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 $z$.

@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