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]]