Standardnormalverteilung

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Allgemein[Bearbeiten]

Die Standardnormalverteilung ist eine spezielle Normalverteilung mit den Parametern \mu=0 und \sigma^2=1. Die Wichtigkeit der Standardnormalverteilung liegt darin, dass man eine Normalverteilung N(\mu; \sigma^2) in eine Standardnormalverteilung transformieren kann mit Z_{}^{} = \tfrac{X-\mu}{\sigma}, insbesondere gilt natürlich F_X(x)=\Phi\left(\tfrac{x-\mu}{\sigma}\right). Ebenfalls gilt \Phi(-x)=1-\Phi(x), so dass sich die folgende Tabellen für \Phi(x) und die Fraktile ergeben.

Auswertung der Verteilungsfunktion \Phi(x)_{}^{} für 0\leq x_{}^{} < 4,0[Bearbeiten]

Beispiele:

  • \Phi(0,12)_{}^{} = \Phi(0,1 + 0,02) = 0,5477584
  • \Phi(-0,12)_{}^{} = 1-\Phi(0,12) = 1-0,5477584=0,4522416


x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000000 0.5039894 0.5079783 0.5119665 0.5159534 0.5199388 0.5239222 0.5279032 0.5318814 0.5358564
0.1 0.5398278 0.5437953 0.5477584 0.5517168 0.5556700 0.5596177 0.5635595 0.5674949 0.5714237 0.5753454
0.2 0.5792597 0.5831662 0.5870644 0.5909541 0.5948349 0.5987063 0.6025681 0.6064199 0.6102612 0.6140919
0.3 0.6179114 0.6217195 0.6255158 0.6293000 0.6330717 0.6368307 0.6405764 0.6443088 0.6480273 0.6517317
0.4 0.6554217 0.6590970 0.6627573 0.6664022 0.6700314 0.6736448 0.6772419 0.6808225 0.6843863 0.6879331
0.5 0.6914625 0.6949743 0.6984682 0.7019440 0.7054015 0.7088403 0.7122603 0.7156612 0.7190427 0.7224047
0.6 0.7257469 0.7290691 0.7323711 0.7356527 0.7389137 0.7421539 0.7453731 0.7485711 0.7517478 0.7549029
0.7 0.7580363 0.7611479 0.7642375 0.7673049 0.7703500 0.7733726 0.7763727 0.7793501 0.7823046 0.7852361
0.8 0.7881446 0.7910299 0.7938919 0.7967306 0.7995458 0.8023375 0.8051055 0.8078498 0.8105703 0.8132671
0.9 0.8159399 0.8185887 0.8212136 0.8238145 0.8263912 0.8289439 0.8314724 0.8339768 0.8364569 0.8389129
1.0 0.8413447 0.8437524 0.8461358 0.8484950 0.8508300 0.8531409 0.8554277 0.8576903 0.8599289 0.8621434
1.1 0.8643339 0.8665005 0.8686431 0.8707619 0.8728568 0.8749281 0.8769756 0.8789995 0.8809999 0.8829768
1.2 0.8849303 0.8868606 0.8887676 0.8906514 0.8925123 0.8943502 0.8961653 0.8979577 0.8997274 0.9014747
1.3 0.9031995 0.9049021 0.9065825 0.9082409 0.9098773 0.9114920 0.9130850 0.9146565 0.9162067 0.9177356
1.4 0.9192433 0.9207302 0.9221962 0.9236415 0.9250663 0.9264707 0.9278550 0.9292191 0.9305634 0.9318879
1.5 0.9331928 0.9344783 0.9357445 0.9369916 0.9382198 0.9394292 0.9406201 0.9417924 0.9429466 0.9440826
1.6 0.9452007 0.9463011 0.9473839 0.9484493 0.9494974 0.9505285 0.9515428 0.9525403 0.9535213 0.9544860
1.7 0.9554345 0.9563671 0.9572838 0.9581849 0.9590705 0.9599408 0.9607961 0.9616364 0.9624620 0.9632730
1.8 0.9640697 0.9648521 0.9656205 0.9663750 0.9671159 0.9678432 0.9685572 0.9692581 0.9699460 0.9706210
1.9 0.9712834 0.9719334 0.9725711 0.9731966 0.9738102 0.9744119 0.9750021 0.9755808 0.9761482 0.9767045
2.0 0.9772499 0.9777844 0.9783083 0.9788217 0.9793248 0.9798178 0.9803007 0.9807738 0.9812372 0.9816911
2.1 0.9821356 0.9825708 0.9829970 0.9834142 0.9838226 0.9842224 0.9846137 0.9849966 0.9853713 0.9857379
2.2 0.9860966 0.9864474 0.9867906 0.9871263 0.9874545 0.9877755 0.9880894 0.9883962 0.9886962 0.9889893
2.3 0.9892759 0.9895559 0.9898296 0.9900969 0.9903581 0.9906133 0.9908625 0.9911060 0.9913437 0.9915758
2.4 0.9918025 0.9920237 0.9922397 0.9924506 0.9926564 0.9928572 0.9930531 0.9932443 0.9934309 0.9936128
2.5 0.9937903 0.9939634 0.9941323 0.9942969 0.9944574 0.9946139 0.9947664 0.9949151 0.9950600 0.9952012
2.6 0.9953388 0.9954729 0.9956035 0.9957308 0.9958547 0.9959754 0.9960930 0.9962074 0.9963189 0.9964274
2.7 0.9965330 0.9966358 0.9967359 0.9968333 0.9969280 0.9970202 0.9971099 0.9971972 0.9972821 0.9973646
2.8 0.9974449 0.9975229 0.9975988 0.9976726 0.9977443 0.9978140 0.9978818 0.9979476 0.9980116 0.9980738
2.9 0.9981342 0.9981929 0.9982498 0.9983052 0.9983589 0.9984111 0.9984618 0.9985110 0.9985588 0.9986051
3.0 0.9986501 0.9986938 0.9987361 0.9987772 0.9988171 0.9988558 0.9988933 0.9989297 0.9989650 0.9989992
3.1 0.9990324 0.9990646 0.9990957 0.9991260 0.9991553 0.9991836 0.9992112 0.9992378 0.9992636 0.9992886
3.2 0.9993129 0.9993363 0.9993590 0.9993810 0.9994024 0.9994230 0.9994429 0.9994623 0.9994810 0.9994991
3.3 0.9995166 0.9995335 0.9995499 0.9995658 0.9995811 0.9995959 0.9996103 0.9996242 0.9996376 0.9996505
3.4 0.9996631 0.9996752 0.9996869 0.9996982 0.9997091 0.9997197 0.9997299 0.9997398 0.9997493 0.9997585
3.5 0.9997674 0.9997759 0.9997842 0.9997922 0.9997999 0.9998074 0.9998146 0.9998215 0.9998282 0.9998347
3.6 0.9998409 0.9998469 0.9998527 0.9998583 0.9998637 0.9998689 0.9998739 0.9998787 0.9998834 0.9998879
3.7 0.9998922 0.9998964 0.9999004 0.9999043 0.9999080 0.9999116 0.9999150 0.9999184 0.9999216 0.9999247
3.8 0.9999277 0.9999305 0.9999333 0.9999359 0.9999385 0.9999409 0.9999433 0.9999456 0.9999478 0.9999499
3.9 0.9999519 0.9999539 0.9999557 0.9999575 0.9999593 0.9999609 0.9999625 0.9999641 0.9999655 0.9999670

Auswertung der inversen Verteilungsfunktion \Phi^{-1}(p) (Quantilsfunktion) für 0,5\leq p<1[Bearbeiten]

Beispiele:

  • 0,95 = 0,9 + 0,050 = \Phi(x) \Longleftrightarrow x=1.644854
  • 0,34=\Phi(x) \Longleftrightarrow 1-0,34=0,66=\Phi(-x) \Longleftrightarrow -x=0.4124631
p 0.5 0.6 0.7 0.8 0.9
0.000 0.0000000 0.2533471 0.5244005 0.8416212 1.2815516
0.005 0.0125335 0.2663106 0.5388360 0.8596174 1.3105791
0.010 0.0250689 0.2793190 0.5533847 0.8778963 1.3407550
0.015 0.0376083 0.2923749 0.5680515 0.8964734 1.3722038
0.020 0.0501536 0.3054808 0.5828415 0.9153651 1.4050716
0.025 0.0627068 0.3186394 0.5977601 0.9345893 1.4395315
0.030 0.0752699 0.3318533 0.6128130 0.9541653 1.4757910
0.035 0.0878448 0.3451255 0.6280060 0.9741139 1.5141019
0.040 0.1004337 0.3584588 0.6433454 0.9944579 1.5547736
0.045 0.1130385 0.3718561 0.6588377 1.0152220 1.5981931
0.050 0.1256613 0.3853205 0.6744898 1.0364334 1.6448536
0.055 0.1383042 0.3988551 0.6903088 1.0581216 1.6953977
0.060 0.1509692 0.4124631 0.7063026 1.0803193 1.7506861
0.065 0.1636585 0.4261480 0.7224791 1.1030626 1.8119107
0.070 0.1763742 0.4399132 0.7388468 1.1263911 1.8807936
0.075 0.1891184 0.4537622 0.7554150 1.1503494 1.9599640
0.080 0.2018935 0.4676988 0.7721932 1.1749868 2.0537489
0.085 0.2147016 0.4817268 0.7891917 1.2003589 2.1700904
0.090 0.2275450 0.4958503 0.8064212 1.2265281 2.3263479
0.095 0.2404260 0.5100735 0.8238936 1.2535654 2.5758293