Normaalijakauma

Normaalijakauman kertymäfunktio

\begin{align*} Φ(\mathit{a}) = \mathit{P}(\mathit{X} ≤ \mathit{a}), Φ(-\mathit{a}) = 1 – Φ(\mathit{a}) \end{align*} \begin{eqnarray*} Φ(x)=\frac{1}{\sqrt{2π}} \displaystyle \int_{-\infty}^{ x } e^{ \frac{t^{2}}{2} }dt \end{eqnarray*}
Normaalijakauman kertymäfunktion taulukko.
x0123456789
0.0.5000.5040.5080.5120.5160.5199.5239.5279.5319.5359
0.1.5398.5438.5478.5517.5557.5596.5636.5675.5714.5753
0.2.5793.5832.5871.5910.5948.5987.6026.6064.6103.6141
0.3.6179.6217.6255.6293.6331.6368.6406.6443.6480.6517
0.4.6554.6591.6628.6664.6700.6736.6772.6808.6844.6879
0.5.6915.6950.6985.7019.7054.7088.7123.7157.7190.7224
0.6.7257.7291.7324.7357.7389.7422.7454.7486.7517.7549
0.7.7580.7611.7642.7673.7704.7734.7764.7794.7823.7852
0.8.7881.7910.7939.7967.7995.8023.8051.8078.8106.8133
0.9.8159.8186.8212.8238.8264.8289.8315.8340.8365.8389
1.0.8413.8438.8461.8485.8508.8531.8554.8577.8599.8621
1.1.8643.8665.8686.8708.8729.8749.8770.8790.8810.8830
1.2.8849.8869.8888.8907.8925.8944.8962.8980.8997.9015
1.3.9032.9049.9066.9082.9099.9115.9131.9147.9162.9177
1.4.9192.9207.9222.9236.9251.9265.9279.9292.9306.9319
1.5.9332.9345.9357.9370.9382.9394.9406.9418.9429.9441
1.6.9452.9463.9474.9484.9495.9505.9515.9525.9535.9545
1.7.9554.9564.9573.9582.9591.9599.9608.9616.9625.9633
1.8.9641.9649.9656.9664.9671.9678.9686.9693.9699.9706
1.9.9713.9719.9726.9732.9738.9744.9750.9756.9761.9767
2.0.9772.9778.9783.9788.9793.9798.9803.9808.9812.9817
2.1.9821.9826.9830.9834.9838.9842.9846.9850.9854.9857
2.2.9861.9864.9868.9871.9875.9878.9881.9884.9887.9890
2.3.9893.9896.9898.9901.9904.9906.9909.9911.9913.9916
2.4.9918.9920.9922.9925.9927.9929.9931.9932.9934.9936
2.5.9938.9940.9941.9943.9945.9946.9948.9949.9951.9952
2.6.9953.9955.9956.9957.9959.9960.9961.9962.9963.9964
2.7.9965.9966.9967.9968.9969.9970.9971.9972.9973.9974
2.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.9981
2.9.9981.9982.9983.9983.9984.9984.9985.9985.9986.9986
3.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990
3.1.9990.9991.9991.9991.9992.9992.9992.9992.9993.9993
3.2.9993.9993.9994.9994.9994.9994.9994.9995.9995.9995
3.3.9995.9995.9995.9996.9996.9996.9996.9996.9996.9997
3.4.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998
x0123456789
Φ (x)0.900.950.9750.9900.9950.9990.99990.99995
x1.28161.64491.962.32642.57583.29053.71903.8906

Esimerkkejä taulukon käyttöön

1. P(X ≤ 1,35) = Φ(1,35) ≈ 0,9115

2. P(-0,55 ≤ X ≤ 1,50) 
= Φ(1,50) – Φ (-0,55) = Φ(1,50) – 1 + Φ(0,55) ≈ 0,9332–1+0,7088 = 0,6420

Normaalijakauman tiheysfunktio

\begin{align*} \phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \end{align*}