Expected value of the ratio

Question

Suppose $X_1, X_2, ..., X_n$ are i.i.d.r.v., where $X_i>0$ and $EX_i=\mu$, $Var(X_i)=\sigma^2$. Find $E\left(\frac{X_1+X_2+...+X_m}{X_1+X_2+...+X_n}\right)$, if $m<n$.

Any ideas how to even start this problem?

Answer 1

$S_n = X_1 + \dots + X_n$

$\mathbb E$ is linear: $1=\mathbb E(\frac {S_n} {S_n}) = n\mathbb E(\frac{X_1} S) $

Therefore: $\mathbb E(\frac{X_1} {S_n})= \frac 1n $

$E(\frac{S_m} {S_n}) = m\mathbb E(\frac{X_1} S) = \frac mn$