,896,252],{},[53,898,252],{"encoding":55},[22,900,902],{"className":901,"ariaHidden":61},[60],[22,903,905,908],{"className":904},[65],[22,906],{"className":907,"style":233},[69],[22,909,252],{"className":910,"style":267},[74,78]," are independent, normally distributed random variables with a mean of 0 and a variance of 1. Let ",[22,913,915,935],{"className":914,"translate":26},[25],[22,916,918],{"className":917},[30],[32,919,920],{"xmlns":34},[36,921,922,932],{},[39,923,924,926,928,930],{},[45,925,15],{},[289,927,301],{"stretchy":300},[45,929,221],{},[289,931,315],{"stretchy":300},[53,933,934],{"encoding":55},"p(X)",[22,936,938],{"className":937,"ariaHidden":61},[60],[22,939,941,945,948,951,954],{"className":940},[65],[22,942],{"className":943,"style":944},[69],"height:1em;vertical-align:-0.25em;",[22,946,15],{"className":947},[74,78],[22,949,301],{"className":950},[378],[22,952,221],{"className":953,"style":237},[74,78],[22,955,315],{"className":956},[429],[22,958,960,980],{"className":959,"translate":26},[25],[22,961,963],{"className":962},[30],[32,964,965],{"xmlns":34},[36,966,967,977],{},[39,968,969,971,973,975],{},[45,970,15],{},[289,972,301],{"stretchy":300},[45,974,252],{},[289,976,315],{"stretchy":300},[53,978,979],{"encoding":55},"p(Y)",[22,981,983],{"className":982,"ariaHidden":61},[60],[22,984,986,989,992,995,998],{"className":985},[65],[22,987],{"className":988,"style":944},[69],[22,990,15],{"className":991},[74,78],[22,993,301],{"className":994},[378],[22,996,252],{"className":997,"style":267},[74,78],[22,999,315],{"className":1000},[429]," be their probability density functions. Thus, we have:",[22,1003,1005],{"className":1004,"translate":26},[272],[22,1006,1008,1109],{"className":1007,"translate":26},[25],[22,1009,1011],{"className":1010},[30],[32,1012,1013],{"xmlns":34,"display":281},[36,1014,1015,1106],{},[39,1016,1017,1019,1021,1023,1025,1027,1040,1060,1063,1066,1068,1070,1072,1074,1076,1088],{},[45,1018,15],{},[289,1020,301],{"stretchy":300},[45,1022,221],{},[289,1024,315],{"stretchy":300},[289,1026,291],{},[1028,1029,1030,1032],"mfrac",{},[49,1031,51],{},[317,1033,1034],{},[39,1035,1036,1038],{},[49,1037,149],{},[45,1039,306],{},[1041,1042,1043,1046],"msup",{},[45,1044,1045],{},"e",[39,1047,1048,1050],{},[289,1049,323],{},[1028,1051,1052,1058],{},[1041,1053,1054,1056],{},[45,1055,221],{},[49,1057,149],{},[49,1059,149],{},[289,1061,1062],{"separator":61},",",[355,1064],{"linebreak":1065},"newline",[45,1067,15],{},[289,1069,301],{"stretchy":300},[45,1071,252],{},[289,1073,315],{"stretchy":300},[289,1075,291],{},[1028,1077,1078,1080],{},[49,1079,51],{},[317,1081,1082],{},[39,1083,1084,1086],{},[49,1085,149],{},[45,1087,306],{},[1041,1089,1090,1092],{},[45,1091,1045],{},[39,1093,1094,1096],{},[289,1095,323],{},[1028,1097,1098,1104],{},[1041,1099,1100,1102],{},[45,1101,252],{},[49,1103,149],{},[49,1105,149],{},[53,1107,1108],{"encoding":55},"p(X) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{X^2}{2}}, \\\\ p(Y) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{Y^2}{2}}",[22,1110,1112,1139,1414,1417,1444],{"className":1111,"ariaHidden":61},[60],[22,1113,1115,1118,1121,1124,1127,1130,1133,1136],{"className":1114},[65],[22,1116],{"className":1117,"style":944},[69],[22,1119,15],{"className":1120},[74,78],[22,1122,301],{"className":1123},[378],[22,1125,221],{"className":1126,"style":237},[74,78],[22,1128,315],{"className":1129},[429],[22,1131],{"className":1132,"style":356},[355],[22,1134,291],{"className":1135},[360],[22,1137],{"className":1138,"style":356},[355],[22,1140,1142,1146,1273,1410],{"className":1141},[65],[22,1143],{"className":1144,"style":1145},[69],"height:2.2514em;vertical-align:-0.93em;",[22,1147,1149,1153,1270],{"className":1148},[74],[22,1150],{"className":1151},[378,1152],"nulldelimiter",[22,1154,1156],{"className":1155},[1028],[22,1157,1159,1261],{"className":1158},[87,88],[22,1160,1162,1258],{"className":1161},[92],[22,1163,1166,1235,1246],{"className":1164,"style":1165},[96],"height:1.3214em;",[22,1167,1169,1173],{"style":1168},"top:-2.2028em;",[22,1170],{"className":1171,"style":1172},[104],"height:3em;",[22,1174,1176],{"className":1175},[74],[22,1177,1179],{"className":1178},[74,433],[22,1180,1182,1226],{"className":1181},[87,88],[22,1183,1185,1223],{"className":1184},[92],[22,1186,1189,1206],{"className":1187,"style":1188},[96],"height:0.9072em;",[22,1190,1193,1196],{"className":1191,"style":1192},[447],"top:-3em;",[22,1194],{"className":1195,"style":1172},[104],[22,1197,1200,1203],{"className":1198,"style":1199},[74],"padding-left:0.833em;",[22,1201,149],{"className":1202},[74],[22,1204,306],{"className":1205,"style":385},[74,78],[22,1207,1209,1212],{"style":1208},"top:-2.8672em;",[22,1210],{"className":1211,"style":1172},[104],[22,1213,1216],{"className":1214,"style":1215},[522],"min-width:0.853em;height:1.08em;",[525,1217,1220],{"xmlns":527,"width":528,"height":1218,"viewBox":1219,"preserveAspectRatio":531},"1.08em","0 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1172},[104],[22,1540,1542],{"className":1541},[74],[22,1543,51],{"className":1544},[74],[22,1546,120],{"className":1547},[119],[22,1549,1551],{"className":1550},[92],[22,1552,1554],{"className":1553,"style":1267},[96],[22,1555],{},[22,1557],{"className":1558},[429,1152],[22,1560,1562,1565],{"className":1561},[74],[22,1563,1045],{"className":1564},[74,78],[22,1566,1568],{"className":1567},[83],[22,1569,1571],{"className":1570},[87],[22,1572,1574],{"className":1573},[92],[22,1575,1577],{"className":1576,"style":1291},[96],[22,1578,1579,1582],{"style":1294},[22,1580],{"className":1581,"style":1172},[104],[22,1583,1585],{"className":1584},[109,110,111,112],[22,1586,1588,1591],{"className":1587},[74,112],[22,1589,323],{"className":1590},[74,112],[22,1592,1594,1597,1679],{"className":1593},[74,112],[22,1595],{"className":1596},[378,1152,109,1313,1314],[22,1598,1600],{"className":1599},[1028],[22,1601,1603,1671],{"className":1602},[87,88],[22,1604,1606,1668],{"className":1605},[92],[22,1607,1609,1623,1631],{"className":1608,"style":1327},[96],[22,1610,1611,1614],{"style":1330},[22,1612],{"className":1613,"style":1172},[104],[22,1615,1617],{"className":1616},[109,1313,1337,112],[22,1618,1620],{"className":1619},[74,112],[22,1621,149],{"className":1622},[74,112],[22,1624,1625,1628],{"style":1346},[22,1626],{"className":1627,"style":1172},[104],[22,1629],{"className":1630,"style":1353},[1244,112],[22,1632,1633,1636],{"style":1356},[22,1634],{"className":1635,"style":1172},[104],[22,1637,1639],{"className":1638},[109,1313,1337,112],[22,1640,1642],{"className":1641},[74,112],[22,1643,1645,1648],{"className":1644},[74,112],[22,1646,252],{"className":1647,"style":267},[74,78,112],[22,1649,1651],{"className":1650},[83],[22,1652,1654],{"className":1653},[87],[22,1655,1657],{"className":1656},[92],[22,1658,1660],{"className":1659,"style":1384},[96],[22,1661,1662,1665],{"style":1387},[22,1663],{"className":1664,"style":1391},[104],[22,1666,149],{"className":1667},[74,112],[22,1669,120],{"className":1670},[119],[22,1672,1674],{"className":1673},[92],[22,1675,1677],{"className":1676,"style":1404},[96],[22,1678],{},[22,1680],{"className":1681},[429,1152,109,1313,1314],[15,1683,1684,1685,130,1713,1741],{},"Since ",[22,1686,1688,1701],{"className":1687,"translate":26},[25],[22,1689,1691],{"className":1690},[30],[32,1692,1693],{"xmlns":34},[36,1694,1695,1699],{},[39,1696,1697],{},[45,1698,221],{},[53,1700,221],{"encoding":55},[22,1702,1704],{"className":1703,"ariaHidden":61},[60],[22,1705,1707,1710],{"className":1706},[65],[22,1708],{"className":1709,"style":233},[69],[22,1711,221],{"className":1712,"style":237},[74,78],[22,1714,1716,1729],{"className":1715,"translate":26},[25],[22,1717,1719],{"className":1718},[30],[32,1720,1721],{"xmlns":34},[36,1722,1723,1727],{},[39,1724,1725],{},[45,1726,252],{},[53,1728,252],{"encoding":55},[22,1730,1732],{"className":1731,"ariaHidden":61},[60],[22,1733,1735,1738],{"className":1734},[65],[22,1736],{"className":1737,"style":233},[69],[22,1739,252],{"className":1740,"style":267},[74,78]," are independent, their joint probability density function is:",[22,1743,1745],{"className":1744,"translate":26},[272],[22,1746,1748,1813],{"className":1747,"translate":26},[25],[22,1749,1751],{"className":1750},[30],[32,1752,1753],{"xmlns":34,"display":281},[36,1754,1755,1810],{},[39,1756,1757,1759,1761,1763,1765,1767,1769,1771,1781],{},[45,1758,15],{},[289,1760,301],{"stretchy":300},[45,1762,221],{},[289,1764,1062],{"separator":61},[45,1766,252],{},[289,1768,315],{"stretchy":300},[289,1770,291],{},[1028,1772,1773,1775],{},[49,1774,51],{},[39,1776,1777,1779],{},[49,1778,149],{},[45,1780,306],{},[1041,1782,1783,1785],{},[45,1784,1045],{},[39,1786,1787,1789],{},[289,1788,323],{},[1028,1790,1791,1808],{},[39,1792,1793,1799,1802],{},[1041,1794,1795,1797],{},[45,1796,221],{},[49,1798,149],{},[289,1800,1801],{},"+",[1041,1803,1804,1806],{},[45,1805,252],{},[49,1807,149],{},[49,1809,149],{},[53,1811,1812],{"encoding":55},"p(X,Y) = \\frac{1}{2\\pi} e^{-\\frac{X^2 + Y^2}{2}}",[22,1814,1816,1852],{"className":1815,"ariaHidden":61},[60],[22,1817,1819,1822,1825,1828,1831,1834,1837,1840,1843,1846,1849],{"className":1818},[65],[22,1820],{"className":1821,"style":944},[69],[22,1823,15],{"className":1824},[74,78],[22,1826,301],{"className":1827},[378],[22,1829,221],{"className":1830,"style":237},[74,78],[22,1832,1062],{"className":1833},[1413],[22,1835],{"className":1836,"style":466},[355],[22,1838,252],{"className":1839,"style":267},[74,78],[22,1841,315],{"className":1842},[429],[22,1844],{"className":1845,"style":356},[355],[22,1847,291],{"className":1848},[360],[22,1850],{"className":1851,"style":356},[355],[22,1853,1855,1859,1926],{"className":1854},[65],[22,1856],{"className":1857,"style":1858},[69],"height:2.0074em;vertical-align:-0.686em;",[22,1860,1862,1865,1923],{"className":1861},[74],[22,1863],{"className":1864},[378,1152],[22,1866,1868],{"className":1867},[1028],[22,1869,1871,1914],{"className":1870},[87,88],[22,1872,1874,1911],{"className":1873},[92],[22,1875,1877,1892,1900],{"className":1876,"style":1165},[96],[22,1878,1880,1883],{"style":1879},"top:-2.314em;",[22,1881],{"className":1882,"style":1172},[104],[22,1884,1886,1889],{"className":1885},[74],[22,1887,149],{"className":1888},[74],[22,1890,306],{"className":1891,"style":385},[74,78],[22,1893,1894,1897],{"style":1237},[22,1895],{"className":1896,"style":1172},[104],[22,1898],{"className":1899,"style":1245},[1244],[22,1901,1902,1905],{"style":1248},[22,1903],{"className":1904,"style":1172},[104],[22,1906,1908],{"className":1907},[74],[22,1909,51],{"className":1910},[74],[22,1912,120],{"className":1913},[119],[22,1915,1917],{"className":1916},[92],[22,1918,1921],{"className":1919,"style":1920},[96],"height:0.686em;",[22,1922],{},[22,1924],{"className":1925},[429,1152],[22,1927,1929,1932],{"className":1928},[74],[22,1930,1045],{"className":1931},[74,78],[22,1933,1935],{"className":1934},[83],[22,1936,1938],{"className":1937},[87],[22,1939,1941],{"className":1940},[92],[22,1942,1944],{"className":1943,"style":1291},[96],[22,1945,1946,1949],{"style":1294},[22,1947],{"className":1948,"style":1172},[104],[22,1950,1952],{"className":1951},[109,110,111,112],[22,1953,1955,1958],{"className":1954},[74,112],[22,1956,323],{"className":1957},[74,112],[22,1959,1961,1964,2076],{"className":1960},[74,112],[22,1962],{"className":1963},[378,1152,109,1313,1314],[22,1965,1967],{"className":1966},[1028],[22,1968,1970,2068],{"className":1969},[87,88],[22,1971,1973,2065],{"className":1972},[92],[22,1974,1976,1990,1998],{"className":1975,"style":1327},[96],[22,1977,1978,1981],{"style":1330},[22,1979],{"className":1980,"style":1172},[104],[22,1982,1984],{"className":1983},[109,1313,1337,112],[22,1985,1987],{"className":1986},[74,112],[22,1988,149],{"className":1989},[74,112],[22,1991,1992,1995],{"style":1346},[22,1993],{"className":1994,"style":1172},[104],[22,1996],{"className":1997,"style":1353},[1244,112],[22,1999,2000,2003],{"style":1356},[22,2001],{"className":2002,"style":1172},[104],[22,2004,2006],{"className":2005},[109,1313,1337,112],[22,2007,2009,2035,2039],{"className":2008},[74,112],[22,2010,2012,2015],{"className":2011},[74,112],[22,2013,221],{"className":2014,"style":237},[74,78,112],[22,2016,2018],{"className":2017},[83],[22,2019,2021],{"className":2020},[87],[22,2022,2024],{"className":2023},[92],[22,2025,2027],{"className":2026,"style":1384},[96],[22,2028,2029,2032],{"style":1387},[22,2030],{"className":2031,"style":1391},[104],[22,2033,149],{"className":2034},[74,112],[22,2036,1801],{"className":2037},[2038,112],"mbin",[22,2040,2042,2045],{"className":2041},[74,112],[22,2043,252],{"className":2044,"style":267},[74,78,112],[22,2046,2048],{"className":2047},[83],[22,2049,2051],{"className":2050},[87],[22,2052,2054],{"className":2053},[92],[22,2055,2057],{"className":2056,"style":1384},[96],[22,2058,2059,2062],{"style":1387},[22,2060],{"className":2061,"style":1391},[104],[22,2063,149],{"className":2064},[74,112],[22,2066,120],{"className":2067},[119],[22,2069,2071],{"className":2070},[92],[22,2072,2074],{"className":2073,"style":1404},[96],[22,2075],{},[22,2077],{"className":2078},[429,1152,109,1313,1314],[15,2080,2081],{},"We now perform a coordinate transformation by defining:",[22,2083,2085],{"className":2084,"translate":26},[272],[22,2086,2088,2138],{"className":2087,"translate":26},[25],[22,2089,2091],{"className":2090},[30],[32,2092,2093],{"xmlns":34,"display":281},[36,2094,2095,2135],{},[39,2096,2097,2099,2101,2104,2106,2108,2110,2113,2115,2117,2119,2121,2123,2125,2127,2129,2131,2133],{},[45,2098,221],{},[289,2100,291],{},[45,2102,2103],{},"R",[45,2105,294],{},[289,2107,297],{},[289,2109,301],{"stretchy":300},[45,2111,2112],{},"θ",[289,2114,315],{"stretchy":300},[289,2116,1062],{"separator":61},[355,2118],{"linebreak":1065},[45,2120,252],{},[289,2122,291],{},[45,2124,2103],{},[45,2126,569],{},[289,2128,297],{},[289,2130,301],{"stretchy":300},[45,2132,2112],{},[289,2134,315],{"stretchy":300},[53,2136,2137],{"encoding":55},"X = R \\cos(\\theta), \\\\ Y = R \\sin(\\theta)",[22,2139,2141,2159,2188,2191,2209],{"className":2140,"ariaHidden":61},[60],[22,2142,2144,2147,2150,2153,2156],{"className":2143},[65],[22,2145],{"className":2146,"style":233},[69],[22,2148,221],{"className":2149,"style":237},[74,78],[22,2151],{"className":2152,"style":356},[355],[22,2154,291],{"className":2155},[360],[22,2157],{"className":2158,"style":356},[355],[22,2160,2162,2165,2169,2172,2175,2178,2182,2185],{"className":2161},[65],[22,2163],{"className":2164,"style":944},[69],[22,2166,2103],{"className":2167,"style":2168},[74,78],"margin-right:0.00773em;",[22,2170],{"className":2171,"style":466},[355],[22,2173,294],{"className":2174},[374],[22,2176,301],{"className":2177},[378],[22,2179,2112],{"className":2180,"style":2181},[74,78],"margin-right:0.02778em;",[22,2183,315],{"className":2184},[429],[22,2186,1062],{"className":2187},[1413],[22,2189],{"className":2190},[355,1065],[22,2192,2194,2197,2200,2203,2206],{"className":2193},[65],[22,2195],{"className":2196,"style":233},[69],[22,2198,252],{"className":2199,"style":267},[74,78],[22,2201],{"className":2202,"style":356},[355],[22,2204,291],{"className":2205},[360],[22,2207],{"className":2208,"style":356},[355],[22,2210,2212,2215,2218,2221,2224,2227,2230],{"className":2211},[65],[22,2213],{"className":2214,"style":944},[69],[22,2216,2103],{"className":2217,"style":2168},[74,78],[22,2219],{"className":2220,"style":466},[355],[22,2222,569],{"className":2223},[374],[22,2225,301],{"className":2226},[378],[22,2228,2112],{"className":2229,"style":2181},[74,78],[22,2231,315],{"className":2232},[429],[15,2234,2235],{},"Thus, the joint distribution in polar coordinates becomes:",[22,2237,2239],{"className":2238,"translate":26},[272],[22,2240,2242,2328],{"className":2241,"translate":26},[25],[22,2243,2245],{"className":2244},[30],[32,2246,2247],{"xmlns":34,"display":281},[36,2248,2249,2325],{},[39,2250,2251,2264,2276,2286,2304,2306,2310,2313,2315,2317,2319,2321,2323],{},[2252,2253,2254,2257,2260],"msubsup",{},[289,2255,2256],{},"∫",[49,2258,2259],{},"0",[45,2261,2263],{"mathvariant":2262},"normal","∞",[2252,2265,2266,2268,2270],{},[289,2267,2256],{},[49,2269,2259],{},[39,2271,2272,2274],{},[49,2273,149],{},[45,2275,306],{},[1028,2277,2278,2280],{},[49,2279,51],{},[39,2281,2282,2284],{},[49,2283,149],{},[45,2285,306],{},[1041,2287,2288,2290],{},[45,2289,1045],{},[39,2291,2292,2294],{},[289,2293,323],{},[1028,2295,2296,2302],{},[1041,2297,2298,2300],{},[45,2299,2103],{},[49,2301,149],{},[49,2303,149],{},[45,2305,2103],{},[2307,2308,2309],"mtext",{}," ",[45,2311,2312],{},"d",[45,2314,2112],{},[2307,2316,2309],{},[45,2318,2312],{},[45,2320,2103],{},[289,2322,291],{},[49,2324,51],{},[53,2326,2327],{"encoding":55},"\\int_{0}^{\\infty} \\int_{0}^{2\\pi} \\frac{1}{2\\pi} e^{-\\frac{R^2}{2}} R \\, d\\theta \\, dR = 1",[22,2329,2331,2687],{"className":2330,"ariaHidden":61},[60],[22,2332,2334,2338,2402,2405,2466,2469,2534,2657,2660,2663,2666,2669,2672,2675,2678,2681,2684],{"className":2333},[65],[22,2335],{"className":2336,"style":2337},[69],"height:2.476em;vertical-align:-0.9119em;",[22,2339,2341,2347],{"className":2340},[374],[22,2342,2256],{"className":2343,"style":2346},[374,2344,2345],"op-symbol","large-op","margin-right:0.44445em;position:relative;top:-0.0011em;",[22,2348,2350],{"className":2349},[83],[22,2351,2353,2393],{"className":2352},[87,88],[22,2354,2356,2390],{"className":2355},[92],[22,2357,2360,2375],{"className":2358,"style":2359},[96],"height:1.4143em;",[22,2361,2363,2366],{"style":2362},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[22,2364],{"className":2365,"style":105},[104],[22,2367,2369],{"className":2368},[109,110,111,112],[22,2370,2372],{"className":2371},[74,112],[22,2373,2259],{"className":2374},[74,112],[22,2376,2378,2381],{"style":2377},"top:-3.8129em;margin-right:0.05em;",[22,2379],{"className":2380,"style":105},[104],[22,2382,2384],{"className":2383},[109,110,111,112],[22,2385,2387],{"className":2386},[74,112],[22,2388,2263],{"className":2389},[74,112],[22,2391,120],{"className":2392},[119],[22,2394,2396],{"className":2395},[92],[22,2397,2400],{"className":2398,"style":2399},[96],"height:0.9119em;",[22,2401],{},[22,2403],{"className":2404,"style":466},[355],[22,2406,2408,2411],{"className":2407},[374],[22,2409,2256],{"className":2410,"style":2346},[374,2344,2345],[22,2412,2414],{"className":2413},[83],[22,2415,2417,2458],{"className":2416},[87,88],[22,2418,2420,2455],{"className":2419},[92],[22,2421,2424,2438],{"className":2422,"style":2423},[96],"height:1.564em;",[22,2425,2426,2429],{"style":2362},[22,2427],{"className":2428,"style":105},[104],[22,2430,2432],{"className":2431},[109,110,111,112],[22,2433,2435],{"className":2434},[74,112],[22,2436,2259],{"className":2437},[74,112],[22,2439,2440,2443],{"style":2377},[22,2441],{"className":2442,"style":105},[104],[22,2444,2446],{"className":2445},[109,110,111,112],[22,2447,2449,2452],{"className":2448},[74,112],[22,2450,149],{"className":2451},[74,112],[22,2453,306],{"className":2454,"style":385},[74,78,112],[22,2456,120],{"className":2457},[119],[22,2459,2461],{"className":2460},[92],[22,2462,2464],{"className":2463,"style":2399},[96],[22,2465],{},[22,2467],{"className":2468,"style":466},[355],[22,2470,2472,2475,2531],{"className":2471},[74],[22,2473],{"className":2474},[378,1152],[22,2476,2478],{"className":2477},[1028],[22,2479,2481,2523],{"className":2480},[87,88],[22,2482,2484,2520],{"className":2483},[92],[22,2485,2487,2501,2509],{"className":2486,"style":1165},[96],[22,2488,2489,2492],{"style":1879},[22,2490],{"className":2491,"style":1172},[104],[22,2493,2495,2498],{"className":2494},[74],[22,2496,149],{"className":2497},[74],[22,2499,306],{"className":2500,"style":385},[74,78],[22,2502,2503,2506],{"style":1237},[22,2504],{"className":2505,"style":1172},[104],[22,2507],{"className":2508,"style":1245},[1244],[22,2510,2511,2514],{"style":1248},[22,2512],{"className":2513,"style":1172},[104],[22,2515,2517],{"className":2516},[74],[22,2518,51],{"className":2519},[74],[22,2521,120],{"className":2522},[119],[22,2524,2526],{"className":2525},[92],[22,2527,2529],{"className":2528,"style":1920},[96],[22,2530],{},[22,2532],{"className":2533},[429,1152],[22,2535,2537,2540],{"className":2536},[74],[22,2538,1045],{"className":2539},[74,78],[22,2541,2543],{"className":2542},[83],[22,2544,2546],{"className":2545},[87],[22,2547,2549],{"className":2548},[92],[22,2550,2552],{"className":2551,"style":1291},[96],[22,2553,2554,2557],{"style":1294},[22,2555],{"className":2556,"style":1172},[104],[22,2558,2560],{"className":2559},[109,110,111,112],[22,2561,2563,2566],{"className":2562},[74,112],[22,2564,323],{"className":2565},[74,112],[22,2567,2569,2572,2654],{"className":2568},[74,112],[22,2570],{"className":2571},[378,1152,109,1313,1314],[22,2573,2575],{"className":2574},[1028],[22,2576,2578,2646],{"className":2577},[87,88],[22,2579,2581,2643],{"className":2580},[92],[22,2582,2584,2598,2606],{"className":2583,"style":1327},[96],[22,2585,2586,2589],{"style":1330},[22,2587],{"className":2588,"style":1172},[104],[22,2590,2592],{"className":2591},[109,1313,1337,112],[22,2593,2595],{"className":2594},[74,112],[22,2596,149],{"className":2597},[74,112],[22,2599,2600,2603],{"style":1346},[22,2601],{"className":2602,"style":1172},[104],[22,2604],{"className":2605,"style":1353},[1244,112],[22,2607,2608,2611],{"style":1356},[22,2609],{"className":2610,"style":1172},[104],[22,2612,2614],{"className":2613},[109,1313,1337,112],[22,2615,2617],{"className":2616},[74,112],[22,2618,2620,2623],{"className":2619},[74,112],[22,2621,2103],{"className":2622,"style":2168},[74,78,112],[22,2624,2626],{"className":2625},[83],[22,2627,2629],{"className":2628},[87],[22,2630,2632],{"className":2631},[92],[22,2633,2635],{"className":2634,"style":1384},[96],[22,2636,2637,2640],{"style":1387},[22,2638],{"className":2639,"style":1391},[104],[22,2641,149],{"className":2642},[74,112],[22,2644,120],{"className":2645},[119],[22,2647,2649],{"className":2648},[92],[22,2650,2652],{"className":2651,"style":1404},[96],[22,2653],{},[22,2655],{"className":2656},[429,1152,109,1313,1314],[22,2658,2103],{"className":2659,"style":2168},[74,78],[22,2661],{"className":2662,"style":466},[355],[22,2664,2312],{"className":2665},[74,78],[22,2667,2112],{"className":2668,"style":2181},[74,78],[22,2670],{"className":2671,"style":466},[355],[22,2673,2312],{"className":2674},[74,78],[22,2676,2103],{"className":2677,"style":2168},[74,78],[22,2679],{"className":2680,"style":356},[355],[22,2682,291],{"className":2683},[360],[22,2685],{"className":2686,"style":356},[355],[22,2688,2690,2694],{"className":2689},[65],[22,2691],{"className":2692,"style":2693},[69],"height:0.6444em;",[22,2695,51],{"className":2696},[74],[15,2698,2699,2700,130,2728,2758,2759,130,2833,2904],{},"From this, we can derive the distribution functions for ",[22,2701,2703,2716],{"className":2702,"translate":26},[25],[22,2704,2706],{"className":2705},[30],[32,2707,2708],{"xmlns":34},[36,2709,2710,2714],{},[39,2711,2712],{},[45,2713,2103],{},[53,2715,2103],{"encoding":55},[22,2717,2719],{"className":2718,"ariaHidden":61},[60],[22,2720,2722,2725],{"className":2721},[65],[22,2723],{"className":2724,"style":233},[69],[22,2726,2103],{"className":2727,"style":2168},[74,78],[22,2729,2731,2745],{"className":2730,"translate":26},[25],[22,2732,2734],{"className":2733},[30],[32,2735,2736],{"xmlns":34},[36,2737,2738,2742],{},[39,2739,2740],{},[45,2741,2112],{},[53,2743,2744],{"encoding":55},"\\theta",[22,2746,2748],{"className":2747,"ariaHidden":61},[60],[22,2749,2751,2755],{"className":2750},[65],[22,2752],{"className":2753,"style":2754},[69],"height:0.6944em;",[22,2756,2112],{"className":2757,"style":2181},[74,78],", denoted as ",[22,2760,2762,2781],{"className":2761,"translate":26},[25],[22,2763,2765],{"className":2764},[30],[32,2766,2767],{"xmlns":34},[36,2768,2769,2778],{},[39,2770,2771],{},[42,2772,2773,2776],{},[45,2774,2775],{},"P",[45,2777,2103],{},[53,2779,2780],{"encoding":55},"P_R",[22,2782,2784],{"className":2783,"ariaHidden":61},[60],[22,2785,2787,2790],{"className":2786},[65],[22,2788],{"className":2789,"style":70},[69],[22,2791,2793,2797],{"className":2792},[74],[22,2794,2775],{"className":2795,"style":2796},[74,78],"margin-right:0.13889em;",[22,2798,2800],{"className":2799},[83],[22,2801,2803,2825],{"className":2802},[87,88],[22,2804,2806,2822],{"className":2805},[92],[22,2807,2810],{"className":2808,"style":2809},[96],"height:0.3283em;",[22,2811,2813,2816],{"style":2812},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[22,2814],{"className":2815,"style":105},[104],[22,2817,2819],{"className":2818},[109,110,111,112],[22,2820,2103],{"className":2821,"style":2168},[74,78,112],[22,2823,120],{"className":2824},[119],[22,2826,2828],{"className":2827},[92],[22,2829,2831],{"className":2830,"style":127},[96],[22,2832],{},[22,2834,2836,2854],{"className":2835,"translate":26},[25],[22,2837,2839],{"className":2838},[30],[32,2840,2841],{"xmlns":34},[36,2842,2843,2851],{},[39,2844,2845],{},[42,2846,2847,2849],{},[45,2848,2775],{},[45,2850,2112],{},[53,2852,2853],{"encoding":55},"P_\\theta",[22,2855,2857],{"className":2856,"ariaHidden":61},[60],[22,2858,2860,2863],{"className":2859},[65],[22,2861],{"className":2862,"style":70},[69],[22,2864,2866,2869],{"className":2865},[74],[22,2867,2775],{"className":2868,"style":2796},[74,78],[22,2870,2872],{"className":2871},[83],[22,2873,2875,2896],{"className":2874},[87,88],[22,2876,2878,2893],{"className":2877},[92],[22,2879,2882],{"className":2880,"style":2881},[96],"height:0.3361em;",[22,2883,2884,2887],{"style":2812},[22,2885],{"className":2886,"style":105},[104],[22,2888,2890],{"className":2889},[109,110,111,112],[22,2891,2112],{"className":2892,"style":2181},[74,78,112],[22,2894,120],{"className":2895},[119],[22,2897,2899],{"className":2898},[92],[22,2900,2902],{"className":2901,"style":127},[96],[22,2903],{},":",[22,2906,2908],{"className":2907,"translate":26},[272],[22,2909,2911,3031],{"className":2910,"translate":26},[25],[22,2912,2914],{"className":2913},[30],[32,2915,2916],{"xmlns":34,"display":281},[36,2917,2918,3028],{},[39,2919,2920,2926,2928,2930,2933,2936,2938,2940,2942,2950,2962,2972,2990,2992,2994,2996,2998,3000,3002,3004,3006,3008,3010],{},[42,2921,2922,2924],{},[45,2923,2775],{},[45,2925,2103],{},[289,2927,301],{"stretchy":300},[45,2929,2103],{},[289,2931,2932],{},"≤",[45,2934,2935],{},"r",[289,2937,315],{"stretchy":300},[289,2939,291],{},[355,2941],{"linebreak":1065},[2252,2943,2944,2946,2948],{},[289,2945,2256],{},[49,2947,2259],{},[45,2949,2935],{},[2252,2951,2952,2954,2956],{},[289,2953,2256],{},[49,2955,2259],{},[39,2957,2958,2960],{},[49,2959,149],{},[45,2961,306],{},[1028,2963,2964,2966],{},[49,2965,51],{},[39,2967,2968,2970],{},[49,2969,149],{},[45,2971,306],{},[1041,2973,2974,2976],{},[45,2975,1045],{},[39,2977,2978,2980],{},[289,2979,323],{},[1028,2981,2982,2988],{},[1041,2983,2984,2986],{},[45,2985,2103],{},[49,2987,149],{},[49,2989,149],{},[45,2991,2103],{},[2307,2993,2309],{},[45,2995,2312],{},[45,2997,2112],{},[2307,2999,2309],{},[45,3001,2312],{},[45,3003,2103],{},[289,3005,291],{},[49,3007,51],{},[289,3009,323],{},[1041,3011,3012,3014],{},[45,3013,1045],{},[39,3015,3016,3018],{},[289,3017,323],{},[1028,3019,3020,3026],{},[1041,3021,3022,3024],{},[45,3023,2935],{},[49,3025,149],{},[49,3027,149],{},[53,3029,3030],{"encoding":55},"P_R(R \\leq r) = \\\\ \\int_{0}^{r} \\int_{0}^{2\\pi} \\frac{1}{2\\pi} e^{-\\frac{R^2}{2}} R \\, d\\theta \\, dR = 1 - 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\\leq \\phi) = \\\\ \\int_{0}^{\\phi} \\int_{0}^{\\infty} \\frac{1}{2\\pi} e^{-\\frac{R^2}{2}} R \\, dR \\, d\\theta = \\frac{\\phi}{2\\pi}",[22,3722,3724,3785,3803,3806,4152],{"className":3723,"ariaHidden":61},[60],[22,3725,3727,3730,3770,3773,3776,3779,3782],{"className":3726},[65],[22,3728],{"className":3729,"style":944},[69],[22,3731,3733,3736],{"className":3732},[74],[22,3734,2775],{"className":3735,"style":2796},[74,78],[22,3737,3739],{"className":3738},[83],[22,3740,3742,3762],{"className":3741},[87,88],[22,3743,3745,3759],{"className":3744},[92],[22,3746,3748],{"className":3747,"style":2881},[96],[22,3749,3750,3753],{"style":2812},[22,3751],{"className":3752,"style":105},[104],[22,3754,3756],{"className":3755},[109,110,111,112],[22,3757,2112],{"className":3758,"style":2181},[74,78,112],[22,3760,120],{"className":3761},[119],[22,3763,3765],{"className":3764},[92],[22,3766,3768],{"className":3767,"style":127},[96],[22,3769],{},[22,3771,301],{"className":3772},[378],[22,3774,2112],{"className":3775,"style":2181},[74,78],[22,3777],{"className":3778,"style":356},[355],[22,3780,2932],{"className":3781},[360],[22,3783],{"className":3784,"style":356},[355],[22,3786,3788,3791,3794,3797,3800],{"className":3787},[65],[22,3789],{"className":3790,"style":944},[69],[22,3792,3641],{"className":3793},[74,78],[22,3795,315],{"className":3796},[429],[22,3798],{"className":3799,"style":356},[355],[22,3801,291],{"className":3802},[360],[22,3804],{"className":3805},[355,1065],[22,3807,3809,3813,3871,3874,3931,3934,3999,4122,4125,4128,4131,4134,4137,4140,4143,4146,4149],{"className":3808},[65],[22,3810],{"className":3811,"style":3812},[69],"height:2.511em;vertical-align:-0.9119em;",[22,3814,3816,3819],{"className":3815},[374],[22,3817,2256],{"className":3818,"style":2346},[374,2344,2345],[22,3820,3822],{"className":3821},[83],[22,3823,3825,3863],{"className":3824},[87,88],[22,3826,3828,3860],{"className":3827},[92],[22,3829,3832,3846],{"className":3830,"style":3831},[96],"height:1.599em;",[22,3833,3834,3837],{"style":2362},[22,3835],{"className":3836,"style":105},[104],[22,3838,3840],{"className":3839},[109,110,111,112],[22,3841,3843],{"className":3842},[74,112],[22,3844,2259],{"className":3845},[74,112],[22,3847,3848,3851],{"style":2377},[22,3849],{"className":3850,"style":105},[104],[22,3852,3854],{"className":3853},[109,110,111,112],[22,3855,3857],{"className":3856},[74,112],[22,3858,3641],{"className":3859},[74,78,112],[22,3861,120],{"className":3862},[119],[22,3864,3866],{"className":3865},[92],[22,3867,3869],{"className":3868,"style":2399},[96],[22,3870],{},[22,3872],{"className":3873,"style":466},[355],[22,3875,3877,3880],{"className":3876},[374],[22,3878,2256],{"className":3879,"style":2346},[374,2344,2345],[22,3881,3883],{"className":3882},[83],[22,3884,3886,3923],{"className":3885},[87,88],[22,3887,3889,3920],{"className":3888},[92],[22,3890,3892,3906],{"className":3891,"style":2359},[96],[22,3893,3894,3897],{"style":2362},[22,3895],{"className":3896,"style":105},[104],[22,3898,3900],{"className":3899},[109,110,111,112],[22,3901,3903],{"className":3902},[74,112],[22,3904,2259],{"className":3905},[74,112],[22,3907,3908,3911],{"style":2377},[22,3909],{"className":3910,"style":105},[104],[22,3912,3914],{"className":3913},[109,110,111,112],[22,3915,3917],{"className":3916},[74,112],[22,3918,2263],{"className":3919},[74,112],[22,3921,120],{"className":3922},[119],[22,3924,3926],{"className":3925},[92],[22,3927,3929],{"className":3928,"style":2399},[96],[22,3930],{},[22,3932],{"className":3933,"style":466},[355],[22,3935,3937,3940,3996],{"className":3936},[74],[22,3938],{"className":3939},[378,1152],[22,3941,3943],{"className":3942},[1028],[22,3944,3946,3988],{"className":3945},[87,88],[22,3947,3949,3985],{"className":3948},[92],[22,3950,3952,3966,3974],{"className":3951,"style":1165},[96],[22,3953,3954,3957],{"style":1879},[22,3955],{"className":3956,"style":1172},[104],[22,3958,3960,3963],{"className":3959},[74],[22,3961,149],{"className":3962},[74],[22,3964,306],{"className":3965,"style":385},[74,78],[22,3967,3968,3971],{"style":1237},[22,3969],{"className":3970,"style":1172},[104],[22,3972],{"className":3973,"style":1245},[1244],[22,3975,3976,3979],{"style":1248},[22,3977],{"className":3978,"style":1172},[104],[22,3980,3982],{"className":3981},[74],[22,3983,51],{"className":3984},[74],[22,3986,120],{"className":3987},[119],[22,3989,3991],{"className":3990},[92],[22,3992,3994],{"className":3993,"style":1920},[96],[22,3995],{},[22,3997],{"className":3998},[429,1152],[22,4000,4002,4005],{"className":4001},[74],[22,4003,1045],{"className":4004},[74,78],[22,4006,4008],{"className":4007},[83],[22,4009,4011],{"className":4010},[87],[22,4012,4014],{"className":4013},[92],[22,4015,4017],{"className":4016,"style":1291},[96],[22,4018,4019,4022],{"style":1294},[22,4020],{"className":4021,"style":1172},[104],[22,4023,4025],{"className":4024},[109,110,111,112],[22,4026,4028,4031],{"className":4027},[74,112],[22,4029,323],{"className":4030},[74,112],[22,4032,4034,4037,4119],{"className":4033},[74,112],[22,4035],{"className":4036},[378,1152,109,1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is evident that ",[22,4229,4231,4244],{"className":4230,"translate":26},[25],[22,4232,4234],{"className":4233},[30],[32,4235,4236],{"xmlns":34},[36,4237,4238,4242],{},[39,4239,4240],{},[45,4241,2112],{},[53,4243,2744],{"encoding":55},[22,4245,4247],{"className":4246,"ariaHidden":61},[60],[22,4248,4250,4253],{"className":4249},[65],[22,4251],{"className":4252,"style":2754},[69],[22,4254,2112],{"className":4255,"style":2181},[74,78]," follows a uniform distribution over ",[22,4258,4260,4286],{"className":4259,"translate":26},[25],[22,4261,4263],{"className":4262},[30],[32,4264,4265],{"xmlns":34},[36,4266,4267,4283],{},[39,4268,4269,4272,4274,4276,4278,4280],{},[289,4270,4271],{"stretchy":300},"[",[49,4273,2259],{},[289,4275,1062],{"separator":61},[49,4277,149],{},[45,4279,306],{},[289,4281,4282],{"stretchy":300},"]",[53,4284,4285],{"encoding":55},"[0, 2\\pi]",[22,4287,4289],{"className":4288,"ariaHidden":61},[60],[22,4290,4292,4295,4298,4301,4304,4307,4310,4313],{"className":4291},[65],[22,4293],{"className":4294,"style":944},[69],[22,4296,4271],{"className":4297},[378],[22,4299,2259],{"className":4300},[74],[22,4302,1062],{"className":4303},[1413],[22,4305],{"className":4306,"style":466},[355],[22,4308,149],{"className":4309},[74],[22,4311,306],{"className":4312,"style":385},[74,78],[22,4314,4282],{"className":4315},[429],". Let",[22,4318,4320],{"className":4319,"translate":26},[272],[22,4321,4323,4366],{"className":4322,"translate":26},[25],[22,4324,4326],{"className":4325},[30],[32,4327,4328],{"xmlns":34,"display":281},[36,4329,4330,4363],{},[39,4331,4332,4339,4341,4343,4345],{},[42,4333,4334,4337],{},[45,4335,4336],{},"F",[45,4338,2103],{},[289,4340,291],{},[49,4342,51],{},[289,4344,323],{},[1041,4346,4347,4349],{},[45,4348,1045],{},[39,4350,4351,4353],{},[289,4352,323],{},[1028,4354,4355,4361],{},[1041,4356,4357,4359],{},[45,4358,2935],{},[49,4360,149],{},[49,4362,149],{},[53,4364,4365],{"encoding":55},"F_R = 1 - e^{-\\frac{r^2}{2}}",[22,4367,4369,4424,4442],{"className":4368,"ariaHidden":61},[60],[22,4370,4372,4375,4415,4418,4421],{"className":4371},[65],[22,4373],{"className":4374,"style":70},[69],[22,4376,4378,4381],{"className":4377},[74],[22,4379,4336],{"className":4380,"style":2796},[74,78],[22,4382,4384],{"className":4383},[83],[22,4385,4387,4407],{"className":4386},[87,88],[22,4388,4390,4404],{"className":4389},[92],[22,4391,4393],{"className":4392,"style":2809},[96],[22,4394,4395,4398],{"style":2812},[22,4396],{"className":4397,"style":105},[104],[22,4399,4401],{"className":4400},[109,110,111,112],[22,4402,2103],{"className":4403,"style":2168},[74,78,112],[22,4405,120],{"className":4406},[119],[22,4408,4410],{"className":4409},[92],[22,4411,4413],{"className":4412,"style":127},[96],[22,4414],{},[22,4416],{"className":4417,"style":356},[355],[22,4419,291],{"className":4420},[360],[22,4422],{"className":4423,"style":356},[355],[22,4425,4427,4430,4433,4436,4439],{"className":4426},[65],[22,4428],{"className":4429,"style":3469},[69],[22,4431,51],{"className":4432},[74],[22,4434],{"className":4435,"style":3476},[355],[22,4437,323],{"className":4438},[2038],[22,4440],{"className":4441,"style":3476},[355],[22,4443,4445,4448],{"className":4444},[65],[22,4446],{"className":4447,"style":1291},[69],[22,4449,4451,4454],{"className":4450},[74],[22,4452,1045],{"className":4453},[74,78],[22,4455,4457],{"className":4456},[83],[22,4458,4460],{"className":4459},[87],[22,4461,4463],{"className":4462},[92],[22,4464,4466],{"className":4465,"style":1291},[96],[22,4467,4468,4471],{"style":1294},[22,4469],{"className":4470,"style":1172},[104],[22,4472,4474],{"className":4473},[109,110,111,112],[22,4475,4477,4480],{"className":4476},[74,112],[22,4478,323],{"className":4479},[74,112],[22,4481,4483,4486,4568],{"className":4482},[74,112],[22,4484],{"className":4485},[378,1152,109,1313,1314],[22,4487,4489],{"className":4488},[1028],[22,4490,4492,4560],{"className":4491},[87,88],[22,4493,4495,4557],{"className":4494},[92],[22,4496,4498,4512,4520],{"className":4497,"style":1327},[96],[22,4499,4500,4503],{"style":1330},[22,4501],{"className":4502,"style":1172},[104],[22,4504,4506],{"className":4505},[109,1313,1337,112],[22,4507,4509],{"className":4508},[74,112],[22,4510,149],{"className":4511},[74,112],[22,4513,4514,4517],{"style":1346},[22,4515],{"className":4516,"style":1172},[104],[22,4518],{"className":4519,"style":1353},[1244,112],[22,4521,4522,4525],{"style":1356},[22,4523],{"className":4524,"style":1172},[104],[22,4526,4528],{"className":4527},[109,1313,1337,112],[22,4529,4531],{"className":4530},[74,112],[22,4532,4534,4537],{"className":4533},[74,112],[22,4535,2935],{"className":4536,"style":2181},[74,78,112],[22,4538,4540],{"className":4539},[83],[22,4541,4543],{"className":4542},[87],[22,4544,4546],{"className":4545},[92],[22,4547,4549],{"className":4548,"style":1384},[96],[22,4550,4551,4554],{"style":1387},[22,4552],{"className":4553,"style":1391},[104],[22,4555,149],{"className":4556},[74,112],[22,4558,120],{"className":4559},[119],[22,4561,4563],{"className":4562},[92],[22,4564,4566],{"className":4565,"style":1404},[96],[22,4567],{},[22,4569],{"className":4570},[429,1152,109,1313,1314],[15,4572,4573],{},"The inverse function is given by:",[22,4575,4577],{"className":4576,"translate":26},[272],[22,4578,4580,4641],{"className":4579,"translate":26},[25],[22,4581,4583],{"className":4582},[30],[32,4584,4585],{"xmlns":34,"display":281},[36,4586,4587,4638],{},[39,4588,4589,4591,4593,4605,4607,4610,4612,4614,4616],{},[45,4590,2103],{},[289,4592,291],{},[2252,4594,4595,4597,4599],{},[45,4596,4336],{},[45,4598,2103],{},[39,4600,4601,4603],{},[289,4602,323],{},[49,4604,51],{},[289,4606,301],{"stretchy":300},[45,4608,4609],{},"z",[289,4611,315],{"stretchy":300},[355,4613],{"linebreak":1065},[289,4615,291],{},[317,4617,4618],{},[39,4619,4620,4622,4624,4626,4628,4630,4632,4634,4636],{},[289,4621,323],{},[49,4623,149],{},[45,4625,328],{},[289,4627,297],{},[289,4629,301],{"stretchy":300},[49,4631,51],{},[289,4633,323],{},[45,4635,4609],{},[289,4637,315],{"stretchy":300},[53,4639,4640],{"encoding":55},"R = F_R^{-1}(z) \\\\ = \\sqrt{-2 \\ln(1 - z)}",[22,4642,4644,4662,4740,4743,4756],{"className":4643,"ariaHidden":61},[60],[22,4645,4647,4650,4653,4656,4659],{"className":4646},[65],[22,4648],{"className":4649,"style":233},[69],[22,4651,2103],{"className":4652,"style":2168},[74,78],[22,4654],{"className":4655,"style":356},[355],[22,4657,291],{"className":4658},[360],[22,4660],{"className":4661,"style":356},[355],[22,4663,4665,4669,4730,4733,4737],{"className":4664},[65],[22,4666],{"className":4667,"style":4668},[69],"height:1.1478em;vertical-align:-0.2837em;",[22,4670,4672,4675],{"className":4671},[74],[22,4673,4336],{"className":4674,"style":2796},[74,78],[22,4676,4678],{"className":4677},[83],[22,4679,4681,4721],{"className":4680},[87,88],[22,4682,4684,4718],{"className":4683},[92],[22,4685,4688,4700],{"className":4686,"style":4687},[96],"height:0.8641em;",[22,4689,4691,4694],{"style":4690},"top:-2.4163em;margin-left:-0.1389em;margin-right:0.05em;",[22,4692],{"className":4693,"style":105},[104],[22,4695,4697],{"className":4696},[109,110,111,112],[22,4698,2103],{"className":4699,"style":2168},[74,78,112],[22,4701,4703,4706],{"style":4702},"top:-3.113em;margin-right:0.05em;",[22,4704],{"className":4705,"style":105},[104],[22,4707,4709],{"className":4708},[109,110,111,112],[22,4710,4712,4715],{"className":4711},[74,112],[22,4713,323],{"className":4714},[74,112],[22,4716,51],{"className":4717},[74,112],[22,4719,120],{"className":4720},[119],[22,4722,4724],{"className":4723},[92],[22,4725,4728],{"className":4726,"style":4727},[96],"height:0.2837em;",[22,4729],{},[22,4731,301],{"className":4732},[378],[22,4734,4609],{"className":4735,"style":4736},[74,78],"margin-right:0.04398em;",[22,4738,315],{"className":4739},[429],[22,4741],{"className":4742},[355,1065],[22,4744,4746,4750,4753],{"className":4745},[65],[22,4747],{"className":4748,"style":4749},[69],"height:0.3669em;",[22,4751,291],{"className":4752},[360],[22,4754],{"className":4755,"style":356},[355],[22,4757,4759,4763],{"className":4758},[65],[22,4760],{"className":4761,"style":4762},[69],"height:1.24em;vertical-align:-0.2561em;",[22,4764,4766],{"className":4765},[74,433],[22,4767,4769,4834],{"className":4768},[87,88],[22,4770,4772,4831],{"className":4771},[92],[22,4773,4776,4818],{"className":4774,"style":4775},[96],"height:0.9839em;",[22,4777,4779,4782],{"className":4778,"style":448},[447],[22,4780],{"className":4781,"style":452},[104],[22,4783,4785,4788,4791,4794,4797,4800,4803,4806,4809,4812,4815],{"className":4784,"style":456},[74],[22,4786,323],{"className":4787},[74],[22,4789,149],{"className":4790},[74],[22,4792],{"className":4793,"style":466},[355],[22,4795,328],{"className":4796},[374],[22,4798,301],{"className":4799},[378],[22,4801,51],{"className":4802},[74],[22,4804],{"className":4805,"style":3476},[355],[22,4807,323],{"className":4808},[2038],[22,4810],{"className":4811,"style":3476},[355],[22,4813,4609],{"className":4814,"style":4736},[74,78],[22,4816,315],{"className":4817},[429],[22,4819,4821,4824],{"style":4820},"top:-2.9439em;",[22,4822],{"className":4823,"style":452},[104],[22,4825,4827],{"className":4826,"style":523},[522],[525,4828,4829],{"xmlns":527,"width":528,"height":529,"viewBox":530,"preserveAspectRatio":531},[533,4830],{"d":535},[22,4832,120],{"className":4833},[119],[22,4835,4837],{"className":4836},[92],[22,4838,4841],{"className":4839,"style":4840},[96],"height:0.2561em;",[22,4842],{},[15,4844,4845,4846,4875,4876,4928,4929,4957,4958,5043,5044,130,5113,5182,5183,5234],{},"When ",[22,4847,4849,4862],{"className":4848,"translate":26},[25],[22,4850,4852],{"className":4851},[30],[32,4853,4854],{"xmlns":34},[36,4855,4856,4860],{},[39,4857,4858],{},[45,4859,4609],{},[53,4861,4609],{"encoding":55},[22,4863,4865],{"className":4864,"ariaHidden":61},[60],[22,4866,4868,4872],{"className":4867},[65],[22,4869],{"className":4870,"style":4871},[69],"height:0.4306em;",[22,4873,4609],{"className":4874,"style":4736},[74,78]," is uniformly distributed over ",[22,4877,4879,4901],{"className":4878,"translate":26},[25],[22,4880,4882],{"className":4881},[30],[32,4883,4884],{"xmlns":34},[36,4885,4886,4898],{},[39,4887,4888,4890,4892,4894,4896],{},[289,4889,4271],{"stretchy":300},[49,4891,2259],{},[289,4893,1062],{"separator":61},[49,4895,51],{},[289,4897,4282],{"stretchy":300},[53,4899,4900],{"encoding":55},"[0,1]",[22,4902,4904],{"className":4903,"ariaHidden":61},[60],[22,4905,4907,4910,4913,4916,4919,4922,4925],{"className":4906},[65],[22,4908],{"className":4909,"style":944},[69],[22,4911,4271],{"className":4912},[378],[22,4914,2259],{"className":4915},[74],[22,4917,1062],{"className":4918},[1413],[22,4920],{"className":4921,"style":466},[355],[22,4923,51],{"className":4924},[74],[22,4926,4282],{"className":4927},[429],", the distribution function of ",[22,4930,4932,4945],{"className":4931,"translate":26},[25],[22,4933,4935],{"className":4934},[30],[32,4936,4937],{"xmlns":34},[36,4938,4939,4943],{},[39,4940,4941],{},[45,4942,2103],{},[53,4944,2103],{"encoding":55},[22,4946,4948],{"className":4947,"ariaHidden":61},[60],[22,4949,4951,4954],{"className":4950},[65],[22,4952],{"className":4953,"style":233},[69],[22,4955,2103],{"className":4956,"style":2168},[74,78]," is ",[22,4959,4961,4985],{"className":4960,"translate":26},[25],[22,4962,4964],{"className":4963},[30],[32,4965,4966],{"xmlns":34},[36,4967,4968,4982],{},[39,4969,4970,4976,4978,4980],{},[42,4971,4972,4974],{},[45,4973,4336],{},[45,4975,2103],{},[289,4977,301],{"stretchy":300},[45,4979,2935],{},[289,4981,315],{"stretchy":300},[53,4983,4984],{"encoding":55},"F_R(r)",[22,4986,4988],{"className":4987,"ariaHidden":61},[60],[22,4989,4991,4994,5034,5037,5040],{"className":4990},[65],[22,4992],{"className":4993,"style":944},[69],[22,4995,4997,5000],{"className":4996},[74],[22,4998,4336],{"className":4999,"style":2796},[74,78],[22,5001,5003],{"className":5002},[83],[22,5004,5006,5026],{"className":5005},[87,88],[22,5007,5009,5023],{"className":5008},[92],[22,5010,5012],{"className":5011,"style":2809},[96],[22,5013,5014,5017],{"style":2812},[22,5015],{"className":5016,"style":105},[104],[22,5018,5020],{"className":5019},[109,110,111,112],[22,5021,2103],{"className":5022,"style":2168},[74,78,112],[22,5024,120],{"className":5025},[119],[22,5027,5029],{"className":5028},[92],[22,5030,5032],{"className":5031,"style":127},[96],[22,5033],{},[22,5035,301],{"className":5036},[378],[22,5038,2935],{"className":5039,"style":2181},[74,78],[22,5041,315],{"className":5042},[429],". Therefore, we can select two random variables ",[22,5045,5047,5064],{"className":5046,"translate":26},[25],[22,5048,5050],{"className":5049},[30],[32,5051,5052],{"xmlns":34},[36,5053,5054,5062],{},[39,5055,5056],{},[42,5057,5058,5060],{},[45,5059,47],{},[49,5061,51],{},[53,5063,56],{"encoding":55},[22,5065,5067],{"className":5066,"ariaHidden":61},[60],[22,5068,5070,5073],{"className":5069},[65],[22,5071],{"className":5072,"style":70},[69],[22,5074,5076,5079],{"className":5075},[74],[22,5077,47],{"className":5078,"style":79},[74,78],[22,5080,5082],{"className":5081},[83],[22,5083,5085,5105],{"className":5084},[87,88],[22,5086,5088,5102],{"className":5087},[92],[22,5089,5091],{"className":5090,"style":97},[96],[22,5092,5093,5096],{"style":100},[22,5094],{"className":5095,"style":105},[104],[22,5097,5099],{"className":5098},[109,110,111,112],[22,5100,51],{"className":5101},[74,112],[22,5103,120],{"className":5104},[119],[22,5106,5108],{"className":5107},[92],[22,5109,5111],{"className":5110,"style":127},[96],[22,5112],{},[22,5114,5116,5133],{"className":5115,"translate":26},[25],[22,5117,5119],{"className":5118},[30],[32,5120,5121],{"xmlns":34},[36,5122,5123,5131],{},[39,5124,5125],{},[42,5126,5127,5129],{},[45,5128,47],{},[49,5130,149],{},[53,5132,152],{"encoding":55},[22,5134,5136],{"className":5135,"ariaHidden":61},[60],[22,5137,5139,5142],{"className":5138},[65],[22,5140],{"className":5141,"style":70},[69],[22,5143,5145,5148],{"className":5144},[74],[22,5146,47],{"className":5147,"style":79},[74,78],[22,5149,5151],{"className":5150},[83],[22,5152,5154,5174],{"className":5153},[87,88],[22,5155,5157,5171],{"className":5156},[92],[22,5158,5160],{"className":5159,"style":97},[96],[22,5161,5162,5165],{"style":100},[22,5163],{"className":5164,"style":105},[104],[22,5166,5168],{"className":5167},[109,110,111,112],[22,5169,149],{"className":5170},[74,112],[22,5172,120],{"className":5173},[119],[22,5175,5177],{"className":5176},[92],[22,5178,5180],{"className":5179,"style":127},[96],[22,5181],{},", uniformly distributed over ",[22,5184,5186,5207],{"className":5185,"translate":26},[25],[22,5187,5189],{"className":5188},[30],[32,5190,5191],{"xmlns":34},[36,5192,5193,5205],{},[39,5194,5195,5197,5199,5201,5203],{},[289,5196,4271],{"stretchy":300},[49,5198,2259],{},[289,5200,1062],{"separator":61},[49,5202,51],{},[289,5204,4282],{"stretchy":300},[53,5206,4900],{"encoding":55},[22,5208,5210],{"className":5209,"ariaHidden":61},[60],[22,5211,5213,5216,5219,5222,5225,5228,5231],{"className":5212},[65],[22,5214],{"className":5215,"style":944},[69],[22,5217,4271],{"className":5218},[378],[22,5220,2259],{"className":5221},[74],[22,5223,1062],{"className":5224},[1413],[22,5226],{"className":5227,"style":466},[355],[22,5229,51],{"className":5230},[74],[22,5232,4282],{"className":5233},[429],", such that:",[22,5236,5238],{"className":5237,"translate":26},[272],[22,5239,5241,5311],{"className":5240,"translate":26},[25],[22,5242,5244],{"className":5243},[30],[32,5245,5246],{"xmlns":34,"display":281},[36,5247,5248,5308],{},[39,5249,5250,5252,5254,5256,5258,5264,5266,5268,5270,5272,5274,5276,5282,5284,5286,5288,5290],{},[45,5251,2112],{},[289,5253,291],{},[49,5255,149],{},[45,5257,306],{},[42,5259,5260,5262],{},[45,5261,47],{},[49,5263,51],{},[289,5265,1062],{"separator":61},[355,5267],{"linebreak":1065},[49,5269,51],{},[289,5271,323],{},[45,5273,4609],{},[289,5275,291],{},[42,5277,5278,5280],{},[45,5279,47],{},[49,5281,149],{},[289,5283,1062],{"separator":61},[355,5285],{"linebreak":1065},[45,5287,2103],{},[289,5289,291],{},[317,5291,5292],{},[39,5293,5294,5296,5298,5300,5302],{},[289,5295,323],{},[49,5297,149],{},[45,5299,328],{},[289,5301,297],{},[42,5303,5304,5306],{},[45,5305,47],{},[49,5307,149],{},[53,5309,5310],{"encoding":55},"\\theta = 2\\pi U_1, \\\\ 1 - z = U_2, \\\\ R = \\sqrt{-2 \\ln U_2}",[22,5312,5314,5332,5388,5391,5409,5427,5476,5479,5497],{"className":5313,"ariaHidden":61},[60],[22,5315,5317,5320,5323,5326,5329],{"className":5316},[65],[22,5318],{"className":5319,"style":2754},[69],[22,5321,2112],{"className":5322,"style":2181},[74,78],[22,5324],{"className":5325,"style":356},[355],[22,5327,291],{"className":5328},[360],[22,5330],{"className":5331,"style":356},[355],[22,5333,5335,5339,5342,5345,5385],{"className":5334},[65],[22,5336],{"className":5337,"style":5338},[69],"height:0.8778em;vertical-align:-0.1944em;",[22,5340,149],{"className":5341},[74],[22,5343,306],{"className":5344,"style":385},[74,78],[22,5346,5348,5351],{"className":5347},[74],[22,5349,47],{"className":5350,"style":79},[74,78],[22,5352,5354],{"className":5353},[83],[22,5355,5357,5377],{"className":5356},[87,88],[22,5358,5360,5374],{"className":5359},[92],[22,5361,5363],{"className":5362,"style":97},[96],[22,5364,5365,5368],{"style":100},[22,5366],{"className":5367,"style":105},[104],[22,5369,5371],{"className":5370},[109,110,111,112],[22,5372,51],{"className":5373},[74,112],[22,5375,120],{"className":5376},[119],[22,5378,5380],{"className":5379},[92],[22,5381,5383],{"className":5382,"style":127},[96],[22,5384],{},[22,5386,1062],{"className":5387},[1413],[22,5389],{"className":5390},[355,1065],[22,5392,5394,5397,5400,5403,5406],{"className":5393},[65],[22,5395],{"className":5396,"style":3469},[69],[22,5398,51],{"className":5399},[74],[22,5401],{"className":5402,"style":3476},[355],[22,5404,323],{"className":5405},[2038],[22,5407],{"className":5408,"style":3476},[355],[22,5410,5412,5415,5418,5421,5424],{"className":5411},[65],[22,5413],{"className":5414,"style":4871},[69],[22,5416,4609],{"className":5417,"style":4736},[74,78],[22,5419],{"className":5420,"style":356},[355],[22,5422,291],{"className":5423},[360],[22,5425],{"className":5426,"style":356},[355],[22,5428,5430,5433,5473],{"className":5429},[65],[22,5431],{"className":5432,"style":5338},[69],[22,5434,5436,5439],{"className":5435},[74],[22,5437,47],{"className":5438,"style":79},[74,78],[22,5440,5442],{"className":5441},[83],[22,5443,5445,5465],{"className":5444},[87,88],[22,5446,5448,5462],{"className":5447},[92],[22,5449,5451],{"className":5450,"style":97},[96],[22,5452,5453,5456],{"style":100},[22,5454],{"className":5455,"style":105},[104],[22,5457,5459],{"className":5458},[109,110,111,112],[22,5460,149],{"className":5461},[74,112],[22,5463,120],{"className":5464},[119],[22,5466,5468],{"className":5467},[92],[22,5469,5471],{"className":5470,"style":127},[96],[22,5472],{},[22,5474,1062],{"className":5475},[1413],[22,5477],{"className":5478},[355,1065],[22,5480,5482,5485,5488,5491,5494],{"className":5481},[65],[22,5483],{"className":5484,"style":233},[69],[22,5486,2103],{"className":5487,"style":2168},[74,78],[22,5489],{"className":5490,"style":356},[355],[22,5492,291],{"className":5493},[360],[22,5495],{"className":5496,"style":356},[355],[22,5498,5500,5504],{"className":5499},[65],[22,5501],{"className":5502,"style":5503},[69],"height:1.24em;vertical-align:-0.2339em;",[22,5505,5507],{"className":5506},[74,433],[22,5508,5510,5595],{"className":5509},[87,88],[22,5511,5513,5592],{"className":5512},[92],[22,5514,5516,5580],{"className":5515,"style":443},[96],[22,5517,5519,5522],{"className":5518,"style":448},[447],[22,5520],{"className":5521,"style":452},[104],[22,5523,5525,5528,5531,5534,5537,5540],{"className":5524,"style":456},[74],[22,5526,323],{"className":5527},[74],[22,5529,149],{"className":5530},[74],[22,5532],{"className":5533,"style":466},[355],[22,5535,328],{"className":5536},[374],[22,5538],{"className":5539,"style":466},[355],[22,5541,5543,5546],{"className":5542},[74],[22,5544,47],{"className":5545,"style":79},[74,78],[22,5547,5549],{"className":5548},[83],[22,5550,5552,5572],{"className":5551},[87,88],[22,5553,5555,5569],{"className":5554},[92],[22,5556,5558],{"className":5557,"style":97},[96],[22,5559,5560,5563],{"style":100},[22,5561],{"className":5562,"style":105},[104],[22,5564,5566],{"className":5565},[109,110,111,112],[22,5567,149],{"className":5568},[74,112],[22,5570,120],{"className":5571},[119],[22,5573,5575],{"className":5574},[92],[22,5576,5578],{"className":5577,"style":127},[96],[22,5579],{},[22,5581,5582,5585],{"style":515},[22,5583],{"className":5584,"style":452},[104],[22,5586,5588],{"className":5587,"style":523},[522],[525,5589,5590],{"xmlns":527,"width":528,"height":529,"viewBox":530,"preserveAspectRatio":531},[533,5591],{"d":535},[22,5593,120],{"className":5594},[119],[22,5596,5598],{"className":5597},[92],[22,5599,5601],{"className":5600,"style":545},[96],[22,5602],{},[15,5604,5605],{},"Substituting these into the earlier expressions, we obtain:",[22,5607,5609],{"className":5608,"translate":26},[272],[22,5610,5612,5659],{"className":5611,"translate":26},[25],[22,5613,5615],{"className":5614},[30],[32,5616,5617],{"xmlns":34,"display":281},[36,5618,5619,5657],{},[39,5620,5621,5623,5625,5627,5629,5631,5633,5635,5637,5639,5641,5643,5645,5647,5649,5651,5653,5655],{},[45,5622,221],{},[289,5624,291],{},[45,5626,2103],{},[45,5628,294],{},[289,5630,297],{},[289,5632,301],{"stretchy":300},[45,5634,2112],{},[289,5636,315],{"stretchy":300},[289,5638,1062],{"separator":61},[355,5640],{"linebreak":1065},[45,5642,252],{},[289,5644,291],{},[45,5646,2103],{},[45,5648,569],{},[289,5650,297],{},[289,5652,301],{"stretchy":300},[45,5654,2112],{},[289,5656,315],{"stretchy":300},[53,5658,2137],{"encoding":55},[22,5660,5662,5680,5707,5710,5728],{"className":5661,"ariaHidden":61},[60],[22,5663,5665,5668,5671,5674,5677],{"className":5664},[65],[22,5666],{"className":5667,"style":233},[69],[22,5669,221],{"className":5670,"style":237},[74,78],[22,5672],{"className":5673,"style":356},[355],[22,5675,291],{"className":5676},[360],[22,5678],{"className":5679,"style":356},[355],[22,5681,5683,5686,5689,5692,5695,5698,5701,5704],{"className":5682},[65],[22,5684],{"className":5685,"style":944},[69],[22,5687,2103],{"className":5688,"style":2168},[74,78],[22,5690],{"className":5691,"style":466},[355],[22,5693,294],{"className":5694},[374],[22,5696,301],{"className":5697},[378],[22,5699,2112],{"className":5700,"style":2181},[74,78],[22,5702,315],{"className":5703},[429],[22,5705,1062],{"className":5706},[1413],[22,5708],{"className":5709},[355,1065],[22,5711,5713,5716,5719,5722,5725],{"className":5712},[65],[22,5714],{"className":5715,"style":233},[69],[22,5717,252],{"className":5718,"style":267},[74,78],[22,5720],{"className":5721,"style":356},[355],[22,5723,291],{"className":5724},[360],[22,5726],{"className":5727,"style":356},[355],[22,5729,5731,5734,5737,5740,5743,5746,5749],{"className":5730},[65],[22,5732],{"className":5733,"style":944},[69],[22,5735,2103],{"className":5736,"style":2168},[74,78],[22,5738],{"className":5739,"style":466},[355],[22,5741,569],{"className":5742},[374],[22,5744,301],{"className":5745},[378],[22,5747,2112],{"className":5748,"style":2181},[74,78],[22,5750,315],{"className":5751},[429],[15,5753,5754,5755,130,5783,5811],{},"Thus, the original expressions for ",[22,5756,5758,5771],{"className":5757,"translate":26},[25],[22,5759,5761],{"className":5760},[30],[32,5762,5763],{"xmlns":34},[36,5764,5765,5769],{},[39,5766,5767],{},[45,5768,221],{},[53,5770,221],{"encoding":55},[22,5772,5774],{"className":5773,"ariaHidden":61},[60],[22,5775,5777,5780],{"className":5776},[65],[22,5778],{"className":5779,"style":233},[69],[22,5781,221],{"className":5782,"style":237},[74,78],[22,5784,5786,5799],{"className":5785,"translate":26},[25],[22,5787,5789],{"className":5788},[30],[32,5790,5791],{"xmlns":34},[36,5792,5793,5797],{},[39,5794,5795],{},[45,5796,252],{},[53,5798,252],{"encoding":55},[22,5800,5802],{"className":5801,"ariaHidden":61},[60],[22,5803,5805,5808],{"className":5804},[65],[22,5806],{"className":5807,"style":233},[69],[22,5809,252],{"className":5810,"style":267},[74,78]," are recovered, where they follow a normal distribution with mean 0 and variance 1.",[10,5813,5815],{"id":5814},"verification","Verification",[15,5817,5818,5819,5900,5901,5985],{},"To generate a random variable ",[22,5820,5822,5852],{"className":5821,"translate":26},[25],[22,5823,5825],{"className":5824},[30],[32,5826,5827],{"xmlns":34},[36,5828,5829,5849],{},[39,5830,5831,5833,5836,5839,5841,5843,5845,5847],{},[45,5832,221],{},[289,5834,5835],{},"∼",[45,5837,5838],{},"N",[289,5840,301],{"stretchy":300},[49,5842,2259],{},[289,5844,1062],{"separator":61},[49,5846,51],{},[289,5848,315],{"stretchy":300},[53,5850,5851],{"encoding":55},"X \\sim N(0,1)",[22,5853,5855,5873],{"className":5854,"ariaHidden":61},[60],[22,5856,5858,5861,5864,5867,5870],{"className":5857},[65],[22,5859],{"className":5860,"style":233},[69],[22,5862,221],{"className":5863,"style":237},[74,78],[22,5865],{"className":5866,"style":356},[355],[22,5868,5835],{"className":5869},[360],[22,5871],{"className":5872,"style":356},[355],[22,5874,5876,5879,5882,5885,5888,5891,5894,5897],{"className":5875},[65],[22,5877],{"className":5878,"style":944},[69],[22,5880,5838],{"className":5881,"style":79},[74,78],[22,5883,301],{"className":5884},[378],[22,5886,2259],{"className":5887},[74],[22,5889,1062],{"className":5890},[1413],[22,5892],{"className":5893,"style":466},[355],[22,5895,51],{"className":5896},[74],[22,5898,315],{"className":5899},[429]," following a standard normal distribution, we note that the inverse of the error function ",[22,5902,5904,5933],{"className":5903,"translate":26},[25],[22,5905,5907],{"className":5906},[30],[32,5908,5909],{"xmlns":34},[36,5910,5911,5930],{},[39,5912,5913],{},[1041,5914,5915,5924],{},[39,5916,5917,5919,5921],{},[45,5918,1045],{"mathvariant":2262},[45,5920,2935],{"mathvariant":2262},[45,5922,5923],{"mathvariant":2262},"f",[39,5925,5926,5928],{},[289,5927,323],{},[49,5929,51],{},[53,5931,5932],{"encoding":55},"\\mathrm{erf}^{-1}",[22,5934,5936],{"className":5935,"ariaHidden":61},[60],[22,5937,5939,5943],{"className":5938},[65],[22,5940],{"className":5941,"style":5942},[69],"height:0.8984em;",[22,5944,5946,5955],{"className":5945},[74],[22,5947,5949],{"className":5948},[74],[22,5950,5954],{"className":5951,"style":5953},[74,5952],"mathrm","margin-right:0.07778em;","erf",[22,5956,5958],{"className":5957},[83],[22,5959,5961],{"className":5960},[87],[22,5962,5964],{"className":5963},[92],[22,5965,5967],{"className":5966,"style":5942},[96],[22,5968,5970,5973],{"style":5969},"top:-3.1473em;margin-right:0.05em;",[22,5971],{"className":5972,"style":105},[104],[22,5974,5976],{"className":5975},[109,110,111,112],[22,5977,5979,5982],{"className":5978},[74,112],[22,5980,323],{"className":5981},[74,112],[22,5983,51],{"className":5984},[74,112]," is not elementary. Using a series expansion to approximate this inverse can introduce significant computational error.",[15,5987,5988],{},"Elementary functions, on the other hand, have well-established high-precision algorithms. Thus, we aim to construct a normally distributed random variable from a uniformly distributed one using elementary functions.",[15,5990,5991,5992,6081],{},"Let ",[22,5993,5995,6027],{"className":5994,"translate":26},[25],[22,5996,5998],{"className":5997},[30],[32,5999,6000],{"xmlns":34},[36,6001,6002,6024],{},[39,6003,6004,6006,6008,6011,6014,6016,6018,6020,6022],{},[45,6005,47],{},[289,6007,1062],{"separator":61},[45,6009,6010],{},"V",[289,6012,6013],{},"∈",[289,6015,301],{"stretchy":300},[49,6017,2259],{},[289,6019,1062],{"separator":61},[49,6021,51],{},[289,6023,315],{"stretchy":300},[53,6025,6026],{"encoding":55},"U, V \\in (0,1)",[22,6028,6030,6057],{"className":6029,"ariaHidden":61},[60],[22,6031,6033,6036,6039,6042,6045,6048,6051,6054],{"className":6032},[65],[22,6034],{"className":6035,"style":5338},[69],[22,6037,47],{"className":6038,"style":79},[74,78],[22,6040,1062],{"className":6041},[1413],[22,6043],{"className":6044,"style":466},[355],[22,6046,6010],{"className":6047,"style":267},[74,78],[22,6049],{"className":6050,"style":356},[355],[22,6052,6013],{"className":6053},[360],[22,6055],{"className":6056,"style":356},[355],[22,6058,6060,6063,6066,6069,6072,6075,6078],{"className":6059},[65],[22,6061],{"className":6062,"style":944},[69],[22,6064,301],{"className":6065},[378],[22,6067,2259],{"className":6068},[74],[22,6070,1062],{"className":6071},[1413],[22,6073],{"className":6074,"style":466},[355],[22,6076,51],{"className":6077},[74],[22,6079,315],{"className":6080},[429]," be uniformly distributed random variables. Consider the following transformations:",[22,6083,6085],{"className":6084,"translate":26},[272],[22,6086,6088,6168],{"className":6087,"translate":26},[25],[22,6089,6091],{"className":6090},[30],[32,6092,6093],{"xmlns":34,"display":281},[36,6094,6095,6165],{},[39,6096,6097,6099,6101,6115,6117,6119,6121,6123,6125,6127,6129,6131,6133,6135,6137,6151,6153,6155,6157,6159,6161,6163],{},[45,6098,221],{},[289,6100,291],{},[317,6102,6103],{},[39,6104,6105,6107,6109,6111,6113],{},[289,6106,323],{},[49,6108,149],{},[45,6110,328],{},[289,6112,297],{},[45,6114,47],{},[45,6116,294],{},[289,6118,297],{},[289,6120,301],{"stretchy":300},[49,6122,149],{},[45,6124,306],{},[45,6126,6010],{},[289,6128,315],{"stretchy":300},[289,6130,1062],{"separator":61},[355,6132],{"linebreak":1065},[45,6134,252],{},[289,6136,291],{},[317,6138,6139],{},[39,6140,6141,6143,6145,6147,6149],{},[289,6142,323],{},[49,6144,149],{},[45,6146,328],{},[289,6148,297],{},[45,6150,47],{},[45,6152,569],{},[289,6154,297],{},[289,6156,301],{"stretchy":300},[49,6158,149],{},[45,6160,306],{},[45,6162,6010],{},[289,6164,315],{"stretchy":300},[53,6166,6167],{"encoding":55},"X = \\sqrt{-2\\ln U} \\cos(2\\pi V), \\\\ Y = \\sqrt{-2\\ln U} \\sin(2\\pi V)",[22,6169,6171,6189,6285,6288,6306],{"className":6170,"ariaHidden":61},[60],[22,6172,6174,6177,6180,6183,6186],{"className":6173},[65],[22,6175],{"className":6176,"style":233},[69],[22,6178,221],{"className":6179,"style":237},[74,78],[22,6181],{"className":6182,"style":356},[355],[22,6184,291],{"className":6185},[360],[22,6187],{"className":6188,"style":356},[355],[22,6190,6192,6196,6261,6264,6267,6270,6273,6276,6279,6282],{"className":6191},[65],[22,6193],{"className":6194,"style":6195},[69],"height:1.1894em;vertical-align:-0.25em;",[22,6197,6199],{"className":6198},[74,433],[22,6200,6202,6252],{"className":6201},[87,88],[22,6203,6205,6249],{"className":6204},[92],[22,6206,6209,6236],{"className":6207,"style":6208},[96],"height:0.9394em;",[22,6210,6212,6215],{"className":6211,"style":1192},[447],[22,6213],{"className":6214,"style":1172},[104],[22,6216,6218,6221,6224,6227,6230,6233],{"className":6217,"style":1199},[74],[22,6219,323],{"className":6220},[74],[22,6222,149],{"className":6223},[74],[22,6225],{"className":6226,"style":466},[355],[22,6228,328],{"className":6229},[374],[22,6231],{"className":6232,"style":466},[355],[22,6234,47],{"className":6235,"style":79},[74,78],[22,6237,6239,6242],{"style":6238},"top:-2.8994em;",[22,6240],{"className":6241,"style":1172},[104],[22,6243,6245],{"className":6244,"style":1215},[522],[525,6246,6247],{"xmlns":527,"width":528,"height":1218,"viewBox":1219,"preserveAspectRatio":531},[533,6248],{"d":1222},[22,6250,120],{"className":6251},[119],[22,6253,6255],{"className":6254},[92],[22,6256,6259],{"className":6257,"style":6258},[96],"height:0.1006em;",[22,6260],{},[22,6262],{"className":6263,"style":466},[355],[22,6265,294],{"className":6266},[374],[22,6268,301],{"className":6269},[378],[22,6271,149],{"className":6272},[74],[22,6274,306],{"className":6275,"style":385},[74,78],[22,6277,6010],{"className":6278,"style":267},[74,78],[22,6280,315],{"className":6281},[429],[22,6283,1062],{"className":6284},[1413],[22,6286],{"className":6287},[355,1065],[22,6289,6291,6294,6297,6300,6303],{"className":6290},[65],[22,6292],{"className":6293,"style":233},[69],[22,6295,252],{"className":6296,"style":267},[74,78],[22,6298],{"className":6299,"style":356},[355],[22,6301,291],{"className":6302},[360],[22,6304],{"className":6305,"style":356},[355],[22,6307,6309,6312,6374,6377,6380,6383,6386,6389,6392],{"className":6308},[65],[22,6310],{"className":6311,"style":6195},[69],[22,6313,6315],{"className":6314},[74,433],[22,6316,6318,6366],{"className":6317},[87,88],[22,6319,6321,6363],{"className":6320},[92],[22,6322,6324,6351],{"className":6323,"style":6208},[96],[22,6325,6327,6330],{"className":6326,"style":1192},[447],[22,6328],{"className":6329,"style":1172},[104],[22,6331,6333,6336,6339,6342,6345,6348],{"className":6332,"style":1199},[74],[22,6334,323],{"className":6335},[74],[22,6337,149],{"className":6338},[74],[22,6340],{"className":6341,"style":466},[355],[22,6343,328],{"className":6344},[374],[22,6346],{"className":6347,"style":466},[355],[22,6349,47],{"className":6350,"style":79},[74,78],[22,6352,6353,6356],{"style":6238},[22,6354],{"className":6355,"style":1172},[104],[22,6357,6359],{"className":6358,"style":1215},[522],[525,6360,6361],{"xmlns":527,"width":528,"height":1218,"viewBox":1219,"preserveAspectRatio":531},[533,6362],{"d":1222},[22,6364,120],{"className":6365},[119],[22,6367,6369],{"className":6368},[92],[22,6370,6372],{"className":6371,"style":6258},[96],[22,6373],{},[22,6375],{"className":6376,"style":466},[355],[22,6378,569],{"className":6379},[374],[22,6381,301],{"className":6382},[378],[22,6384,149],{"className":6385},[74],[22,6387,306],{"className":6388,"style":385},[74,78],[22,6390,6010],{"className":6391,"style":267},[74,78],[22,6393,315],{"className":6394},[429],[15,6396,6397,6398,130,6426,6454],{},"Next, we verify that ",[22,6399,6401,6414],{"className":6400,"translate":26},[25],[22,6402,6404],{"className":6403},[30],[32,6405,6406],{"xmlns":34},[36,6407,6408,6412],{},[39,6409,6410],{},[45,6411,221],{},[53,6413,221],{"encoding":55},[22,6415,6417],{"className":6416,"ariaHidden":61},[60],[22,6418,6420,6423],{"className":6419},[65],[22,6421],{"className":6422,"style":233},[69],[22,6424,221],{"className":6425,"style":237},[74,78],[22,6427,6429,6442],{"className":6428,"translate":26},[25],[22,6430,6432],{"className":6431},[30],[32,6433,6434],{"xmlns":34},[36,6435,6436,6440],{},[39,6437,6438],{},[45,6439,252],{},[53,6441,252],{"encoding":55},[22,6443,6445],{"className":6444,"ariaHidden":61},[60],[22,6446,6448,6451],{"className":6447},[65],[22,6449],{"className":6450,"style":233},[69],[22,6452,252],{"className":6453,"style":267},[74,78]," are independent and follow a standard normal distribution.",[15,6456,6457],{},"Define:",[22,6459,6461],{"className":6460,"translate":26},[272],[22,6462,6464,6524],{"className":6463,"translate":26},[25],[22,6465,6467],{"className":6466},[30],[32,6468,6469],{"xmlns":34,"display":281},[36,6470,6471,6521],{},[39,6472,6473,6475,6477,6495,6497,6499,6502,6504,6507,6509],{},[45,6474,2103],{},[289,6476,291],{},[317,6478,6479],{},[39,6480,6481,6487,6489],{},[1041,6482,6483,6485],{},[45,6484,221],{},[49,6486,149],{},[289,6488,1801],{},[1041,6490,6491,6493],{},[45,6492,252],{},[49,6494,149],{},[289,6496,1062],{"separator":61},[355,6498],{"linebreak":1065},[45,6500,6501],{"mathvariant":2262},"Θ",[289,6503,291],{},[45,6505,6506],{},"arctan",[289,6508,297],{},[39,6510,6511,6513,6519],{},[289,6512,301],{"fence":61},[1028,6514,6515,6517],{},[45,6516,252],{},[45,6518,221],{},[289,6520,315],{"fence":61},[53,6522,6523],{"encoding":55},"R = \\sqrt{X^2 + Y^2}, \\\\ \\Theta = \\arctan\\left(\\frac{Y}{X}\\right)",[22,6525,6527,6545,6671,6674,6692],{"className":6526,"ariaHidden":61},[60],[22,6528,6530,6533,6536,6539,6542],{"className":6529},[65],[22,6531],{"className":6532,"style":233},[69],[22,6534,2103],{"className":6535,"style":2168},[74,78],[22,6537],{"className":6538,"style":356},[355],[22,6540,291],{"className":6541},[360],[22,6543],{"className":6544,"style":356},[355],[22,6546,6548,6552,6668],{"className":6547},[65],[22,6549],{"className":6550,"style":6551},[69],"height:1.2567em;vertical-align:-0.1944em;",[22,6553,6555],{"className":6554},[74,433],[22,6556,6558,6659],{"className":6557},[87,88],[22,6559,6561,6656],{"className":6560},[92],[22,6562,6565,6643],{"className":6563,"style":6564},[96],"height:1.0623em;",[22,6566,6568,6571],{"className":6567,"style":448},[447],[22,6569],{"className":6570,"style":452},[104],[22,6572,6574,6605,6608,6611,6614],{"className":6573,"style":456},[74],[22,6575,6577,6580],{"className":6576},[74],[22,6578,221],{"className":6579,"style":237},[74,78],[22,6581,6583],{"className":6582},[83],[22,6584,6586],{"className":6585},[87],[22,6587,6589],{"className":6588},[92],[22,6590,6593],{"className":6591,"style":6592},[96],"height:0.7401em;",[22,6594,6596,6599],{"style":6595},"top:-2.989em;margin-right:0.05em;",[22,6597],{"className":6598,"style":105},[104],[22,6600,6602],{"className":6601},[109,110,111,112],[22,6603,149],{"className":6604},[74,112],[22,6606],{"className":6607,"style":3476},[355],[22,6609,1801],{"className":6610},[2038],[22,6612],{"className":6613,"style":3476},[355],[22,6615,6617,6620],{"className":6616},[74],[22,6618,252],{"className":6619,"style":267},[74,78],[22,6621,6623],{"className":6622},[83],[22,6624,6626],{"className":6625},[87],[22,6627,6629],{"className":6628},[92],[22,6630,6632],{"className":6631,"style":6592},[96],[22,6633,6634,6637],{"style":6595},[22,6635],{"className":6636,"style":105},[104],[22,6638,6640],{"className":6639},[109,110,111,112],[22,6641,149],{"className":6642},[74,112],[22,6644,6646,6649],{"style":6645},"top:-3.0223em;",[22,6647],{"className":6648,"style":452},[104],[22,6650,6652],{"className":6651,"style":523},[522],[525,6653,6654],{"xmlns":527,"width":528,"height":529,"viewBox":530,"preserveAspectRatio":531},[533,6655],{"d":535},[22,6657,120],{"className":6658},[119],[22,6660,6662],{"className":6661},[92],[22,6663,6666],{"className":6664,"style":6665},[96],"height:0.1777em;",[22,6667],{},[22,6669,1062],{"className":6670},[1413],[22,6672],{"className":6673},[355,1065],[22,6675,6677,6680,6683,6686,6689],{"className":6676},[65],[22,6678],{"className":6679,"style":233},[69],[22,6681,6501],{"className":6682},[74],[22,6684],{"className":6685,"style":356},[355],[22,6687,291],{"className":6688},[360],[22,6690],{"className":6691,"style":356},[355],[22,6693,6695,6699,6702,6705],{"className":6694},[65],[22,6696],{"className":6697,"style":6698},[69],"height:2.4em;vertical-align:-0.95em;",[22,6700,6506],{"className":6701},[374],[22,6703],{"className":6704,"style":466},[355],[22,6706,6709,6718,6781],{"className":6707},[6708],"minner",[22,6710,6714],{"className":6711,"style":6713},[378,6712],"delimcenter","top:0em;",[22,6715,301],{"className":6716},[6717,111],"delimsizing",[22,6719,6721,6724,6778],{"className":6720},[74],[22,6722],{"className":6723},[378,1152],[22,6725,6727],{"className":6726},[1028],[22,6728,6730,6770],{"className":6729},[87,88],[22,6731,6733,6767],{"className":6732},[92],[22,6734,6737,6748,6756],{"className":6735,"style":6736},[96],"height:1.3603em;",[22,6738,6739,6742],{"style":1879},[22,6740],{"className":6741,"style":1172},[104],[22,6743,6745],{"className":6744},[74],[22,6746,221],{"className":6747,"style":237},[74,78],[22,6749,6750,6753],{"style":1237},[22,6751],{"className":6752,"style":1172},[104],[22,6754],{"className":6755,"style":1245},[1244],[22,6757,6758,6761],{"style":1248},[22,6759],{"className":6760,"style":1172},[104],[22,6762,6764],{"className":6763},[74],[22,6765,252],{"className":6766,"style":267},[74,78],[22,6768,120],{"className":6769},[119],[22,6771,6773],{"className":6772},[92],[22,6774,6776],{"className":6775,"style":1920},[96],[22,6777],{},[22,6779],{"className":6780},[429,1152],[22,6782,6784],{"className":6783,"style":6713},[429,6712],[22,6785,315],{"className":6786},[6717,111],[15,6788,6789,6790,130,6851,6977,6978,130,7006,7035],{},"Note that ",[22,6791,6793,6815],{"className":6792,"translate":26},[25],[22,6794,6796],{"className":6795},[30],[32,6797,6798],{"xmlns":34},[36,6799,6800,6812],{},[39,6801,6802,6804,6806,6808,6810],{},[45,6803,6501],{"mathvariant":2262},[289,6805,291],{},[49,6807,149],{},[45,6809,306],{},[45,6811,6010],{},[53,6813,6814],{"encoding":55},"\\Theta = 2\\pi V",[22,6816,6818,6836],{"className":6817,"ariaHidden":61},[60],[22,6819,6821,6824,6827,6830,6833],{"className":6820},[65],[22,6822],{"className":6823,"style":233},[69],[22,6825,6501],{"className":6826},[74],[22,6828],{"className":6829,"style":356},[355],[22,6831,291],{"className":6832},[360],[22,6834],{"className":6835,"style":356},[355],[22,6837,6839,6842,6845,6848],{"className":6838},[65],[22,6840],{"className":6841,"style":233},[69],[22,6843,149],{"className":6844},[74],[22,6846,306],{"className":6847,"style":385},[74,78],[22,6849,6010],{"className":6850,"style":267},[74,78],[22,6852,6854,6884],{"className":6853,"translate":26},[25],[22,6855,6857],{"className":6856},[30],[32,6858,6859],{"xmlns":34},[36,6860,6861,6881],{},[39,6862,6863,6865,6867],{},[45,6864,2103],{},[289,6866,291],{},[317,6868,6869],{},[39,6870,6871,6873,6875,6877,6879],{},[289,6872,323],{},[49,6874,149],{},[45,6876,328],{},[289,6878,297],{},[45,6880,47],{},[53,6882,6883],{"encoding":55},"R = \\sqrt{-2\\ln U}",[22,6885,6887,6905],{"className":6886,"ariaHidden":61},[60],[22,6888,6890,6893,6896,6899,6902],{"className":6889},[65],[22,6891],{"className":6892,"style":233},[69],[22,6894,2103],{"className":6895,"style":2168},[74,78],[22,6897],{"className":6898,"style":356},[355],[22,6900,291],{"className":6901},[360],[22,6903],{"className":6904,"style":356},[355],[22,6906,6908,6912],{"className":6907},[65],[22,6909],{"className":6910,"style":6911},[69],"height:1.04em;vertical-align:-0.1494em;",[22,6913,6915],{"className":6914},[74,433],[22,6916,6918,6968],{"className":6917},[87,88],[22,6919,6921,6965],{"className":6920},[92],[22,6922,6925,6952],{"className":6923,"style":6924},[96],"height:0.8906em;",[22,6926,6928,6931],{"className":6927,"style":1192},[447],[22,6929],{"className":6930,"style":1172},[104],[22,6932,6934,6937,6940,6943,6946,6949],{"className":6933,"style":1199},[74],[22,6935,323],{"className":6936},[74],[22,6938,149],{"className":6939},[74],[22,6941],{"className":6942,"style":466},[355],[22,6944,328],{"className":6945},[374],[22,6947],{"className":6948,"style":466},[355],[22,6950,47],{"className":6951,"style":79},[74,78],[22,6953,6955,6958],{"style":6954},"top:-2.8506em;",[22,6956],{"className":6957,"style":1172},[104],[22,6959,6961],{"className":6960,"style":1215},[522],[525,6962,6963],{"xmlns":527,"width":528,"height":1218,"viewBox":1219,"preserveAspectRatio":531},[533,6964],{"d":1222},[22,6966,120],{"className":6967},[119],[22,6969,6971],{"className":6970},[92],[22,6972,6975],{"className":6973,"style":6974},[96],"height:0.1494em;",[22,6976],{},". We now show that ",[22,6979,6981,6994],{"className":6980,"translate":26},[25],[22,6982,6984],{"className":6983},[30],[32,6985,6986],{"xmlns":34},[36,6987,6988,6992],{},[39,6989,6990],{},[45,6991,2103],{},[53,6993,2103],{"encoding":55},[22,6995,6997],{"className":6996,"ariaHidden":61},[60],[22,6998,7000,7003],{"className":6999},[65],[22,7001],{"className":7002,"style":233},[69],[22,7004,2103],{"className":7005,"style":2168},[74,78],[22,7007,7009,7023],{"className":7008,"translate":26},[25],[22,7010,7012],{"className":7011},[30],[32,7013,7014],{"xmlns":34},[36,7015,7016,7020],{},[39,7017,7018],{},[45,7019,6501],{"mathvariant":2262},[53,7021,7022],{"encoding":55},"\\Theta",[22,7024,7026],{"className":7025,"ariaHidden":61},[60],[22,7027,7029,7032],{"className":7028},[65],[22,7030],{"className":7031,"style":233},[69],[22,7033,6501],{"className":7034},[74],", under the polar coordinate transformation, are independent and that they yield standard normal random variables.",[15,7037,7038],{},"Consider the probability density function of one variable when the other",[15,7040,7041,7042,7070],{},"is fixed. The probability density function of ",[22,7043,7045,7058],{"className":7044,"translate":26},[25],[22,7046,7048],{"className":7047},[30],[32,7049,7050],{"xmlns":34},[36,7051,7052,7056],{},[39,7053,7054],{},[45,7055,6501],{"mathvariant":2262},[53,7057,7022],{"encoding":55},[22,7059,7061],{"className":7060,"ariaHidden":61},[60],[22,7062,7064,7067],{"className":7063},[65],[22,7065],{"className":7066,"style":233},[69],[22,7068,6501],{"className":7069},[74]," is:",[22,7072,7074],{"className":7073,"translate":26},[272],[22,7075,7077,7113],{"className":7076,"translate":26},[25],[22,7078,7080],{"className":7079},[30],[32,7081,7082],{"xmlns":34,"display":281},[36,7083,7084,7110],{},[39,7085,7086,7092,7094,7096,7098,7100],{},[42,7087,7088,7090],{},[45,7089,2775],{},[45,7091,6501],{"mathvariant":2262},[289,7093,301],{"stretchy":300},[45,7095,6501],{"mathvariant":2262},[289,7097,315],{"stretchy":300},[289,7099,291],{},[1028,7101,7102,7104],{},[49,7103,51],{},[39,7105,7106,7108],{},[49,7107,149],{},[45,7109,306],{},[53,7111,7112],{"encoding":55},"P_\\Theta(\\Theta) = \\frac{1}{2\\pi}",[22,7114,7116,7180],{"className":7115,"ariaHidden":61},[60],[22,7117,7119,7122,7162,7165,7168,7171,7174,7177],{"className":7118},[65],[22,7120],{"className":7121,"style":944},[69],[22,7123,7125,7128],{"className":7124},[74],[22,7126,2775],{"className":7127,"style":2796},[74,78],[22,7129,7131],{"className":7130},[83],[22,7132,7134,7154],{"className":7133},[87,88],[22,7135,7137,7151],{"className":7136},[92],[22,7138,7140],{"className":7139,"style":2809},[96],[22,7141,7142,7145],{"style":2812},[22,7143],{"className":7144,"style":105},[104],[22,7146,7148],{"className":7147},[109,110,111,112],[22,7149,6501],{"className":7150},[74,112],[22,7152,120],{"className":7153},[119],[22,7155,7157],{"className":7156},[92],[22,7158,7160],{"className":7159,"style":127},[96],[22,7161],{},[22,7163,301],{"className":7164},[378],[22,7166,6501],{"className":7167},[74],[22,7169,315],{"className":7170},[429],[22,7172],{"className":7173,"style":356},[355],[22,7175,291],{"className":7176},[360],[22,7178],{"className":7179,"style":356},[355],[22,7181,7183,7186],{"className":7182},[65],[22,7184],{"className":7185,"style":1858},[69],[22,7187,7189,7192,7248],{"className":7188},[74],[22,7190],{"className":7191},[378,1152],[22,7193,7195],{"className":7194},[1028],[22,7196,7198,7240],{"className":7197},[87,88],[22,7199,7201,7237],{"className":7200},[92],[22,7202,7204,7218,7226],{"className":7203,"style":1165},[96],[22,7205,7206,7209],{"style":1879},[22,7207],{"className":7208,"style":1172},[104],[22,7210,7212,7215],{"className":7211},[74],[22,7213,149],{"className":7214},[74],[22,7216,306],{"className":7217,"style":385},[74,78],[22,7219,7220,7223],{"style":1237},[22,7221],{"className":7222,"style":1172},[104],[22,7224],{"className":7225,"style":1245},[1244],[22,7227,7228,7231],{"style":1248},[22,7229],{"className":7230,"style":1172},[104],[22,7232,7234],{"className":7233},[74],[22,7235,51],{"className":7236},[74],[22,7238,120],{"className":7239},[119],[22,7241,7243],{"className":7242},[92],[22,7244,7246],{"className":7245,"style":1920},[96],[22,7247],{},[22,7249],{"className":7250},[429,1152],[15,7252,7253,7254,7282],{},"which is straightforward. For ",[22,7255,7257,7270],{"className":7256,"translate":26},[25],[22,7258,7260],{"className":7259},[30],[32,7261,7262],{"xmlns":34},[36,7263,7264,7268],{},[39,7265,7266],{},[45,7267,2103],{},[53,7269,2103],{"encoding":55},[22,7271,7273],{"className":7272,"ariaHidden":61},[60],[22,7274,7276,7279],{"className":7275},[65],[22,7277],{"className":7278,"style":233},[69],[22,7280,2103],{"className":7281,"style":2168},[74,78],", the cumulative distribution function is:",[22,7284,7286],{"className":7285,"translate":26},[272],[22,7287,7289,7360],{"className":7288,"translate":26},[25],[22,7290,7292],{"className":7291},[30],[32,7293,7294],{"xmlns":34,"display":281},[36,7295,7296,7357],{},[39,7297,7298,7300,7302,7304,7306,7312,7314,7316,7318,7320],{},[45,7299,2775],{},[289,7301,301],{"stretchy":300},[45,7303,2103],{},[289,7305,2932],{},[42,7307,7308,7310],{},[45,7309,2103],{},[49,7311,2259],{},[289,7313,315],{"stretchy":300},[289,7315,291],{},[355,7317],{"linebreak":1065},[45,7319,2775],{},[39,7321,7322,7324,7326,7328,7331,7333,7355],{},[289,7323,301],{"fence":61},[45,7325,47],{},[289,7327,2932],{},[45,7329,7330],{},"exp",[289,7332,297],{},[39,7334,7335,7337,7353],{},[289,7336,301],{"fence":61},[1028,7338,7339,7351],{},[39,7340,7341,7343],{},[289,7342,323],{},[2252,7344,7345,7347,7349],{},[45,7346,2103],{},[49,7348,2259],{},[49,7350,149],{},[49,7352,149],{},[289,7354,315],{"fence":61},[289,7356,315],{"fence":61},[53,7358,7359],{"encoding":55},"P(R \\leq R_0) = \\\\ P\\left(U \\leq \\exp\\left(\\frac{-R_0^2}{2}\\right)\\right)",[22,7361,7363,7387,7443,7446],{"className":7362,"ariaHidden":61},[60],[22,7364,7366,7369,7372,7375,7378,7381,7384],{"className":7365},[65],[22,7367],{"className":7368,"style":944},[69],[22,7370,2775],{"className":7371,"style":2796},[74,78],[22,7373,301],{"className":7374},[378],[22,7376,2103],{"className":7377,"style":2168},[74,78],[22,7379],{"className":7380,"style":356},[355],[22,7382,2932],{"className":7383},[360],[22,7385],{"className":7386,"style":356},[355],[22,7388,7390,7393,7434,7437,7440],{"className":7389},[65],[22,7391],{"className":7392,"style":944},[69],[22,7394,7396,7399],{"className":7395},[74],[22,7397,2103],{"className":7398,"style":2168},[74,78],[22,7400,7402],{"className":7401},[83],[22,7403,7405,7426],{"className":7404},[87,88],[22,7406,7408,7423],{"className":7407},[92],[22,7409,7411],{"className":7410,"style":97},[96],[22,7412,7414,7417],{"style":7413},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[22,7415],{"className":7416,"style":105},[104],[22,7418,7420],{"className":7419},[109,110,111,112],[22,7421,2259],{"className":7422},[74,112],[22,7424,120],{"className":7425},[119],[22,7427,7429],{"className":7428},[92],[22,7430,7432],{"className":7431,"style":127},[96],[22,7433],{},[22,7435,315],{"className":7436},[429],[22,7438],{"className":7439,"style":356},[355],[22,7441,291],{"className":7442},[360],[22,7444],{"className":7445},[355,1065],[22,7447,7449,7453,7456,7459],{"className":7448},[65],[22,7450],{"className":7451,"style":7452},[69],"height:2.4411em;vertical-align:-0.95em;",[22,7454,2775],{"className":7455,"style":2796},[74,78],[22,7457],{"className":7458,"style":466},[355],[22,7460,7462,7468,7471,7474,7477,7480,7483,7486,7619],{"className":7461},[6708],[22,7463,7465],{"className":7464,"style":6713},[378,6712],[22,7466,301],{"className":7467},[6717,111],[22,7469,47],{"className":7470,"style":79},[74,78],[22,7472],{"className":7473,"style":356},[355],[22,7475,2932],{"className":7476},[360],[22,7478],{"className":7479,"style":356},[355],[22,7481,7330],{"className":7482},[374],[22,7484],{"className":7485,"style":466},[355],[22,7487,7489,7495,7613],{"className":7488},[6708],[22,7490,7492],{"className":7491,"style":6713},[378,6712],[22,7493,301],{"className":7494},[6717,111],[22,7496,7498,7501,7610],{"className":7497},[74],[22,7499],{"className":7500},[378,1152],[22,7502,7504],{"className":7503},[1028],[22,7505,7507,7602],{"className":7506},[87,88],[22,7508,7510,7599],{"className":7509},[92],[22,7511,7514,7525,7533],{"className":7512,"style":7513},[96],"height:1.4911em;",[22,7515,7516,7519],{"style":1879},[22,7517],{"className":7518,"style":1172},[104],[22,7520,7522],{"className":7521},[74],[22,7523,149],{"className":7524},[74],[22,7526,7527,7530],{"style":1237},[22,7528],{"className":7529,"style":1172},[104],[22,7531],{"className":7532,"style":1245},[1244],[22,7534,7535,7538],{"style":1248},[22,7536],{"className":7537,"style":1172},[104],[22,7539,7541,7544],{"className":7540},[74],[22,7542,323],{"className":7543},[74],[22,7545,7547,7550],{"className":7546},[74],[22,7548,2103],{"className":7549,"style":2168},[74,78],[22,7551,7553],{"className":7552},[83],[22,7554,7556,7590],{"className":7555},[87,88],[22,7557,7559,7587],{"className":7558},[92],[22,7560,7563,7575],{"className":7561,"style":7562},[96],"height:0.8141em;",[22,7564,7566,7569],{"style":7565},"top:-2.4519em;margin-left:-0.0077em;margin-right:0.05em;",[22,7567],{"className":7568,"style":105},[104],[22,7570,7572],{"className":7571},[109,110,111,112],[22,7573,2259],{"className":7574},[74,112],[22,7576,7578,7581],{"style":7577},"top:-3.063em;margin-right:0.05em;",[22,7579],{"className":7580,"style":105},[104],[22,7582,7584],{"className":7583},[109,110,111,112],[22,7585,149],{"className":7586},[74,112],[22,7588,120],{"className":7589},[119],[22,7591,7593],{"className":7592},[92],[22,7594,7597],{"className":7595,"style":7596},[96],"height:0.2481em;",[22,7598],{},[22,7600,120],{"className":7601},[119],[22,7603,7605],{"className":7604},[92],[22,7606,7608],{"className":7607,"style":1920},[96],[22,7609],{},[22,7611],{"className":7612},[429,1152],[22,7614,7616],{"className":7615,"style":6713},[429,6712],[22,7617,315],{"className":7618},[6717,111],[22,7620,7622],{"className":7621,"style":6713},[429,6712],[22,7623,315],{"className":7624},[6717,111],[15,7626,7627,7628,7070],{},"Thus, the probability density function of ",[22,7629,7631,7644],{"className":7630,"translate":26},[25],[22,7632,7634],{"className":7633},[30],[32,7635,7636],{"xmlns":34},[36,7637,7638,7642],{},[39,7639,7640],{},[45,7641,2103],{},[53,7643,2103],{"encoding":55},[22,7645,7647],{"className":7646,"ariaHidden":61},[60],[22,7648,7650,7653],{"className":7649},[65],[22,7651],{"className":7652,"style":233},[69],[22,7654,2103],{"className":7655,"style":2168},[74,78],[22,7657,7659],{"className":7658,"translate":26},[272],[22,7660,7662,7708],{"className":7661,"translate":26},[25],[22,7663,7665],{"className":7664},[30],[32,7666,7667],{"xmlns":34,"display":281},[36,7668,7669,7705],{},[39,7670,7671,7677,7679,7681,7683,7685,7687],{},[42,7672,7673,7675],{},[45,7674,2775],{},[45,7676,2103],{},[289,7678,301],{"stretchy":300},[45,7680,2103],{},[289,7682,315],{"stretchy":300},[289,7684,291],{},[45,7686,2103],{},[1041,7688,7689,7691],{},[45,7690,1045],{},[39,7692,7693,7695],{},[289,7694,323],{},[1028,7696,7697,7703],{},[1041,7698,7699,7701],{},[45,7700,2103],{},[49,7702,149],{},[49,7704,149],{},[53,7706,7707],{"encoding":55},"P_R(R) = R e^{-\\frac{R^2}{2}}",[22,7709,7711,7775],{"className":7710,"ariaHidden":61},[60],[22,7712,7714,7717,7757,7760,7763,7766,7769,7772],{"className":7713},[65],[22,7715],{"className":7716,"style":944},[69],[22,7718,7720,7723],{"className":7719},[74],[22,7721,2775],{"className":7722,"style":2796},[74,78],[22,7724,7726],{"className":7725},[83],[22,7727,7729,7749],{"className":7728},[87,88],[22,7730,7732,7746],{"className":7731},[92],[22,7733,7735],{"className":7734,"style":2809},[96],[22,7736,7737,7740],{"style":2812},[22,7738],{"className":7739,"style":105},[104],[22,7741,7743],{"className":7742},[109,110,111,112],[22,7744,2103],{"className":7745,"style":2168},[74,78,112],[22,7747,120],{"className":7748},[119],[22,7750,7752],{"className":7751},[92],[22,7753,7755],{"className":7754,"style":127},[96],[22,7756],{},[22,7758,301],{"className":7759},[378],[22,7761,2103],{"className":7762,"style":2168},[74,78],[22,7764,315],{"className":7765},[429],[22,7767],{"className":7768,"style":356},[355],[22,7770,291],{"className":7771},[360],[22,7773],{"className":7774,"style":356},[355],[22,7776,7778,7781,7784],{"className":7777},[65],[22,7779],{"className":7780,"style":1291},[69],[22,7782,2103],{"className":7783,"style":2168},[74,78],[22,7785,7787,7790],{"className":7786},[74],[22,7788,1045],{"className":7789},[74,78],[22,7791,7793],{"className":7792},[83],[22,7794,7796],{"className":7795},[87],[22,7797,7799],{"className":7798},[92],[22,7800,7802],{"className":7801,"style":1291},[96],[22,7803,7804,7807],{"style":1294},[22,7805],{"className":7806,"style":1172},[104],[22,7808,7810],{"className":7809},[109,110,111,112],[22,7811,7813,7816],{"className":7812},[74,112],[22,7814,323],{"className":7815},[74,112],[22,7817,7819,7822,7904],{"className":7818},[74,112],[22,7820],{"className":7821},[378,1152,109,1313,1314],[22,7823,7825],{"className":7824},[1028],[22,7826,7828,7896],{"className":7827},[87,88],[22,7829,7831,7893],{"className":7830},[92],[22,7832,7834,7848,7856],{"className":7833,"style":1327},[96],[22,7835,7836,7839],{"style":1330},[22,7837],{"className":7838,"style":1172},[104],[22,7840,7842],{"className":7841},[109,1313,1337,112],[22,7843,7845],{"className":7844},[74,112],[22,7846,149],{"className":7847},[74,112],[22,7849,7850,7853],{"style":1346},[22,7851],{"className":7852,"style":1172},[104],[22,7854],{"className":7855,"style":1353},[1244,112],[22,7857,7858,7861],{"style":1356},[22,7859],{"className":7860,"style":1172},[104],[22,7862,7864],{"className":7863},[109,1313,1337,112],[22,7865,7867],{"className":7866},[74,112],[22,7868,7870,7873],{"className":7869},[74,112],[22,7871,2103],{"className":7872,"style":2168},[74,78,112],[22,7874,7876],{"className":7875},[83],[22,7877,7879],{"className":7878},[87],[22,7880,7882],{"className":7881},[92],[22,7883,7885],{"className":7884,"style":1384},[96],[22,7886,7887,7890],{"style":1387},[22,7888],{"className":7889,"style":1391},[104],[22,7891,149],{"className":7892},[74,112],[22,7894,120],{"className":7895},[119],[22,7897,7899],{"className":7898},[92],[22,7900,7902],{"className":7901,"style":1404},[96],[22,7903],{},[22,7905],{"className":7906},[429,1152,109,1313,1314],[15,7908,7909,7910,130,7938,7070],{},"Hence, the joint probability density of ",[22,7911,7913,7926],{"className":7912,"translate":26},[25],[22,7914,7916],{"className":7915},[30],[32,7917,7918],{"xmlns":34},[36,7919,7920,7924],{},[39,7921,7922],{},[45,7923,2103],{},[53,7925,2103],{"encoding":55},[22,7927,7929],{"className":7928,"ariaHidden":61},[60],[22,7930,7932,7935],{"className":7931},[65],[22,7933],{"className":7934,"style":233},[69],[22,7936,2103],{"className":7937,"style":2168},[74,78],[22,7939,7941,7954],{"className":7940,"translate":26},[25],[22,7942,7944],{"className":7943},[30],[32,7945,7946],{"xmlns":34},[36,7947,7948,7952],{},[39,7949,7950],{},[45,7951,6501],{"mathvariant":2262},[53,7953,7022],{"encoding":55},[22,7955,7957],{"className":7956,"ariaHidden":61},[60],[22,7958,7960,7963],{"className":7959},[65],[22,7961],{"className":7962,"style":233},[69],[22,7964,6501],{"className":7965},[74],[22,7967,7969],{"className":7968,"translate":26},[272],[22,7970,7972,8028],{"className":7971,"translate":26},[25],[22,7973,7975],{"className":7974},[30],[32,7976,7977],{"xmlns":34,"display":281},[36,7978,7979,8025],{},[39,7980,7981,7983,7985,7987,7989,7991,7993,7995,8005,8007],{},[45,7982,2775],{},[289,7984,301],{"stretchy":300},[45,7986,2103],{},[289,7988,1062],{"separator":61},[45,7990,6501],{"mathvariant":2262},[289,7992,315],{"stretchy":300},[289,7994,291],{},[1028,7996,7997,7999],{},[49,7998,51],{},[39,8000,8001,8003],{},[49,8002,149],{},[45,8004,306],{},[45,8006,2103],{},[1041,8008,8009,8011],{},[45,8010,1045],{},[39,8012,8013,8015],{},[289,8014,323],{},[1028,8016,8017,8023],{},[1041,8018,8019,8021],{},[45,8020,2103],{},[49,8022,149],{},[49,8024,149],{},[53,8026,8027],{"encoding":55},"P(R, \\Theta) = \\frac{1}{2\\pi} R 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the polar coordinates yield:",[22,8268,8270],{"className":8269,"translate":26},[272],[22,8271,8273,8311],{"className":8272,"translate":26},[25],[22,8274,8276],{"className":8275},[30],[32,8277,8278],{"xmlns":34,"display":281},[36,8279,8280,8308],{},[39,8281,8282,8284,8286,8288,8290,8292,8294,8296,8298,8300,8302,8304,8306],{},[45,8283,2103],{},[2307,8285,2309],{},[45,8287,2312],{},[45,8289,2103],{},[2307,8291,2309],{},[45,8293,2312],{},[45,8295,6501],{"mathvariant":2262},[289,8297,291],{},[45,8299,2312],{},[45,8301,221],{},[2307,8303,2309],{},[45,8305,2312],{},[45,8307,252],{},[53,8309,8310],{"encoding":55},"R \\, dR \\, d\\Theta = dX \\, dY",[22,8312,8314,8350],{"className":8313,"ariaHidden":61},[60],[22,8315,8317,8320,8323,8326,8329,8332,8335,8338,8341,8344,8347],{"className":8316},[65],[22,8318],{"className":8319,"style":2754},[69],[22,8321,2103],{"className":8322,"style":2168},[74,78],[22,8324],{"className":8325,"style":466},[355],[22,8327,2312],{"className":8328},[74,78],[22,8330,2103],{"className":8331,"style":2168},[74,78],[22,8333],{"className":8334,"style":466},[355],[22,8336,2312],{"className":8337},[74,78],[22,8339,6501],{"className":8340},[74],[22,8342],{"className":8343,"style":356},[355],[22,8345,291],{"className":8346},[360],[22,8348],{"className":8349,"style":356},[355],[22,8351,8353,8356,8359,8362,8365,8368],{"className":8352},[65],[22,8354],{"className":8355,"style":2754},[69],[22,8357,2312],{"className":8358},[74,78],[22,8360,221],{"className":8361,"style":237},[74,78],[22,8363],{"className":8364,"style":466},[355],[22,8366,2312],{"className":8367},[74,78],[22,8369,252],{"className":8370,"style":267},[74,78],[15,8372,8373,8374,130,8402,2904],{},"we obtain the joint probability density of 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= \\frac{1}{2\\pi} e^{-\\frac{X^2 + Y^2}{2}} \\\\ = \\left(\\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{X^2}{2}}\\right) \\left(\\frac{1}{\\sqrt{2\\pi}} 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{\n",[22,9529,9531,9534,9537,9540,9542],{"class":9432,"line":9530},6,[22,9532,9533],{"class":9436},"  const",[22,9535,9536],{"class":9450}," u",[22,9538,9539],{"class":9436}," =",[22,9541,9441],{"class":9440},[22,9543,9469],{"class":9444},[22,9545,9547,9549,9552,9554,9556],{"class":9432,"line":9546},7,[22,9548,9533],{"class":9436},[22,9550,9551],{"class":9450}," v",[22,9553,9539],{"class":9436},[22,9555,9441],{"class":9440},[22,9557,9469],{"class":9444},[22,9559,9561,9563,9566,9568,9570,9572,9574,9576,9579,9581,9584,9586,9589,9592,9594,9596,9598,9601,9603,9605,9607,9610],{"class":9432,"line":9560},8,[22,9562,9533],{"class":9436},[22,9564,9565],{"class":9450}," x",[22,9567,9539],{"class":9436},[22,9569,9463],{"class":9444},[22,9571,294],{"class":9440},[22,9573,301],{"class":9444},[22,9575,149],{"class":9450},[22,9577,9578],{"class":9436}," *",[22,9580,9463],{"class":9444},[22,9582,9583],{"class":9450},"PI",[22,9585,9578],{"class":9436},[22,9587,9588],{"class":9444}," u) ",[22,9590,9591],{"class":9436},"*",[22,9593,9463],{"class":9444},[22,9595,433],{"class":9440},[22,9597,301],{"class":9444},[22,9599,9600],{"class":9436},"-",[22,9602,149],{"class":9450},[22,9604,9578],{"class":9436},[22,9606,9463],{"class":9444},[22,9608,9609],{"class":9440},"log",[22,9611,9612],{"class":9444},"(v));\n",[22,9614,9616,9618,9621,9623,9625,9627,9629,9631,9633,9635,9637,9639,9641,9643,9645,9647,9649,9651,9653,9655,9657,9659],{"class":9432,"line":9615},9,[22,9617,9533],{"class":9436},[22,9619,9620],{"class":9450}," y",[22,9622,9539],{"class":9436},[22,9624,9463],{"class":9444},[22,9626,569],{"class":9440},[22,9628,301],{"class":9444},[22,9630,149],{"class":9450},[22,9632,9578],{"class":9436},[22,9634,9463],{"class":9444},[22,9636,9583],{"class":9450},[22,9638,9578],{"class":9436},[22,9640,9588],{"class":9444},[22,9642,9591],{"class":9436},[22,9644,9463],{"class":9444},[22,9646,433],{"class":9440},[22,9648,301],{"class":9444},[22,9650,9600],{"class":9436},[22,9652,149],{"class":9450},[22,9654,9578],{"class":9436},[22,9656,9463],{"class":9444},[22,9658,9609],{"class":9440},[22,9660,9612],{"class":9444},[22,9662,9664,9666,9669,9671,9674,9676,9679,9681,9683,9685],{"class":9432,"line":9663},10,[22,9665,9460],{"class":9436},[22,9667,9668],{"class":9444}," [x ",[22,9670,9591],{"class":9436},[22,9672,9673],{"class":9444}," sigma ",[22,9675,1801],{"class":9436},[22,9677,9678],{"class":9444}," mu, y ",[22,9680,9591],{"class":9436},[22,9682,9673],{"class":9444},[22,9684,1801],{"class":9436},[22,9686,9687],{"class":9444}," mu];\n",[22,9689,9691],{"class":9432,"line":9690},11,[22,9692,9475],{"class":9444},[22,9694,9696],{"class":9432,"line":9695},12,[22,9697,9482],{"emptyLinePlaceholder":9481},[22,9699,9701],{"class":9432,"line":9700},13,[22,9702,9704],{"class":9703},"sAwPA","\u002F\u002F Usage example\n",[22,9706,9708,9711,9713,9716,9718,9721,9724,9726,9728,9730,9732,9734,9736],{"class":9432,"line":9707},14,[22,9709,9710],{"class":9436},"const",[22,9712,9517],{"class":9444},[22,9714,9715],{"class":9450},"value1",[22,9717,9503],{"class":9444},[22,9719,9720],{"class":9450},"value2",[22,9722,9723],{"class":9444},"] ",[22,9725,291],{"class":9436},[22,9727,9490],{"class":9440},[22,9729,301],{"class":9444},[22,9731,2259],{"class":9450},[22,9733,9503],{"class":9444},[22,9735,51],{"class":9450},[22,9737,9738],{"class":9444},");\n",[22,9740,9742,9745,9747],{"class":9432,"line":9741},15,[22,9743,9744],{"class":9444},"console.",[22,9746,9609],{"class":9440},[22,9748,9749],{"class":9444},"(value1, value2);\n",[22,9751,9753],{"class":9432,"line":9752},16,[22,9754,9755],{"class":9703},"\u002F\u002F Outputs two normally distributed random numbers\n",[15,9757,9758,9759,9762,9763,9766],{},"To verify that the ",[9428,9760,9761],{},"boxMuller"," function generates numbers that follow a normal distribution, one can generate a large set of random numbers and plot a histogram to compare with a standard normal distribution. You can use libraries such as ",[9428,9764,9765],{},"plotly"," to visualize the histogram and evaluate the output.",[15,9768,9769,9770,9772,9773,9776,9777,9780],{},"Here is a test program that generates a dataset using the ",[9428,9771,9761],{}," function and performs basic statistical tests, along with plotting the histogram. Assuming you are running in a Node.js environment, you can use ",[9428,9774,9775],{},"plotly-nodejs"," or directly use ",[9428,9778,9779],{},"plotly.js"," in the browser to generate the graph.",[15,9782,9783],{},"First, install the required dependencies:",[9421,9785,9789],{"className":9786,"code":9787,"language":9788,"meta":9426,"style":9426},"language-bash shiki shiki-themes github-dark","npm i plotly.js-dist\n","bash",[9428,9790,9791],{"__ignoreMap":9426},[22,9792,9793,9796,9800],{"class":9432,"line":9433},[22,9794,9795],{"class":9440},"npm",[22,9797,9799],{"class":9798},"sU2Wk"," i",[22,9801,9802],{"class":9798}," plotly.js-dist\n",[15,9804,9805],{},"Then, execute the following code:",[9421,9807,9809],{"className":9423,"code":9808,"language":9425,"meta":9426,"style":9426},"import * as fs from \"fs\";\nimport * as plotly from \"plotly.js-dist\";\n\n\u002F\u002F Generate random numbers following a normal distribution using Box-Muller\nfunction randInterval(): number {\n  return Math.random();\n}\n\nfunction boxMuller(mu: number, sigma: number): [number, number] {\n  const u = randInterval();\n  const v = randInterval();\n  const x = Math.cos(2 * Math.PI * u) * Math.sqrt(-2 * Math.log(v));\n  const y = Math.sin(2 * Math.PI * u) * Math.sqrt(-2 * Math.log(v));\n  return [x * sigma + mu, y * sigma + mu];\n}\n\n\u002F\u002F Generate n normally distributed random numbers\nfunction generateNormalDistribution(\n  mu: number,\n  sigma: number,\n  n: number\n): number[] {\n  const values: number[] = [];\n  for (let i = 0; i \u003C n \u002F 2; i++) {\n    const [value1, value2] = boxMuller(mu, sigma);\n    values.push(value1, value2);\n  }\n  return values;\n}\n\n\u002F\u002F Generate the data\nconst mu = 0;\nconst sigma = 1;\nconst sampleSize = 10000; \u002F\u002F Sample size\nconst values = generateNormalDistribution(mu, sigma, sampleSize);\n\n\u002F\u002F Calculate the mean and variance of the generated random numbers\nconst mean = values.reduce((acc, val) => acc + val, 0) \u002F values.length;\nconst variance =\n  values.reduce((acc, val) => acc + (val - mean) ** 2, 0) \u002F values.length;\n\nconsole.log(`Mean: ${mean}`);\nconsole.log(`Variance: ${variance}`);\n\n\u002F\u002F Plot the histogram\nconst trace = {\n  x: values,\n  type: \"histogram\",\n  xbins: {\n    size: 0.1, \u002F\u002F Bin size for the histogram\n  },\n  marker: {\n    color: \"blue\",\n  },\n  opacity: 0.7,\n  name: \"Box-Muller Distribution\",\n};\n\nconst data = [trace];\n\nconst layout = {\n  title: \"Generated Normal Distribution\",\n  xaxis: { title: \"Value\" },\n  yaxis: { title: \"Frequency\" },\n  bargap: 0.05,\n};\n\nconst graphOptions = {\n  filename: \"box-muller-distribution\",\n  fileopt: \"overwrite\",\n};\nfs.writeFileSync(\"box-muller.html\", plotly.plot(data, layout, graphOptions));\n\nconsole.log(\"Box-Muller distribution data saved as box-muller.html.\");\n",[9428,9810,9811,9833,9851,9855,9860,9874,9884,9888,9892,9928,9940,9952,9998,10044,10066,10070,10074,10080,10091,10104,10116,10127,10139,10159,10203,10226,10237,10243,10251,10256,10261,10267,10281,10296,10315,10329,10334,10340,10394,10405,10456,10461,10481,10500,10505,10511,10523,10529,10540,10546,10560,10566,10572,10583,10588,10599,10610,10616,10621,10634,10639,10651,10662,10674,10685,10696,10701,10706,10718,10729,10740,10745,10768,10773],{"__ignoreMap":9426},[22,9812,9813,9816,9818,9821,9824,9827,9830],{"class":9432,"line":9433},[22,9814,9815],{"class":9436},"import",[22,9817,9578],{"class":9450},[22,9819,9820],{"class":9436}," as",[22,9822,9823],{"class":9444}," fs ",[22,9825,9826],{"class":9436},"from",[22,9828,9829],{"class":9798}," \"fs\"",[22,9831,9832],{"class":9444},";\n",[22,9834,9835,9837,9839,9841,9844,9846,9849],{"class":9432,"line":9457},[22,9836,9815],{"class":9436},[22,9838,9578],{"class":9450},[22,9840,9820],{"class":9436},[22,9842,9843],{"class":9444}," plotly ",[22,9845,9826],{"class":9436},[22,9847,9848],{"class":9798}," \"plotly.js-dist\"",[22,9850,9832],{"class":9444},[22,9852,9853],{"class":9432,"line":9472},[22,9854,9482],{"emptyLinePlaceholder":9481},[22,9856,9857],{"class":9432,"line":9478},[22,9858,9859],{"class":9703},"\u002F\u002F Generate random numbers following a normal distribution using Box-Muller\n",[22,9861,9862,9864,9866,9868,9870,9872],{"class":9432,"line":9485},[22,9863,9437],{"class":9436},[22,9865,9441],{"class":9440},[22,9867,9445],{"class":9444},[22,9869,2904],{"class":9436},[22,9871,9451],{"class":9450},[22,9873,9454],{"class":9444},[22,9875,9876,9878,9880,9882],{"class":9432,"line":9530},[22,9877,9460],{"class":9436},[22,9879,9463],{"class":9444},[22,9881,9466],{"class":9440},[22,9883,9469],{"class":9444},[22,9885,9886],{"class":9432,"line":9546},[22,9887,9475],{"class":9444},[22,9889,9890],{"class":9432,"line":9560},[22,9891,9482],{"emptyLinePlaceholder":9481},[22,9893,9894,9896,9898,9900,9902,9904,9906,9908,9910,9912,9914,9916,9918,9920,9922,9924,9926],{"class":9432,"line":9615},[22,9895,9437],{"class":9436},[22,9897,9490],{"class":9440},[22,9899,301],{"class":9444},[22,9901,9496],{"class":9495},[22,9903,2904],{"class":9436},[22,9905,9451],{"class":9450},[22,9907,9503],{"class":9444},[22,9909,9506],{"class":9495},[22,9911,2904],{"class":9436},[22,9913,9451],{"class":9450},[22,9915,315],{"class":9444},[22,9917,2904],{"class":9436},[22,9919,9517],{"class":9444},[22,9921,9520],{"class":9450},[22,9923,9503],{"class":9444},[22,9925,9520],{"class":9450},[22,9927,9527],{"class":9444},[22,9929,9930,9932,9934,9936,9938],{"class":9432,"line":9663},[22,9931,9533],{"class":9436},[22,9933,9536],{"class":9450},[22,9935,9539],{"class":9436},[22,9937,9441],{"class":9440},[22,9939,9469],{"class":9444},[22,9941,9942,9944,9946,9948,9950],{"class":9432,"line":9690},[22,9943,9533],{"class":9436},[22,9945,9551],{"class":9450},[22,9947,9539],{"class":9436},[22,9949,9441],{"class":9440},[22,9951,9469],{"class":9444},[22,9953,9954,9956,9958,9960,9962,9964,9966,9968,9970,9972,9974,9976,9978,9980,9982,9984,9986,9988,9990,9992,9994,9996],{"class":9432,"line":9695},[22,9955,9533],{"class":9436},[22,9957,9565],{"class":9450},[22,9959,9539],{"class":9436},[22,9961,9463],{"class":9444},[22,9963,294],{"class":9440},[22,9965,301],{"class":9444},[22,9967,149],{"class":9450},[22,9969,9578],{"class":9436},[22,9971,9463],{"class":9444},[22,9973,9583],{"class":9450},[22,9975,9578],{"class":9436},[22,9977,9588],{"class":9444},[22,9979,9591],{"class":9436},[22,9981,9463],{"class":9444},[22,9983,433],{"class":9440},[22,9985,301],{"class":9444},[22,9987,9600],{"class":9436},[22,9989,149],{"class":9450},[22,9991,9578],{"class":9436},[22,9993,9463],{"class":9444},[22,9995,9609],{"class":9440},[22,9997,9612],{"class":9444},[22,9999,10000,10002,10004,10006,10008,10010,10012,10014,10016,10018,10020,10022,10024,10026,10028,10030,10032,10034,10036,10038,10040,10042],{"class":9432,"line":9700},[22,10001,9533],{"class":9436},[22,10003,9620],{"class":9450},[22,10005,9539],{"class":9436},[22,10007,9463],{"class":9444},[22,10009,569],{"class":9440},[22,10011,301],{"class":9444},[22,10013,149],{"class":9450},[22,10015,9578],{"class":9436},[22,10017,9463],{"class":9444},[22,10019,9583],{"class":9450},[22,10021,9578],{"class":9436},[22,10023,9588],{"class":9444},[22,10025,9591],{"class":9436},[22,10027,9463],{"class":9444},[22,10029,433],{"class":9440},[22,10031,301],{"class":9444},[22,10033,9600],{"class":9436},[22,10035,149],{"class":9450},[22,10037,9578],{"class":9436},[22,10039,9463],{"class":9444},[22,10041,9609],{"class":9440},[22,10043,9612],{"class":9444},[22,10045,10046,10048,10050,10052,10054,10056,10058,10060,10062,10064],{"class":9432,"line":9707},[22,10047,9460],{"class":9436},[22,10049,9668],{"class":9444},[22,10051,9591],{"class":9436},[22,10053,9673],{"class":9444},[22,10055,1801],{"class":9436},[22,10057,9678],{"class":9444},[22,10059,9591],{"class":9436},[22,10061,9673],{"class":9444},[22,10063,1801],{"class":9436},[22,10065,9687],{"class":9444},[22,10067,10068],{"class":9432,"line":9741},[22,10069,9475],{"class":9444},[22,10071,10072],{"class":9432,"line":9752},[22,10073,9482],{"emptyLinePlaceholder":9481},[22,10075,10077],{"class":9432,"line":10076},17,[22,10078,10079],{"class":9703},"\u002F\u002F Generate n normally distributed random numbers\n",[22,10081,10083,10085,10088],{"class":9432,"line":10082},18,[22,10084,9437],{"class":9436},[22,10086,10087],{"class":9440}," generateNormalDistribution",[22,10089,10090],{"class":9444},"(\n",[22,10092,10094,10097,10099,10101],{"class":9432,"line":10093},19,[22,10095,10096],{"class":9495},"  mu",[22,10098,2904],{"class":9436},[22,10100,9451],{"class":9450},[22,10102,10103],{"class":9444},",\n",[22,10105,10107,10110,10112,10114],{"class":9432,"line":10106},20,[22,10108,10109],{"class":9495},"  sigma",[22,10111,2904],{"class":9436},[22,10113,9451],{"class":9450},[22,10115,10103],{"class":9444},[22,10117,10119,10122,10124],{"class":9432,"line":10118},21,[22,10120,10121],{"class":9495},"  n",[22,10123,2904],{"class":9436},[22,10125,10126],{"class":9450}," number\n",[22,10128,10130,10132,10134,10136],{"class":9432,"line":10129},22,[22,10131,315],{"class":9444},[22,10133,2904],{"class":9436},[22,10135,9451],{"class":9450},[22,10137,10138],{"class":9444},"[] {\n",[22,10140,10142,10144,10147,10149,10151,10154,10156],{"class":9432,"line":10141},23,[22,10143,9533],{"class":9436},[22,10145,10146],{"class":9450}," values",[22,10148,2904],{"class":9436},[22,10150,9451],{"class":9450},[22,10152,10153],{"class":9444},"[] ",[22,10155,291],{"class":9436},[22,10157,10158],{"class":9444}," [];\n",[22,10160,10162,10165,10168,10171,10174,10176,10179,10182,10185,10188,10191,10194,10197,10200],{"class":9432,"line":10161},24,[22,10163,10164],{"class":9436},"  for",[22,10166,10167],{"class":9444}," (",[22,10169,10170],{"class":9436},"let",[22,10172,10173],{"class":9444}," i ",[22,10175,291],{"class":9436},[22,10177,10178],{"class":9450}," 0",[22,10180,10181],{"class":9444},"; i ",[22,10183,10184],{"class":9436},"\u003C",[22,10186,10187],{"class":9444}," n ",[22,10189,10190],{"class":9436},"\u002F",[22,10192,10193],{"class":9450}," 2",[22,10195,10196],{"class":9444},"; i",[22,10198,10199],{"class":9436},"++",[22,10201,10202],{"class":9444},") {\n",[22,10204,10206,10209,10211,10213,10215,10217,10219,10221,10223],{"class":9432,"line":10205},25,[22,10207,10208],{"class":9436},"    const",[22,10210,9517],{"class":9444},[22,10212,9715],{"class":9450},[22,10214,9503],{"class":9444},[22,10216,9720],{"class":9450},[22,10218,9723],{"class":9444},[22,10220,291],{"class":9436},[22,10222,9490],{"class":9440},[22,10224,10225],{"class":9444},"(mu, sigma);\n",[22,10227,10229,10232,10235],{"class":9432,"line":10228},26,[22,10230,10231],{"class":9444},"    values.",[22,10233,10234],{"class":9440},"push",[22,10236,9749],{"class":9444},[22,10238,10240],{"class":9432,"line":10239},27,[22,10241,10242],{"class":9444},"  }\n",[22,10244,10246,10248],{"class":9432,"line":10245},28,[22,10247,9460],{"class":9436},[22,10249,10250],{"class":9444}," values;\n",[22,10252,10254],{"class":9432,"line":10253},29,[22,10255,9475],{"class":9444},[22,10257,10259],{"class":9432,"line":10258},30,[22,10260,9482],{"emptyLinePlaceholder":9481},[22,10262,10264],{"class":9432,"line":10263},31,[22,10265,10266],{"class":9703},"\u002F\u002F Generate the data\n",[22,10268,10270,10272,10275,10277,10279],{"class":9432,"line":10269},32,[22,10271,9710],{"class":9436},[22,10273,10274],{"class":9450}," mu",[22,10276,9539],{"class":9436},[22,10278,10178],{"class":9450},[22,10280,9832],{"class":9444},[22,10282,10284,10286,10289,10291,10294],{"class":9432,"line":10283},33,[22,10285,9710],{"class":9436},[22,10287,10288],{"class":9450}," sigma",[22,10290,9539],{"class":9436},[22,10292,10293],{"class":9450}," 1",[22,10295,9832],{"class":9444},[22,10297,10299,10301,10304,10306,10309,10312],{"class":9432,"line":10298},34,[22,10300,9710],{"class":9436},[22,10302,10303],{"class":9450}," sampleSize",[22,10305,9539],{"class":9436},[22,10307,10308],{"class":9450}," 10000",[22,10310,10311],{"class":9444},"; ",[22,10313,10314],{"class":9703},"\u002F\u002F Sample size\n",[22,10316,10318,10320,10322,10324,10326],{"class":9432,"line":10317},35,[22,10319,9710],{"class":9436},[22,10321,10146],{"class":9450},[22,10323,9539],{"class":9436},[22,10325,10087],{"class":9440},[22,10327,10328],{"class":9444},"(mu, sigma, sampleSize);\n",[22,10330,10332],{"class":9432,"line":10331},36,[22,10333,9482],{"emptyLinePlaceholder":9481},[22,10335,10337],{"class":9432,"line":10336},37,[22,10338,10339],{"class":9703},"\u002F\u002F Calculate the mean and variance of the generated random numbers\n",[22,10341,10343,10345,10348,10350,10353,10356,10359,10362,10364,10367,10370,10373,10376,10378,10381,10383,10385,10387,10389,10392],{"class":9432,"line":10342},38,[22,10344,9710],{"class":9436},[22,10346,10347],{"class":9450}," mean",[22,10349,9539],{"class":9436},[22,10351,10352],{"class":9444}," 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