# How to find the mean [E(x)] of this function. Please i know i can find variance after this is shown

Then how do i find the mean of $$\int_0^\infty\frac4{\sqrt{\pi} b^3}x^2e^{-x^2/b^2}dx;$$ if so do how do i do this using the gamma distribution? is there a short cut or do i have to do integration by parts?please explain.

I proved it is a PDF using polar coordinates but wonder if there is not a quicker way using the gamma distribtution. my book and the internet have not been helpful

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Use change of variables. Let $y=x^2$ and then it becomes a gamma distribution.
You probably know how to integrate $\int_0^\infty e^{-x^2} dx$. This is indeed quite the same after two times partial integration.