Moments of Gamma distribution

Question

I know that if $X$ follows $\Gamma(\alpha,\beta)$, then $E(X)=\alpha/\beta$ and $E(X^2)=\alpha(\alpha+1)/\beta^2$, so that $Var(X)=\alpha/\beta^2$. However, I'm curious about the shape of functions $h(x)$ such that $h(X)=\alpha^2/\beta$. Obviously, $h(x)=\alpha x$ is an example. Are there any other ones, and how would we find them?

Answer 1

This problem asks for h(x) such that

$\displaystyle\frac{\alpha^2}{\beta}=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\int_{0}^{\infty}x^{\alpha-1}h(x)e^{-\beta x} dx$ This is equivalent to $\displaystyle\frac{\alpha^2}{\beta}=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\mathscr{L}(x^{\alpha-1}h(x)) (\beta) $

Then

$\displaystyle h(x) =x^{-\alpha+1}\Gamma(\alpha)\alpha^{2}\mathscr{L}^{-1}(\beta^{-1-\alpha}) $

$\displaystyle \mathscr{L}^{-1}(\beta^{-1-\alpha})=\frac{x^{\alpha}} {\Gamma(\alpha+1)}$

Then by properties of Laplace transform we can uniquely determine h(x) to be $\displaystyle h(x) =\alpha x$

Thus there are no other functions that have this property.