Given a random variable $X$ which is distributed gamma with shape $\alpha$ and rate $\lambda$, for which the variance is known, how does one calculate $\text{Var}(\frac{1}{X})$? I am hoping not to have to derive the inverse-gamma distribution and then calculate its expectation and variance; is there a trick which can be used given that the variance of $X$ is known?
Variance of inverse gamma distribution
gamma distributiongamma functionprobability distributionsvariance
Best Answer
$Eg(X)=\int g(x)f_X(x)dx$. You don't have to find the distribution of $g(X)$ to find $Eg(X)$. So take $g(x)=\frac 1 {x^{2}}$ and see what happens to the integral. Once you know $E\frac 1{X{2}}$ and $E \frac 1 X$ you can see what the variance is.