[{"data":1,"prerenderedAt":3592},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fnormalverteilung":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fnormalverteilung":3579},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":556,"category":441,"checks":3514,"conditions":3536,"conditionsHtml":3537,"deprecated":3538,"description":3508,"extension":3539,"figure":3538,"figureCaption":3538,"figureHtml":3538,"formulaId":3540,"latex":3541,"latexHtml":3542,"level":3543,"meta":3544,"navigation":3545,"path":3546,"redirectFrom":3547,"related":3548,"seo":3553,"stem":3554,"subject":587,"summary":3555,"tags":3556,"variables":3561,"__hash__":3578},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fstochastics\u002Fnormalverteilung.md","Normalverteilung",[550,551,552,553,554,555],"Gaußsche Glockenkurve","Glockenkurve Formel","Dichtefunktion Normalverteilung","Gauß-Verteilung","phi von x","N(μ, σ²)",{"type":557,"value":558,"toc":3507},"minimark",[559,564,658,773,1202,1213,1505,1613,1617,1725,2011,2119,2690,2872,2876],[560,561,563],"h2",{"id":562},"erklärung","Erklärung",[565,566,567,568,572,573,576,577,624,625,657],"p",{},"Viele Größen streuen zufällig um einen Mittelwert, z.B. Körpergrößen, Messfehler oder Füllmengen. Ihre Verteilung hat oft die Form einer ",[569,570,571],"strong",{},"Glockenkurve",". Eine Zufallsgröße mit genau dieser Form heißt ",[569,574,575],{},"normalverteilt"," mit den Parametern ",[578,579,582,605],"span",{"className":580},[581],"katex",[578,583,586],{"className":584},[585],"katex-mathml",[587,588,590],"math",{"xmlns":589},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[591,592,593,600],"semantics",{},[594,595,596],"mrow",{},[597,598,599],"mi",{},"μ",[601,602,604],"annotation",{"encoding":603},"application\u002Fx-tex","\\mu",[578,606,610],{"className":607,"ariaHidden":609},[608],"katex-html","true",[578,611,614,619],{"className":612},[613],"base",[578,615],{"className":616,"style":618},[617],"strut","height:0.625em;vertical-align:-0.1944em;",[578,620,599],{"className":621},[622,623],"mord","mathnormal"," und ",[578,626,628,643],{"className":627},[581],[578,629,631],{"className":630},[585],[587,632,633],{"xmlns":589},[591,634,635,640],{},[594,636,637],{},[597,638,639],{},"σ",[601,641,642],{"encoding":603},"\\sigma",[578,644,646],{"className":645,"ariaHidden":609},[608],[578,647,649,653],{"className":648},[613],[578,650],{"className":651,"style":652},[617],"height:0.4306em;",[578,654,639],{"className":655,"style":656},[622,623],"margin-right:0.0359em;",".",[565,659,660,661,664,665,772],{},"Die Formel beschreibt die ",[569,662,663],{},"Dichtefunktion"," ",[578,666,668,696],{"className":667},[581],[578,669,671],{"className":670},[585],[587,672,673],{"xmlns":589},[591,674,675,693],{},[594,676,677],{},[678,679,680,683],"msub",{},[597,681,682],{},"φ",[594,684,685,687,691],{},[597,686,599],{},[688,689,690],"mo",{"separator":609},",",[597,692,639],{},[601,694,695],{"encoding":603},"\\varphi_{\\mu,\\sigma}",[578,697,699],{"className":698,"ariaHidden":609},[608],[578,700,702,706],{"className":701},[613],[578,703],{"className":704,"style":705},[617],"height:0.7167em;vertical-align:-0.2861em;",[578,707,709,712],{"className":708},[622],[578,710,682],{"className":711},[622,623],[578,713,716],{"className":714},[715],"msupsub",[578,717,721,763],{"className":718},[719,720],"vlist-t","vlist-t2",[578,722,725,758],{"className":723},[724],"vlist-r",[578,726,730],{"className":727,"style":729},[728],"vlist","height:0.1514em;",[578,731,733,738],{"style":732},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[578,734],{"className":735,"style":737},[736],"pstrut","height:2.7em;",[578,739,745],{"className":740},[741,742,743,744],"sizing","reset-size6","size3","mtight",[578,746,748,751,755],{"className":747},[622,744],[578,749,599],{"className":750},[622,623,744],[578,752,690],{"className":753},[754,744],"mpunct",[578,756,639],{"className":757,"style":656},[622,623,744],[578,759,762],{"className":760},[761],"vlist-s","​",[578,764,766],{"className":765},[724],[578,767,770],{"className":768,"style":769},[728],"height:0.2861em;",[578,771],{},":",[774,775,776,1030,1170],"ul",{},[777,778,779,780,836,837,657],"li",{},"Sie ist symmetrisch zur Geraden ",[578,781,783,803],{"className":782},[581],[578,784,786],{"className":785},[585],[587,787,788],{"xmlns":589},[591,789,790,800],{},[594,791,792,795,798],{},[597,793,794],{},"x",[688,796,797],{},"=",[597,799,599],{},[601,801,802],{"encoding":603},"x = \\mu",[578,804,806,827],{"className":805,"ariaHidden":609},[608],[578,807,809,812,815,820,824],{"className":808},[613],[578,810],{"className":811,"style":652},[617],[578,813,794],{"className":814},[622,623],[578,816],{"className":817,"style":819},[818],"mspace","margin-right:0.2778em;",[578,821,797],{"className":822},[823],"mrel",[578,825],{"className":826,"style":819},[818],[578,828,830,833],{"className":829},[613],[578,831],{"className":832,"style":618},[617],[578,834,599],{"className":835},[622,623]," und hat dort ihren Hochpunkt mit dem Wert ",[578,838,840,874],{"className":839},[581],[578,841,843],{"className":842},[585],[587,844,845],{"xmlns":589},[591,846,847,871],{},[594,848,849],{},[850,851,852,856],"mfrac",{},[853,854,855],"mn",{},"1",[594,857,858,860],{},[597,859,639],{},[861,862,863],"msqrt",{},[594,864,865,868],{},[853,866,867],{},"2",[597,869,870],{},"π",[601,872,873],{"encoding":603},"\\frac{1}{\\sigma\\sqrt{2\\pi}}",[578,875,877],{"className":876,"ariaHidden":609},[608],[578,878,880,884],{"className":879},[613],[578,881],{"className":882,"style":883},[617],"height:1.3831em;vertical-align:-0.538em;",[578,885,887,892,1026],{"className":886},[622],[578,888],{"className":889},[890,891],"mopen","nulldelimiter",[578,893,895],{"className":894},[850],[578,896,898,1017],{"className":897},[719,720],[578,899,901,1014],{"className":900},[724],[578,902,905,988,999],{"className":903,"style":904},[728],"height:0.8451em;",[578,906,908,912],{"style":907},"top:-2.551em;",[578,909],{"className":910,"style":911},[736],"height:3em;",[578,913,915],{"className":914},[741,742,743,744],[578,916,918,921],{"className":917},[622,744],[578,919,639],{"className":920,"style":656},[622,623,744],[578,922,925],{"className":923},[622,924,744],"sqrt",[578,926,928,979],{"className":927},[719,720],[578,929,931,976],{"className":930},[724],[578,932,935,953],{"className":933,"style":934},[728],"height:0.9128em;",[578,936,940,943],{"className":937,"style":939},[938],"svg-align","top:-3em;",[578,941],{"className":942,"style":911},[736],[578,944,947,950],{"className":945,"style":946},[622,744],"padding-left:0.833em;",[578,948,867],{"className":949},[622,744],[578,951,870],{"className":952,"style":656},[622,623,744],[578,954,956,959],{"style":955},"top:-2.8728em;",[578,957],{"className":958,"style":911},[736],[578,960,964],{"className":961,"style":963},[962,744],"hide-tail","min-width:0.853em;height:1.08em;",[965,966,972],"svg",{"xmlns":967,"width":968,"height":969,"viewBox":970,"preserveAspectRatio":971},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[973,974],"path",{"d":975},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[578,977,762],{"className":978},[761],[578,980,982],{"className":981},[724],[578,983,986],{"className":984,"style":985},[728],"height:0.1272em;",[578,987],{},[578,989,991,994],{"style":990},"top:-3.23em;",[578,992],{"className":993,"style":911},[736],[578,995],{"className":996,"style":998},[997],"frac-line","border-bottom-width:0.04em;",[578,1000,1002,1005],{"style":1001},"top:-3.394em;",[578,1003],{"className":1004,"style":911},[736],[578,1006,1008],{"className":1007},[741,742,743,744],[578,1009,1011],{"className":1010},[622,744],[578,1012,855],{"className":1013},[622,744],[578,1015,762],{"className":1016},[761],[578,1018,1020],{"className":1019},[724],[578,1021,1024],{"className":1022,"style":1023},[728],"height:0.538em;",[578,1025],{},[578,1027],{"className":1028},[1029,891],"mclose",[777,1031,1032,1033,624,1088,1140,1141,1169],{},"Ihre Wendestellen liegen bei ",[578,1034,1036,1055],{"className":1035},[581],[578,1037,1039],{"className":1038},[585],[587,1040,1041],{"xmlns":589},[591,1042,1043,1052],{},[594,1044,1045,1047,1050],{},[597,1046,599],{},[688,1048,1049],{},"−",[597,1051,639],{},[601,1053,1054],{"encoding":603},"\\mu - \\sigma",[578,1056,1058,1079],{"className":1057,"ariaHidden":609},[608],[578,1059,1061,1065,1068,1072,1076],{"className":1060},[613],[578,1062],{"className":1063,"style":1064},[617],"height:0.7778em;vertical-align:-0.1944em;",[578,1066,599],{"className":1067},[622,623],[578,1069],{"className":1070,"style":1071},[818],"margin-right:0.2222em;",[578,1073,1049],{"className":1074},[1075],"mbin",[578,1077],{"className":1078,"style":1071},[818],[578,1080,1082,1085],{"className":1081},[613],[578,1083],{"className":1084,"style":652},[617],[578,1086,639],{"className":1087,"style":656},[622,623],[578,1089,1091,1110],{"className":1090},[581],[578,1092,1094],{"className":1093},[585],[587,1095,1096],{"xmlns":589},[591,1097,1098,1107],{},[594,1099,1100,1102,1105],{},[597,1101,599],{},[688,1103,1104],{},"+",[597,1106,639],{},[601,1108,1109],{"encoding":603},"\\mu + \\sigma",[578,1111,1113,1131],{"className":1112,"ariaHidden":609},[608],[578,1114,1116,1119,1122,1125,1128],{"className":1115},[613],[578,1117],{"className":1118,"style":1064},[617],[578,1120,599],{"className":1121},[622,623],[578,1123],{"className":1124,"style":1071},[818],[578,1126,1104],{"className":1127},[1075],[578,1129],{"className":1130,"style":1071},[818],[578,1132,1134,1137],{"className":1133},[613],[578,1135],{"className":1136,"style":652},[617],[578,1138,639],{"className":1139,"style":656},[622,623],". Je größer ",[578,1142,1144,1157],{"className":1143},[581],[578,1145,1147],{"className":1146},[585],[587,1148,1149],{"xmlns":589},[591,1150,1151,1155],{},[594,1152,1153],{},[597,1154,639],{},[601,1156,642],{"encoding":603},[578,1158,1160],{"className":1159,"ariaHidden":609},[608],[578,1161,1163,1166],{"className":1162},[613],[578,1164],{"className":1165,"style":652},[617],[578,1167,639],{"className":1168,"style":656},[622,623],", desto flacher und breiter ist die Kurve.",[777,1171,1172,1173,657],{},"Die gesamte Fläche unter der Kurve ist ",[578,1174,1176,1189],{"className":1175},[581],[578,1177,1179],{"className":1178},[585],[587,1180,1181],{"xmlns":589},[591,1182,1183,1187],{},[594,1184,1185],{},[853,1186,855],{},[601,1188,855],{"encoding":603},[578,1190,1192],{"className":1191,"ariaHidden":609},[608],[578,1193,1195,1199],{"className":1194},[613],[578,1196],{"className":1197,"style":1198},[617],"height:0.6444em;",[578,1200,855],{"className":1201},[622],[565,1203,1204,1205,1208,1209,1212],{},"Eine normalverteilte Zufallsgröße ist ",[569,1206,1207],{},"stetig",". Wahrscheinlichkeiten sind deshalb ",[569,1210,1211],{},"Flächen"," unter der Kurve:",[578,1214,1217],{"className":1215},[1216],"katex-display",[578,1218,1220,1297],{"className":1219},[581],[578,1221,1223],{"className":1222},[585],[587,1224,1226],{"xmlns":589,"display":1225},"block",[591,1227,1228,1294],{},[594,1229,1230,1233,1237,1240,1243,1246,1248,1251,1254,1256,1266,1278,1280,1282,1284,1288,1292],{},[597,1231,1232],{},"P",[688,1234,1236],{"stretchy":1235},"false","(",[597,1238,1239],{},"a",[688,1241,1242],{},"≤",[597,1244,1245],{},"X",[688,1247,1242],{},[597,1249,1250],{},"b",[688,1252,1253],{"stretchy":1235},")",[688,1255,797],{},[1257,1258,1259,1262,1264],"msubsup",{},[688,1260,1261],{},"∫",[597,1263,1239],{},[597,1265,1250],{},[678,1267,1268,1270],{},[597,1269,682],{},[594,1271,1272,1274,1276],{},[597,1273,599],{},[688,1275,690],{"separator":609},[597,1277,639],{},[688,1279,1236],{"stretchy":1235},[597,1281,794],{},[688,1283,1253],{"stretchy":1235},[1285,1286,1287],"mtext",{}," ",[597,1289,1291],{"mathvariant":1290},"normal","d",[597,1293,794],{},[601,1295,1296],{"encoding":603},"P(a \\leq X \\leq b) = \\int_a^b \\varphi_{\\mu,\\sigma}(x)\\,\\mathrm{d}x",[578,1298,1300,1326,1346,1367],{"className":1299,"ariaHidden":609},[608],[578,1301,1303,1307,1311,1314,1317,1320,1323],{"className":1302},[613],[578,1304],{"className":1305,"style":1306},[617],"height:1em;vertical-align:-0.25em;",[578,1308,1232],{"className":1309,"style":1310},[622,623],"margin-right:0.1389em;",[578,1312,1236],{"className":1313},[890],[578,1315,1239],{"className":1316},[622,623],[578,1318],{"className":1319,"style":819},[818],[578,1321,1242],{"className":1322},[823],[578,1324],{"className":1325,"style":819},[818],[578,1327,1329,1333,1337,1340,1343],{"className":1328},[613],[578,1330],{"className":1331,"style":1332},[617],"height:0.8193em;vertical-align:-0.136em;",[578,1334,1245],{"className":1335,"style":1336},[622,623],"margin-right:0.0785em;",[578,1338],{"className":1339,"style":819},[818],[578,1341,1242],{"className":1342},[823],[578,1344],{"className":1345,"style":819},[818],[578,1347,1349,1352,1355,1358,1361,1364],{"className":1348},[613],[578,1350],{"className":1351,"style":1306},[617],[578,1353,1250],{"className":1354},[622,623],[578,1356,1253],{"className":1357},[1029],[578,1359],{"className":1360,"style":819},[818],[578,1362,797],{"className":1363},[823],[578,1365],{"className":1366,"style":819},[818],[578,1368,1370,1374,1433,1437,1486,1489,1492,1495,1498,1502],{"className":1369},[613],[578,1371],{"className":1372,"style":1373},[617],"height:2.511em;vertical-align:-0.9119em;",[578,1375,1378,1384],{"className":1376},[1377],"mop",[578,1379,1261],{"className":1380,"style":1383},[1377,1381,1382],"op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[578,1385,1387],{"className":1386},[715],[578,1388,1390,1424],{"className":1389},[719,720],[578,1391,1393,1421],{"className":1392},[724],[578,1394,1397,1409],{"className":1395,"style":1396},[728],"height:1.599em;",[578,1398,1400,1403],{"style":1399},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[578,1401],{"className":1402,"style":737},[736],[578,1404,1406],{"className":1405},[741,742,743,744],[578,1407,1239],{"className":1408},[622,623,744],[578,1410,1412,1415],{"style":1411},"top:-3.8129em;margin-right:0.05em;",[578,1413],{"className":1414,"style":737},[736],[578,1416,1418],{"className":1417},[741,742,743,744],[578,1419,1250],{"className":1420},[622,623,744],[578,1422,762],{"className":1423},[761],[578,1425,1427],{"className":1426},[724],[578,1428,1431],{"className":1429,"style":1430},[728],"height:0.9119em;",[578,1432],{},[578,1434],{"className":1435,"style":1436},[818],"margin-right:0.1667em;",[578,1438,1440,1443],{"className":1439},[622],[578,1441,682],{"className":1442},[622,623],[578,1444,1446],{"className":1445},[715],[578,1447,1449,1478],{"className":1448},[719,720],[578,1450,1452,1475],{"className":1451},[724],[578,1453,1455],{"className":1454,"style":729},[728],[578,1456,1457,1460],{"style":732},[578,1458],{"className":1459,"style":737},[736],[578,1461,1463],{"className":1462},[741,742,743,744],[578,1464,1466,1469,1472],{"className":1465},[622,744],[578,1467,599],{"className":1468},[622,623,744],[578,1470,690],{"className":1471},[754,744],[578,1473,639],{"className":1474,"style":656},[622,623,744],[578,1476,762],{"className":1477},[761],[578,1479,1481],{"className":1480},[724],[578,1482,1484],{"className":1483,"style":769},[728],[578,1485],{},[578,1487,1236],{"className":1488},[890],[578,1490,794],{"className":1491},[622,623],[578,1493,1253],{"className":1494},[1029],[578,1496],{"className":1497,"style":1436},[818],[578,1499,1291],{"className":1500},[622,1501],"mathrm",[578,1503,794],{"className":1504},[622,623],[565,1506,1507,1508,1560,1561,1612],{},"Dieses Integral hat keine elementare Stammfunktion. Man bestimmt es mit dem Taschenrechner (Verteilungsfunktion) oder über Standardisierung und eine Tabelle der Standardnormalverteilung (",[578,1509,1511,1530],{"className":1510},[581],[578,1512,1514],{"className":1513},[585],[587,1515,1516],{"xmlns":589},[591,1517,1518,1527],{},[594,1519,1520,1522,1524],{},[597,1521,599],{},[688,1523,797],{},[853,1525,1526],{},"0",[601,1528,1529],{"encoding":603},"\\mu = 0",[578,1531,1533,1551],{"className":1532,"ariaHidden":609},[608],[578,1534,1536,1539,1542,1545,1548],{"className":1535},[613],[578,1537],{"className":1538,"style":618},[617],[578,1540,599],{"className":1541},[622,623],[578,1543],{"className":1544,"style":819},[818],[578,1546,797],{"className":1547},[823],[578,1549],{"className":1550,"style":819},[818],[578,1552,1554,1557],{"className":1553},[613],[578,1555],{"className":1556,"style":1198},[617],[578,1558,1526],{"className":1559},[622],", ",[578,1562,1564,1582],{"className":1563},[581],[578,1565,1567],{"className":1566},[585],[587,1568,1569],{"xmlns":589},[591,1570,1571,1579],{},[594,1572,1573,1575,1577],{},[597,1574,639],{},[688,1576,797],{},[853,1578,855],{},[601,1580,1581],{"encoding":603},"\\sigma = 1",[578,1583,1585,1603],{"className":1584,"ariaHidden":609},[608],[578,1586,1588,1591,1594,1597,1600],{"className":1587},[613],[578,1589],{"className":1590,"style":652},[617],[578,1592,639],{"className":1593,"style":656},[622,623],[578,1595],{"className":1596,"style":819},[818],[578,1598,797],{"className":1599},[823],[578,1601],{"className":1602,"style":819},[818],[578,1604,1606,1609],{"className":1605},[613],[578,1607],{"className":1608,"style":1198},[617],[578,1610,855],{"className":1611},[622],").",[560,1614,1616],{"id":1615},"beispiele","Beispiele",[565,1618,1619,1620,1623,1624,1560,1674,1724],{},"Für die ",[569,1621,1622],{},"Standardnormalverteilung"," (",[578,1625,1627,1644],{"className":1626},[581],[578,1628,1630],{"className":1629},[585],[587,1631,1632],{"xmlns":589},[591,1633,1634,1642],{},[594,1635,1636,1638,1640],{},[597,1637,599],{},[688,1639,797],{},[853,1641,1526],{},[601,1643,1529],{"encoding":603},[578,1645,1647,1665],{"className":1646,"ariaHidden":609},[608],[578,1648,1650,1653,1656,1659,1662],{"className":1649},[613],[578,1651],{"className":1652,"style":618},[617],[578,1654,599],{"className":1655},[622,623],[578,1657],{"className":1658,"style":819},[818],[578,1660,797],{"className":1661},[823],[578,1663],{"className":1664,"style":819},[818],[578,1666,1668,1671],{"className":1667},[613],[578,1669],{"className":1670,"style":1198},[617],[578,1672,1526],{"className":1673},[622],[578,1675,1677,1694],{"className":1676},[581],[578,1678,1680],{"className":1679},[585],[587,1681,1682],{"xmlns":589},[591,1683,1684,1692],{},[594,1685,1686,1688,1690],{},[597,1687,639],{},[688,1689,797],{},[853,1691,855],{},[601,1693,1581],{"encoding":603},[578,1695,1697,1715],{"className":1696,"ariaHidden":609},[608],[578,1698,1700,1703,1706,1709,1712],{"className":1699},[613],[578,1701],{"className":1702,"style":652},[617],[578,1704,639],{"className":1705,"style":656},[622,623],[578,1707],{"className":1708,"style":819},[818],[578,1710,797],{"className":1711},[823],[578,1713],{"className":1714,"style":819},[818],[578,1716,1718,1721],{"className":1717},[613],[578,1719],{"className":1720,"style":1198},[617],[578,1722,855],{"className":1723},[622],") ist der Hochpunkt",[578,1726,1728],{"className":1727},[1216],[578,1729,1731,1781],{"className":1730},[581],[578,1732,1734],{"className":1733},[585],[587,1735,1736],{"xmlns":589,"display":1225},[591,1737,1738,1778],{},[594,1739,1740,1752,1754,1756,1758,1760,1772,1775],{},[678,1741,1742,1744],{},[597,1743,682],{},[594,1745,1746,1748,1750],{},[853,1747,1526],{},[688,1749,690],{"separator":609},[853,1751,855],{},[688,1753,1236],{"stretchy":1235},[853,1755,1526],{},[688,1757,1253],{"stretchy":1235},[688,1759,797],{},[850,1761,1762,1764],{},[853,1763,855],{},[861,1765,1766],{},[594,1767,1768,1770],{},[853,1769,867],{},[597,1771,870],{},[688,1773,1774],{},"≈",[853,1776,1777],{},"0,3989",[601,1779,1780],{"encoding":603},"\\varphi_{0,1}(0) = \\frac{1}{\\sqrt{2\\pi}} \\approx 0{,}3989",[578,1782,1784,1859,1991],{"className":1783,"ariaHidden":609},[608],[578,1785,1787,1791,1841,1844,1847,1850,1853,1856],{"className":1786},[613],[578,1788],{"className":1789,"style":1790},[617],"height:1.0361em;vertical-align:-0.2861em;",[578,1792,1794,1797],{"className":1793},[622],[578,1795,682],{"className":1796},[622,623],[578,1798,1800],{"className":1799},[715],[578,1801,1803,1833],{"className":1802},[719,720],[578,1804,1806,1830],{"className":1805},[724],[578,1807,1810],{"className":1808,"style":1809},[728],"height:0.3011em;",[578,1811,1812,1815],{"style":732},[578,1813],{"className":1814,"style":737},[736],[578,1816,1818],{"className":1817},[741,742,743,744],[578,1819,1821,1824,1827],{"className":1820},[622,744],[578,1822,1526],{"className":1823},[622,744],[578,1825,690],{"className":1826},[754,744],[578,1828,855],{"className":1829},[622,744],[578,1831,762],{"className":1832},[761],[578,1834,1836],{"className":1835},[724],[578,1837,1839],{"className":1838,"style":769},[728],[578,1840],{},[578,1842,1236],{"className":1843},[890],[578,1845,1526],{"className":1846},[622],[578,1848,1253],{"className":1849},[1029],[578,1851],{"className":1852,"style":819},[818],[578,1854,797],{"className":1855},[823],[578,1857],{"className":1858,"style":819},[818],[578,1860,1862,1866,1982,1985,1988],{"className":1861},[613],[578,1863],{"className":1864,"style":1865},[617],"height:2.2514em;vertical-align:-0.93em;",[578,1867,1869,1872,1979],{"className":1868},[622],[578,1870],{"className":1871},[890,891],[578,1873,1875],{"className":1874},[850],[578,1876,1878,1970],{"className":1877},[719,720],[578,1879,1881,1967],{"className":1880},[724],[578,1882,1885,1947,1955],{"className":1883,"style":1884},[728],"height:1.3214em;",[578,1886,1888,1891],{"style":1887},"top:-2.2028em;",[578,1889],{"className":1890,"style":911},[736],[578,1892,1894],{"className":1893},[622],[578,1895,1897],{"className":1896},[622,924],[578,1898,1900,1938],{"className":1899},[719,720],[578,1901,1903,1935],{"className":1902},[724],[578,1904,1907,1922],{"className":1905,"style":1906},[728],"height:0.9072em;",[578,1908,1910,1913],{"className":1909,"style":939},[938],[578,1911],{"className":1912,"style":911},[736],[578,1914,1916,1919],{"className":1915,"style":946},[622],[578,1917,867],{"className":1918},[622],[578,1920,870],{"className":1921,"style":656},[622,623],[578,1923,1925,1928],{"style":1924},"top:-2.8672em;",[578,1926],{"className":1927,"style":911},[736],[578,1929,1931],{"className":1930,"style":963},[962],[965,1932,1933],{"xmlns":967,"width":968,"height":969,"viewBox":970,"preserveAspectRatio":971},[973,1934],{"d":975},[578,1936,762],{"className":1937},[761],[578,1939,1941],{"className":1940},[724],[578,1942,1945],{"className":1943,"style":1944},[728],"height:0.1328em;",[578,1946],{},[578,1948,1949,1952],{"style":990},[578,1950],{"className":1951,"style":911},[736],[578,1953],{"className":1954,"style":998},[997],[578,1956,1958,1961],{"style":1957},"top:-3.677em;",[578,1959],{"className":1960,"style":911},[736],[578,1962,1964],{"className":1963},[622],[578,1965,855],{"className":1966},[622],[578,1968,762],{"className":1969},[761],[578,1971,1973],{"className":1972},[724],[578,1974,1977],{"className":1975,"style":1976},[728],"height:0.93em;",[578,1978],{},[578,1980],{"className":1981},[1029,891],[578,1983],{"className":1984,"style":819},[818],[578,1986,1774],{"className":1987},[823],[578,1989],{"className":1990,"style":819},[818],[578,1992,1994,1998,2001,2007],{"className":1993},[613],[578,1995],{"className":1996,"style":1997},[617],"height:0.8389em;vertical-align:-0.1944em;",[578,1999,1526],{"className":2000},[622],[578,2002,2004],{"className":2003},[622],[578,2005,690],{"className":2006},[754],[578,2008,2010],{"className":2009},[622],"3989",[565,2012,2013,2014,624,2066,2118],{},"Ein Testwert sei normalverteilt mit ",[578,2015,2017,2036],{"className":2016},[581],[578,2018,2020],{"className":2019},[585],[587,2021,2022],{"xmlns":589},[591,2023,2024,2033],{},[594,2025,2026,2028,2030],{},[597,2027,599],{},[688,2029,797],{},[853,2031,2032],{},"100",[601,2034,2035],{"encoding":603},"\\mu = 100",[578,2037,2039,2057],{"className":2038,"ariaHidden":609},[608],[578,2040,2042,2045,2048,2051,2054],{"className":2041},[613],[578,2043],{"className":2044,"style":618},[617],[578,2046,599],{"className":2047},[622,623],[578,2049],{"className":2050,"style":819},[818],[578,2052,797],{"className":2053},[823],[578,2055],{"className":2056,"style":819},[818],[578,2058,2060,2063],{"className":2059},[613],[578,2061],{"className":2062,"style":1198},[617],[578,2064,2032],{"className":2065},[622],[578,2067,2069,2088],{"className":2068},[581],[578,2070,2072],{"className":2071},[585],[587,2073,2074],{"xmlns":589},[591,2075,2076,2085],{},[594,2077,2078,2080,2082],{},[597,2079,639],{},[688,2081,797],{},[853,2083,2084],{},"15",[601,2086,2087],{"encoding":603},"\\sigma = 15",[578,2089,2091,2109],{"className":2090,"ariaHidden":609},[608],[578,2092,2094,2097,2100,2103,2106],{"className":2093},[613],[578,2095],{"className":2096,"style":652},[617],[578,2098,639],{"className":2099,"style":656},[622,623],[578,2101],{"className":2102,"style":819},[818],[578,2104,797],{"className":2105},[823],[578,2107],{"className":2108,"style":819},[818],[578,2110,2112,2115],{"className":2111},[613],[578,2113],{"className":2114,"style":1198},[617],[578,2116,2084],{"className":2117},[622],". Dichte am Hochpunkt und eine Standardabweichung daneben:",[578,2120,2122],{"className":2121},[1216],[578,2123,2125,2223],{"className":2124},[581],[578,2126,2128],{"className":2127},[585],[587,2129,2130],{"xmlns":589,"display":1225},[591,2131,2132,2220],{},[594,2133,2134,2136,2138,2140,2142,2144,2160,2162,2165,2167,2170,2172,2174,2177,2179,2181,2197,2199,2215,2217],{},[597,2135,682],{},[688,2137,1236],{"stretchy":1235},[853,2139,2032],{},[688,2141,1253],{"stretchy":1235},[688,2143,797],{},[850,2145,2146,2148],{},[853,2147,855],{},[594,2149,2150,2152],{},[853,2151,2084],{},[861,2153,2154],{},[594,2155,2156,2158],{},[853,2157,867],{},[597,2159,870],{},[688,2161,1774],{},[853,2163,2164],{},"0,0266",[688,2166,690],{"separator":609},[818,2168],{"width":2169},"2em",[597,2171,682],{},[688,2173,1236],{"stretchy":1235},[853,2175,2176],{},"115",[688,2178,1253],{"stretchy":1235},[688,2180,797],{},[850,2182,2183,2185],{},[853,2184,855],{},[594,2186,2187,2189],{},[853,2188,2084],{},[861,2190,2191],{},[594,2192,2193,2195],{},[853,2194,867],{},[597,2196,870],{},[1285,2198,1287],{},[2200,2201,2202,2205],"msup",{},[597,2203,2204],{},"e",[594,2206,2207,2209],{},[688,2208,1049],{},[850,2210,2211,2213],{},[853,2212,855],{},[853,2214,867],{},[688,2216,1774],{},[853,2218,2219],{},"0,0161",[601,2221,2222],{"encoding":603},"\\varphi(100) = \\frac{1}{15\\sqrt{2\\pi}} \\approx 0{,}0266, \\qquad \\varphi(115) = \\frac{1}{15\\sqrt{2\\pi}} \\, e^{-\\frac{1}{2}} \\approx 0{,}0161",[578,2224,2226,2253,2380,2430,2671],{"className":2225,"ariaHidden":609},[608],[578,2227,2229,2232,2235,2238,2241,2244,2247,2250],{"className":2228},[613],[578,2230],{"className":2231,"style":1306},[617],[578,2233,682],{"className":2234},[622,623],[578,2236,1236],{"className":2237},[890],[578,2239,2032],{"className":2240},[622],[578,2242,1253],{"className":2243},[1029],[578,2245],{"className":2246,"style":819},[818],[578,2248,797],{"className":2249},[823],[578,2251],{"className":2252,"style":819},[818],[578,2254,2256,2259,2371,2374,2377],{"className":2255},[613],[578,2257],{"className":2258,"style":1865},[617],[578,2260,2262,2265,2368],{"className":2261},[622],[578,2263],{"className":2264},[890,891],[578,2266,2268],{"className":2267},[850],[578,2269,2271,2360],{"className":2270},[719,720],[578,2272,2274,2357],{"className":2273},[724],[578,2275,2277,2338,2346],{"className":2276,"style":1884},[728],[578,2278,2279,2282],{"style":1887},[578,2280],{"className":2281,"style":911},[736],[578,2283,2285,2288],{"className":2284},[622],[578,2286,2084],{"className":2287},[622],[578,2289,2291],{"className":2290},[622,924],[578,2292,2294,2330],{"className":2293},[719,720],[578,2295,2297,2327],{"className":2296},[724],[578,2298,2300,2315],{"className":2299,"style":1906},[728],[578,2301,2303,2306],{"className":2302,"style":939},[938],[578,2304],{"className":2305,"style":911},[736],[578,2307,2309,2312],{"className":2308,"style":946},[622],[578,2310,867],{"className":2311},[622],[578,2313,870],{"className":2314,"style":656},[622,623],[578,2316,2317,2320],{"style":1924},[578,2318],{"className":2319,"style":911},[736],[578,2321,2323],{"className":2322,"style":963},[962],[965,2324,2325],{"xmlns":967,"width":968,"height":969,"viewBox":970,"preserveAspectRatio":971},[973,2326],{"d":975},[578,2328,762],{"className":2329},[761],[578,2331,2333],{"className":2332},[724],[578,2334,2336],{"className":2335,"style":1944},[728],[578,2337],{},[578,2339,2340,2343],{"style":990},[578,2341],{"className":2342,"style":911},[736],[578,2344],{"className":2345,"style":998},[997],[578,2347,2348,2351],{"style":1957},[578,2349],{"className":2350,"style":911},[736],[578,2352,2354],{"className":2353},[622],[578,2355,855],{"className":2356},[622],[578,2358,762],{"className":2359},[761],[578,2361,2363],{"className":2362},[724],[578,2364,2366],{"className":2365,"style":1976},[728],[578,2367],{},[578,2369],{"className":2370},[1029,891],[578,2372],{"className":2373,"style":819},[818],[578,2375,1774],{"className":2376},[823],[578,2378],{"className":2379,"style":819},[818],[578,2381,2383,2386,2389,2395,2399,2402,2406,2409,2412,2415,2418,2421,2424,2427],{"className":2382},[613],[578,2384],{"className":2385,"style":1306},[617],[578,2387,1526],{"className":2388},[622],[578,2390,2392],{"className":2391},[622],[578,2393,690],{"className":2394},[754],[578,2396,2398],{"className":2397},[622],"0266",[578,2400,690],{"className":2401},[754],[578,2403],{"className":2404,"style":2405},[818],"margin-right:2em;",[578,2407],{"className":2408,"style":1436},[818],[578,2410,682],{"className":2411},[622,623],[578,2413,1236],{"className":2414},[890],[578,2416,2176],{"className":2417},[622],[578,2419,1253],{"className":2420},[1029],[578,2422],{"className":2423,"style":819},[818],[578,2425,797],{"className":2426},[823],[578,2428],{"className":2429,"style":819},[818],[578,2431,2433,2436,2548,2551,2662,2665,2668],{"className":2432},[613],[578,2434],{"className":2435,"style":1865},[617],[578,2437,2439,2442,2545],{"className":2438},[622],[578,2440],{"className":2441},[890,891],[578,2443,2445],{"className":2444},[850],[578,2446,2448,2537],{"className":2447},[719,720],[578,2449,2451,2534],{"className":2450},[724],[578,2452,2454,2515,2523],{"className":2453,"style":1884},[728],[578,2455,2456,2459],{"style":1887},[578,2457],{"className":2458,"style":911},[736],[578,2460,2462,2465],{"className":2461},[622],[578,2463,2084],{"className":2464},[622],[578,2466,2468],{"className":2467},[622,924],[578,2469,2471,2507],{"className":2470},[719,720],[578,2472,2474,2504],{"className":2473},[724],[578,2475,2477,2492],{"className":2476,"style":1906},[728],[578,2478,2480,2483],{"className":2479,"style":939},[938],[578,2481],{"className":2482,"style":911},[736],[578,2484,2486,2489],{"className":2485,"style":946},[622],[578,2487,867],{"className":2488},[622],[578,2490,870],{"className":2491,"style":656},[622,623],[578,2493,2494,2497],{"style":1924},[578,2495],{"className":2496,"style":911},[736],[578,2498,2500],{"className":2499,"style":963},[962],[965,2501,2502],{"xmlns":967,"width":968,"height":969,"viewBox":970,"preserveAspectRatio":971},[973,2503],{"d":975},[578,2505,762],{"className":2506},[761],[578,2508,2510],{"className":2509},[724],[578,2511,2513],{"className":2512,"style":1944},[728],[578,2514],{},[578,2516,2517,2520],{"style":990},[578,2518],{"className":2519,"style":911},[736],[578,2521],{"className":2522,"style":998},[997],[578,2524,2525,2528],{"style":1957},[578,2526],{"className":2527,"style":911},[736],[578,2529,2531],{"className":2530},[622],[578,2532,855],{"className":2533},[622],[578,2535,762],{"className":2536},[761],[578,2538,2540],{"className":2539},[724],[578,2541,2543],{"className":2542,"style":1976},[728],[578,2544],{},[578,2546],{"className":2547},[1029,891],[578,2549],{"className":2550,"style":1436},[818],[578,2552,2554,2557],{"className":2553},[622],[578,2555,2204],{"className":2556},[622,623],[578,2558,2560],{"className":2559},[715],[578,2561,2563],{"className":2562},[719],[578,2564,2566],{"className":2565},[724],[578,2567,2570],{"className":2568,"style":2569},[728],"height:1.004em;",[578,2571,2573,2576],{"style":2572},"top:-3.413em;margin-right:0.05em;",[578,2574],{"className":2575,"style":911},[736],[578,2577,2579],{"className":2578},[741,742,743,744],[578,2580,2582,2585],{"className":2581},[622,744],[578,2583,1049],{"className":2584},[622,744],[578,2586,2588,2593,2659],{"className":2587},[622,744],[578,2589],{"className":2590},[890,891,741,2591,2592],"reset-size3","size6",[578,2594,2596],{"className":2595},[850],[578,2597,2599,2650],{"className":2598},[719,720],[578,2600,2602,2647],{"className":2601},[724],[578,2603,2606,2622,2632],{"className":2604,"style":2605},[728],"height:0.8443em;",[578,2607,2609,2612],{"style":2608},"top:-2.656em;",[578,2610],{"className":2611,"style":911},[736],[578,2613,2616],{"className":2614},[741,2591,2615,744],"size1",[578,2617,2619],{"className":2618},[622,744],[578,2620,867],{"className":2621},[622,744],[578,2623,2625,2628],{"style":2624},"top:-3.2255em;",[578,2626],{"className":2627,"style":911},[736],[578,2629],{"className":2630,"style":2631},[997,744],"border-bottom-width:0.049em;",[578,2633,2635,2638],{"style":2634},"top:-3.384em;",[578,2636],{"className":2637,"style":911},[736],[578,2639,2641],{"className":2640},[741,2591,2615,744],[578,2642,2644],{"className":2643},[622,744],[578,2645,855],{"className":2646},[622,744],[578,2648,762],{"className":2649},[761],[578,2651,2653],{"className":2652},[724],[578,2654,2657],{"className":2655,"style":2656},[728],"height:0.344em;",[578,2658],{},[578,2660],{"className":2661},[1029,891,741,2591,2592],[578,2663],{"className":2664,"style":819},[818],[578,2666,1774],{"className":2667},[823],[578,2669],{"className":2670,"style":819},[818],[578,2672,2674,2677,2680,2686],{"className":2673},[613],[578,2675],{"className":2676,"style":1997},[617],[578,2678,1526],{"className":2679},[622],[578,2681,2683],{"className":2682},[622],[578,2684,690],{"className":2685},[754],[578,2687,2689],{"className":2688},[622],"0161",[565,2691,2692,2693,624,2722,2750,2751,657],{},"Die Wahrscheinlichkeit für einen Wert zwischen ",[578,2694,2696,2710],{"className":2695},[581],[578,2697,2699],{"className":2698},[585],[587,2700,2701],{"xmlns":589},[591,2702,2703,2708],{},[594,2704,2705],{},[853,2706,2707],{},"85",[601,2709,2707],{"encoding":603},[578,2711,2713],{"className":2712,"ariaHidden":609},[608],[578,2714,2716,2719],{"className":2715},[613],[578,2717],{"className":2718,"style":1198},[617],[578,2720,2707],{"className":2721},[622],[578,2723,2725,2738],{"className":2724},[581],[578,2726,2728],{"className":2727},[585],[587,2729,2730],{"xmlns":589},[591,2731,2732,2736],{},[594,2733,2734],{},[853,2735,2176],{},[601,2737,2176],{"encoding":603},[578,2739,2741],{"className":2740,"ariaHidden":609},[608],[578,2742,2744,2747],{"className":2743},[613],[578,2745],{"className":2746,"style":1198},[617],[578,2748,2176],{"className":2749},[622]," ist die Fläche darunter. Der Taschenrechner liefert ",[578,2752,2754,2787],{"className":2753},[581],[578,2755,2757],{"className":2756},[585],[587,2758,2759],{"xmlns":589},[591,2760,2761,2784],{},[594,2762,2763,2765,2767,2769,2771,2773,2775,2777,2779,2781],{},[597,2764,1232],{},[688,2766,1236],{"stretchy":1235},[853,2768,2707],{},[688,2770,1242],{},[597,2772,1245],{},[688,2774,1242],{},[853,2776,2176],{},[688,2778,1253],{"stretchy":1235},[688,2780,1774],{},[853,2782,2783],{},"0,683",[601,2785,2786],{"encoding":603},"P(85 \\leq X \\leq 115) \\approx 0{,}683",[578,2788,2790,2814,2832,2853],{"className":2789,"ariaHidden":609},[608],[578,2791,2793,2796,2799,2802,2805,2808,2811],{"className":2792},[613],[578,2794],{"className":2795,"style":1306},[617],[578,2797,1232],{"className":2798,"style":1310},[622,623],[578,2800,1236],{"className":2801},[890],[578,2803,2707],{"className":2804},[622],[578,2806],{"className":2807,"style":819},[818],[578,2809,1242],{"className":2810},[823],[578,2812],{"className":2813,"style":819},[818],[578,2815,2817,2820,2823,2826,2829],{"className":2816},[613],[578,2818],{"className":2819,"style":1332},[617],[578,2821,1245],{"className":2822,"style":1336},[622,623],[578,2824],{"className":2825,"style":819},[818],[578,2827,1242],{"className":2828},[823],[578,2830],{"className":2831,"style":819},[818],[578,2833,2835,2838,2841,2844,2847,2850],{"className":2834},[613],[578,2836],{"className":2837,"style":1306},[617],[578,2839,2176],{"className":2840},[622],[578,2842,1253],{"className":2843},[1029],[578,2845],{"className":2846,"style":819},[818],[578,2848,1774],{"className":2849},[823],[578,2851],{"className":2852,"style":819},[818],[578,2854,2856,2859,2862,2868],{"className":2855},[613],[578,2857],{"className":2858,"style":1997},[617],[578,2860,1526],{"className":2861},[622],[578,2863,2865],{"className":2864},[622],[578,2866,690],{"className":2867},[754],[578,2869,2871],{"className":2870},[622],"683",[560,2873,2875],{"id":2874},"typische-fehler","Typische Fehler",[774,2877,2878,3049,3345,3450],{},[777,2879,2880,2881,2956,2957,2960,2961,657],{},"Den Funktionswert als Wahrscheinlichkeit lesen. ",[578,2882,2884,2908],{"className":2883},[581],[578,2885,2887],{"className":2886},[585],[587,2888,2889],{"xmlns":589},[591,2890,2891,2905],{},[594,2892,2893,2895,2897,2899,2901,2903],{},[597,2894,682],{},[688,2896,1236],{"stretchy":1235},[853,2898,2032],{},[688,2900,1253],{"stretchy":1235},[688,2902,1774],{},[853,2904,2164],{},[601,2906,2907],{"encoding":603},"\\varphi(100) \\approx 0{,}0266",[578,2909,2911,2938],{"className":2910,"ariaHidden":609},[608],[578,2912,2914,2917,2920,2923,2926,2929,2932,2935],{"className":2913},[613],[578,2915],{"className":2916,"style":1306},[617],[578,2918,682],{"className":2919},[622,623],[578,2921,1236],{"className":2922},[890],[578,2924,2032],{"className":2925},[622],[578,2927,1253],{"className":2928},[1029],[578,2930],{"className":2931,"style":819},[818],[578,2933,1774],{"className":2934},[823],[578,2936],{"className":2937,"style":819},[818],[578,2939,2941,2944,2947,2953],{"className":2940},[613],[578,2942],{"className":2943,"style":1997},[617],[578,2945,1526],{"className":2946},[622],[578,2948,2950],{"className":2949},[622],[578,2951,690],{"className":2952},[754],[578,2954,2398],{"className":2955},[622]," ist ",[569,2958,2959],{},"keine"," Wahrscheinlichkeit; für jeden einzelnen Wert gilt ",[578,2962,2964,2992],{"className":2963},[581],[578,2965,2967],{"className":2966},[585],[587,2968,2969],{"xmlns":589},[591,2970,2971,2989],{},[594,2972,2973,2975,2977,2979,2981,2983,2985,2987],{},[597,2974,1232],{},[688,2976,1236],{"stretchy":1235},[597,2978,1245],{},[688,2980,797],{},[853,2982,2032],{},[688,2984,1253],{"stretchy":1235},[688,2986,797],{},[853,2988,1526],{},[601,2990,2991],{"encoding":603},"P(X = 100) = 0",[578,2993,2995,3019,3040],{"className":2994,"ariaHidden":609},[608],[578,2996,2998,3001,3004,3007,3010,3013,3016],{"className":2997},[613],[578,2999],{"className":3000,"style":1306},[617],[578,3002,1232],{"className":3003,"style":1310},[622,623],[578,3005,1236],{"className":3006},[890],[578,3008,1245],{"className":3009,"style":1336},[622,623],[578,3011],{"className":3012,"style":819},[818],[578,3014,797],{"className":3015},[823],[578,3017],{"className":3018,"style":819},[818],[578,3020,3022,3025,3028,3031,3034,3037],{"className":3021},[613],[578,3023],{"className":3024,"style":1306},[617],[578,3026,2032],{"className":3027},[622],[578,3029,1253],{"className":3030},[1029],[578,3032],{"className":3033,"style":819},[818],[578,3035,797],{"className":3036},[823],[578,3038],{"className":3039,"style":819},[818],[578,3041,3043,3046],{"className":3042},[613],[578,3044],{"className":3045,"style":1198},[617],[578,3047,1526],{"className":3048},[622],[777,3050,3051,3112,3113,3141,3142,3232,3233,3236,3237,2956,3295,657],{},[578,3052,3054,3072],{"className":3053},[581],[578,3055,3057],{"className":3056},[585],[587,3058,3059],{"xmlns":589},[591,3060,3061,3069],{},[594,3062,3063],{},[2200,3064,3065,3067],{},[597,3066,639],{},[853,3068,867],{},[601,3070,3071],{"encoding":603},"\\sigma^2",[578,3073,3075],{"className":3074,"ariaHidden":609},[608],[578,3076,3078,3082],{"className":3077},[613],[578,3079],{"className":3080,"style":3081},[617],"height:0.8141em;",[578,3083,3085,3088],{"className":3084},[622],[578,3086,639],{"className":3087,"style":656},[622,623],[578,3089,3091],{"className":3090},[715],[578,3092,3094],{"className":3093},[719],[578,3095,3097],{"className":3096},[724],[578,3098,3100],{"className":3099,"style":3081},[728],[578,3101,3103,3106],{"style":3102},"top:-3.063em;margin-right:0.05em;",[578,3104],{"className":3105,"style":737},[736],[578,3107,3109],{"className":3108},[741,742,743,744],[578,3110,867],{"className":3111},[622,744]," statt ",[578,3114,3116,3129],{"className":3115},[581],[578,3117,3119],{"className":3118},[585],[587,3120,3121],{"xmlns":589},[591,3122,3123,3127],{},[594,3124,3125],{},[597,3126,639],{},[601,3128,642],{"encoding":603},[578,3130,3132],{"className":3131,"ariaHidden":609},[608],[578,3133,3135,3138],{"className":3134},[613],[578,3136],{"className":3137,"style":652},[617],[578,3139,639],{"className":3140,"style":656},[622,623]," einsetzen. Die Angabe ",[578,3143,3145,3174],{"className":3144},[581],[578,3146,3148],{"className":3147},[585],[587,3149,3150],{"xmlns":589},[591,3151,3152,3171],{},[594,3153,3154,3157,3159,3161,3163,3169],{},[597,3155,3156],{},"N",[688,3158,1236],{"stretchy":1235},[597,3160,599],{},[688,3162,690],{"separator":609},[2200,3164,3165,3167],{},[597,3166,639],{},[853,3168,867],{},[688,3170,1253],{"stretchy":1235},[601,3172,3173],{"encoding":603},"N(\\mu, \\sigma^2)",[578,3175,3177],{"className":3176,"ariaHidden":609},[608],[578,3178,3180,3184,3188,3191,3194,3197,3200,3229],{"className":3179},[613],[578,3181],{"className":3182,"style":3183},[617],"height:1.0641em;vertical-align:-0.25em;",[578,3185,3156],{"className":3186,"style":3187},[622,623],"margin-right:0.109em;",[578,3189,1236],{"className":3190},[890],[578,3192,599],{"className":3193},[622,623],[578,3195,690],{"className":3196},[754],[578,3198],{"className":3199,"style":1436},[818],[578,3201,3203,3206],{"className":3202},[622],[578,3204,639],{"className":3205,"style":656},[622,623],[578,3207,3209],{"className":3208},[715],[578,3210,3212],{"className":3211},[719],[578,3213,3215],{"className":3214},[724],[578,3216,3218],{"className":3217,"style":3081},[728],[578,3219,3220,3223],{"style":3102},[578,3221],{"className":3222,"style":737},[736],[578,3224,3226],{"className":3225},[741,742,743,744],[578,3227,867],{"className":3228},[622,744],[578,3230,1253],{"className":3231},[1029]," nennt die ",[569,3234,3235],{},"Varianz",". Bei ",[578,3238,3240,3265],{"className":3239},[581],[578,3241,3243],{"className":3242},[585],[587,3244,3245],{"xmlns":589},[591,3246,3247,3262],{},[594,3248,3249,3251,3253,3255,3257,3260],{},[597,3250,3156],{},[688,3252,1236],{"stretchy":1235},[853,3254,2032],{},[688,3256,690],{"separator":609},[853,3258,3259],{},"225",[688,3261,1253],{"stretchy":1235},[601,3263,3264],{"encoding":603},"N(100, 225)",[578,3266,3268],{"className":3267,"ariaHidden":609},[608],[578,3269,3271,3274,3277,3280,3283,3286,3289,3292],{"className":3270},[613],[578,3272],{"className":3273,"style":1306},[617],[578,3275,3156],{"className":3276,"style":3187},[622,623],[578,3278,1236],{"className":3279},[890],[578,3281,2032],{"className":3282},[622],[578,3284,690],{"className":3285},[754],[578,3287],{"className":3288,"style":1436},[818],[578,3290,3259],{"className":3291},[622],[578,3293,1253],{"className":3294},[1029],[578,3296,3298,3315],{"className":3297},[581],[578,3299,3301],{"className":3300},[585],[587,3302,3303],{"xmlns":589},[591,3304,3305,3313],{},[594,3306,3307,3309,3311],{},[597,3308,639],{},[688,3310,797],{},[853,3312,2084],{},[601,3314,2087],{"encoding":603},[578,3316,3318,3336],{"className":3317,"ariaHidden":609},[608],[578,3319,3321,3324,3327,3330,3333],{"className":3320},[613],[578,3322],{"className":3323,"style":652},[617],[578,3325,639],{"className":3326,"style":656},[622,623],[578,3328],{"className":3329,"style":819},[818],[578,3331,797],{"className":3332},[823],[578,3334],{"className":3335,"style":819},[818],[578,3337,3339,3342],{"className":3338},[613],[578,3340],{"className":3341,"style":1198},[617],[578,3343,2084],{"className":3344},[622],[777,3346,3347,3348,3449],{},"Das Minus im Exponenten oder den Faktor ",[578,3349,3351,3369],{"className":3350},[581],[578,3352,3354],{"className":3353},[585],[587,3355,3356],{"xmlns":589},[591,3357,3358,3366],{},[594,3359,3360],{},[850,3361,3362,3364],{},[853,3363,855],{},[853,3365,867],{},[601,3367,3368],{"encoding":603},"\\frac{1}{2}",[578,3370,3372],{"className":3371,"ariaHidden":609},[608],[578,3373,3375,3379],{"className":3374},[613],[578,3376],{"className":3377,"style":3378},[617],"height:1.1901em;vertical-align:-0.345em;",[578,3380,3382,3385,3446],{"className":3381},[622],[578,3383],{"className":3384},[890,891],[578,3386,3388],{"className":3387},[850],[578,3389,3391,3437],{"className":3390},[719,720],[578,3392,3394,3434],{"className":3393},[724],[578,3395,3397,3412,3420],{"className":3396,"style":904},[728],[578,3398,3400,3403],{"style":3399},"top:-2.655em;",[578,3401],{"className":3402,"style":911},[736],[578,3404,3406],{"className":3405},[741,742,743,744],[578,3407,3409],{"className":3408},[622,744],[578,3410,867],{"className":3411},[622,744],[578,3413,3414,3417],{"style":990},[578,3415],{"className":3416,"style":911},[736],[578,3418],{"className":3419,"style":998},[997],[578,3421,3422,3425],{"style":1001},[578,3423],{"className":3424,"style":911},[736],[578,3426,3428],{"className":3427},[741,742,743,744],[578,3429,3431],{"className":3430},[622,744],[578,3432,855],{"className":3433},[622,744],[578,3435,762],{"className":3436},[761],[578,3438,3440],{"className":3439},[724],[578,3441,3444],{"className":3442,"style":3443},[728],"height:0.345em;",[578,3445],{},[578,3447],{"className":3448},[1029,891]," vergessen.",[777,3451,3452,3453,657],{},"Bei diskreten Größen (z.B. Anzahlen) die Normalverteilung ohne Prüfung verwenden. Für die Näherung einer Binomialverteilung braucht man ",[578,3454,3456,3476],{"className":3455},[581],[578,3457,3459],{"className":3458},[585],[587,3460,3461],{"xmlns":589},[591,3462,3463,3473],{},[594,3464,3465,3467,3470],{},[597,3466,639],{},[688,3468,3469],{},">",[853,3471,3472],{},"3",[601,3474,3475],{"encoding":603},"\\sigma > 3",[578,3477,3479,3498],{"className":3478,"ariaHidden":609},[608],[578,3480,3482,3486,3489,3492,3495],{"className":3481},[613],[578,3483],{"className":3484,"style":3485},[617],"height:0.5782em;vertical-align:-0.0391em;",[578,3487,639],{"className":3488,"style":656},[622,623],[578,3490],{"className":3491,"style":819},[818],[578,3493,3469],{"className":3494},[823],[578,3496],{"className":3497,"style":819},[818],[578,3499,3501,3504],{"className":3500},[613],[578,3502],{"className":3503,"style":1198},[617],[578,3505,3472],{"className":3506},[622],{"title":3508,"searchDepth":3509,"depth":3509,"links":3510},"",2,[3511,3512,3513],{"id":562,"depth":3509,"text":563},{"id":1615,"depth":3509,"text":1616},{"id":2874,"depth":3509,"text":2875},[3515,3518,3521,3524,3527,3530,3533],{"lhs":3516,"rhs":3517},"exp(-0.5*((u + x - u)\u002Fs)^2) \u002F (s*sqrt(2*pi))","exp(-0.5*((u - x - u)\u002Fs)^2) \u002F (s*sqrt(2*pi))",{"lhs":3519,"rhs":3520},"exp(-0.5*((u - u)\u002Fs)^2) \u002F (s*sqrt(2*pi))","1\u002F(s*sqrt(2*pi))",{"lhs":3522,"rhs":3523},"round(1\u002Fsqrt(2*pi), 4)","0.3989",{"lhs":3525,"rhs":3526},"round(1\u002F(15*sqrt(2*pi)), 4)","0.0266",{"lhs":3528,"rhs":3529},"exp(-0.5*((115 - 100)\u002F15)^2) \u002F (15*sqrt(2*pi))","exp(-0.5) \u002F (15*sqrt(2*pi))",{"lhs":3531,"rhs":3532},"round(exp(-0.5) \u002F (15*sqrt(2*pi)), 4)","0.0161",{"lhs":3534,"rhs":3535},"round(0.5*(1 + erf((115 - 100)\u002F(15*sqrt(2)))) - 0.5*(1 + erf((85 - 100)\u002F(15*sqrt(2)))), 3)","0.683","\\mu \\in \\mathbb{R},\\ \\sigma > 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu \\in \\mathbb{R},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8833em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>σ\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sigma > 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","normal-distribution","\\varphi_{\\mu,\\sigma}(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} \\, e^{-\\frac{1}{2}\\left(\\frac{x - \\mu}{\\sigma}\\right)^2}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>φ\u003C\u002Fmi>\u003Cmrow>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>σ\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmrow>\u003Cmi>σ\u003C\u002Fmi>\u003Cmsqrt>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmi>π\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmfrac>\u003Cmsup>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>μ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>σ\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi_{\\mu,\\sigma}(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} \\, e^{-\\frac{1}{2}\\left(\\frac{x - \\mu}{\\sigma}\\right)^2}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.2514em;vertical-align:-0.93em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3214em;\">\u003Cspan style=\"top:-2.2028em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9072em;\">\u003Cspan class=\"svg-align\" style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:0.833em;\">\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.8672em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1328em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.93em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">e\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1857em;\">\u003Cspan style=\"top:-3.413em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen nulldelimiter katex-sizing reset-size3 size6\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8443em;\">\u003Cspan style=\"top:-2.656em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2255em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.384em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.344em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter katex-sizing reset-size3 size6\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"minner mtight\">\u003Cspan class=\"minner mtight\">\u003Cspan class=\"mopen katex-sizing reset-size3 size6 mtight delimcenter\" style=\"top:0.075em;\">\u003Cspan class=\"mtight\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen nulldelimiter katex-sizing reset-size3 size6\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.879em;\">\u003Cspan style=\"top:-2.656em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2255em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line mtight\" style=\"border-bottom-width:0.049em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.4624em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.344em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter katex-sizing reset-size3 size6\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose katex-sizing reset-size3 size6 mtight delimcenter\" style=\"top:0.075em;\">\u003Cspan class=\"mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1039em;\">\u003Cspan style=\"top:-3.1436em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fstochastics\u002Fnormalverteilung",[],[3549,3550,3551,3552],"standardization","sigma-rules","expected-value","binomial-distribution",{"title":548,"description":3508},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fnormalverteilung","Die Dichte der Normalverteilung ist die Gaußsche Glockenkurve mit Hochpunkt bei μ und Breite σ.",[3557,3558,3559,3560],"gaußsche glockenkurve","dichtefunktion","stetige verteilung","wahrscheinlichkeitsverteilung",[3562,3566,3569,3572,3575],{"symbol":3563,"meaning":3564,"symbolHtml":3565},"\\varphi_{\\mu,\\sigma}(x)","Wert der Dichtefunktion an der Stelle x","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>φ\u003C\u002Fmi>\u003Cmrow>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>σ\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varphi_{\\mu,\\sigma}(x)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">φ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":794,"meaning":3567,"symbolHtml":3568},"beliebige reelle Zahl, ein möglicher Wert der Zufallsgröße","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":604,"meaning":3570,"symbolHtml":3571},"Erwartungswert, Lage des Hochpunkts der Glockenkurve","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>μ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":642,"meaning":3573,"symbolHtml":3574},"Standardabweichung, bestimmt die Breite der Kurve","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>σ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sigma\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":2204,"meaning":3576,"symbolHtml":3577},"Eulersche Zahl, etwa 2,718","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>e\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">e\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">e\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","saUTiUXP_YAvKAZPhsosa1_-E1pAItT5rfXe4E1oRTc",[3580,3583,3586,3589],{"stem":3581,"title":3582,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fbinomialverteilung","Binomialverteilung",{"stem":3584,"title":3585,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Ferwartungswert","Erwartungswert",{"stem":3587,"title":3588,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fsigma-regeln","Sigma-Regeln",{"stem":3590,"title":3591,"category":441},"de\u002Fformulas\u002Fmath\u002Fstochastics\u002Fstandardisierung","Standardisierung",1793824286919]