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I'm trying to learn about the Gaussian Normal distribution using Mathematica.

So I defined my Gaussian like this

Gaus[x_, mu_, sigma_] = 1/(sigma*Sqrt[2 \[Pi]]) E^(-(1/2) ((x - mu)/sigma)^2);

I just tried to integrate it

Integrate[Gaus[x, mu, sigma], x, Assumptions -> {sigma > 0}]

which gives me

-(1/2) Erf[(mu - x)/(Sqrt[2] sigma)]

I plot it using a manipulate block

Manipulate[Column[{
   Style["Gaussian Normal", Bold],
   Plot[Gaus[x, mu, sigma], {x, -5, 5}, ImageSize -> Large, 
   PlotRange -> Full],
   Style["Gaussian Integral", Bold],
   Plot[-(1/2) Erf[(mu - x)/(Sqrt[2] sigma)], {x, -5, 5}, 
   ImageSize -> Large, PlotRange -> Full]
   }],
 {mu, -5, 5, 1}, {sigma, 0.5, 2, 0.5}]

This is what I get for mean=1 and stdev=1:

Gaussian and Error function plot

I don't understand why the erf function is going through (0/0) and is not shifted up by 1/2.

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  • $\begingroup$ The correct integral is just CDF[NormalDistribution[mu, sigma], x] $\endgroup$ – Bob Hanlon Jun 22 '20 at 15:59
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The antiderivative Integrate isn't unique.

Try

Integrate[Gaus[x, mu, sigma], {x, -Infinity, x},Assumptions -> {sigma > 0}]
(*1/2 (1 + Erf[(-mu + x)/(Sqrt[2] sigma)])*)

to get the expected result!

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