Covariance of Gamma Transformations

Question

If I have two independent random variables: $X$ is governed by Gamma$(a,b)$ and $Y$ is governed by Gamma$(c,b)$, then how can I find the Cov($Z,X$), where $Z = X/Y$?

I tried using properties of covariance, but the transformation was messing me up. Cov(Z,X) = E(ZX) - E(X)E(Z)

Answer 1

Note that$$\begin{align}\operatorname{Cov}(Z,\,X)&=E[ZX]-E[Z]E[X]\\&=E[X^2/Y]-E[X/Y]E[X]\\&=E[X^2]E[1/Y]-E[X]^2E[1/Y]\\&=\operatorname{Var}(X)E[1/Y].\end{align}$$I leave it to you to write the final expression in terms of $a,\,b,\,c$.

Answer 2

Also note that $\mathbb{E}\Big[\frac{1}{Y}\Big]$ is known as $\frac{1}{Y}\sim \text{Inverse Gamma}$

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Inverse Gamma Distribution