Stochastics

Normal Distribution

For a   N(μ,σ2) distributed random variable X with given expected value μ and variance  σ2, You may calculate

 

- density function    

- and distribution function

The diagram of the density function f(x) is often called Gaussian curve, or bell shaped curve. The distribution function  Φ(x) is designated as Gaussian error function, because, according to Gauss, this distribution is assumed for the random errors in astronomical observations.

Expected value  μ  and variance  σ2 are entered. For  μ=0  and  σ=1  you receive the standardized normal distribution.

Example:

My = 5       Sigma = .75

  x            f(x)         Phi(x))
----------   ----------    ----------
2            0,00017844   0,00003167
2,33333333   0,00095649   0,00018859
2,66666666   0,00420802   0,00093192
2,99999999   0,01519465   0,00383038
3,33333332   0,04503153   0,01313415
3,66666665   0,10953585   0,03772017
3,99999998   0,21868009   0,09121120
4,33333331   0,35832381   0,18703139
4,66666664   0,48189843   0,32836063
4,99999997   0,53192304   0,49999998
5,3333333    0,48189845   0,67163934
5,66666663   0,35832383   0,81296859
5,99999996   0,21868012   0,90878878
6,33333329   0,10953586   0,96227982
6,66666662   0,04503154   0,98686585
6,99999995   0,01519465   0,99616962
7,33333328   0,00420802   0,99906808
7,66666661   0,00095649   0,99981141
7,99999994   0,00017844   0,99996833

See also:

Wikipedia: Normal distribution
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