# Obtain the mean and variance of the below PDF?

Posted 1 year ago
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 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]] 
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Posted 1 year ago
 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 ?
Posted 1 year ago
 Just an attempt , I think the constant i have added is work over range (-infinity , infinity)
Posted 1 year ago
 @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
Posted 1 year ago
 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.
 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 $z$.
 But The value of $\mu$ are in $(0,1)$, I have set that in Stackexchange post