Given that we have $\displaystyle X_{1}, ... , X_{n} \sim iid G$ where $\displaystyle G$ is unknown.

Find$\displaystyle V(\overline{X} / S)$ Where $\displaystyle X$ is the sample Mean, $\displaystyle S^2$ is the sample variance and $\displaystyle V( \cdot )$ is the variance.

So...

$\displaystyle V(\overline{X} / S) = E[\overline{X}^2 / S^2] - E[\overline{X} / S]^2$

Since $\displaystyle \overline{X}$ and $\displaystyle S^2$ are independent,

$\displaystyle V(\overline{X} / S) = E[\overline{X}^2]E[1 / S^2] - E[\overline{X} / S]^2$

We were having a real hard time finding $\displaystyle E[1 / S^2]$. Also, can the $\displaystyle E[\overline{X} / S]^2$ be broken up in a similar way to the other term?

Is there another way to approach this problem?