Obtain the mean and variance of the below PDF?

Questions
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1

I have tried in Wolfram Cloud to get the Mean value and Variance of this PDF :

\int_{1}^{\infty } \frac{3 \zeta (3) \sqrt{10 \sqrt{2} \pi } \exp \left(-\frac{\pi \sqrt{2 \sigma } (z+\sigma )^2 \text{erf}\left(\frac{\mu (z-\sigma )^2}{\sqrt{\sqrt{2} \pi \sigma }}\right)}{\mu }\right)}{(2\pi^2 \log (\pi )) } \, d\sigma

and \mu lie in (0,1), But i didn’t come up to get the Mean and Variance of that distribution.

Here is my Code for Mean and Variance .:

\[ScriptCapitalD] =
ProbabilityDistribution[ 
3*Zeta[3]/(Pi*Log[Pi])*Sqrt[10Sqrt [ 2]*Pi]/(2*Pi)*Exp[-Pi*Sqrt[2*\[Sigma]](z+\[Sigma])^2
* Erf[\[Mu](z-\[Sigma])^2/Sqrt[Sqrt[2]*Pi*\[Sigma]] ]/\[Mu]], {\[Sigma],-Infinity, Infinity}];
Variance[\[ScriptCapitalD]]

The Same Code For Mean :

\[ScriptCapitalD] =
ProbabilityDistribution[ 
3*Zeta[3]/(Pi*Log[Pi])*Sqrt[10Sqrt [ 2]*Pi]/(2*Pi)*Exp[-Pi*Sqrt[2*\[Sigma]](z+\[Sigma])^2
* Erf[\[Mu](z-\[Sigma])^2/Sqrt[Sqrt[2]*Pi*\[Sigma]] ]/\[Mu]], {\[Sigma],-Infinity, Infinity}];
Mean[\[ScriptCapitalD]]
2

Did you mean to integrate \sigma from -\infty as \sigma shows up under a square root in your equation? That begs the question as to if this is a legitimate pdf. Also, does this mean you were able to determine the constant of integration in the question you posted at Mathematica StackExchange ?

3

Just an attempt , I think the constant i have added is work over range (-infinity , infinity)

4

@Jim Baldwin Thank you so much for your attentionn , it were a wrong typo the integrand is still over sigma from 1 to Infinity , Now it fixed

5

So you determined the constant of integration to make it a legitimate pdf? If so, please add that as answer to your question on Mathematica StackExchange. I’m curious as to how you determined that.

6

So you determined the constant of integration to make it a legitimate pdf? If so, please add that as answer to your question on Mathematica StackExchange. I’m curious as to how you determined that.

But I’m not convinced that you have a pdf that integrates to 1. Consider setting z to 4.75 and \mu to 4.

NIntegrate[PDF[\[ScriptCapitalD] /. {\[Mu] -> 4, z -> 4.75}, \[Sigma]], {\[Sigma], 1, 200}]
(* 0.128714 *)

Please clarify if you are interested in the random variable \sigma or z or a joint pdf for \sigma and z.

7

But The value of \mu are in (0,1) , I have set that in Stackexchange post