# How do I calculate the ratio of a distribution of random variables [duplicate]

I'm quite lost with regards to solving this problem. Any help is appreciated. Thanks.

Suppose $X_1, X_2, . . .$ are independent discrete random variables, having the same distribution, and $E[X_i]>0$, for each $i$. Is it true that for any two positive integers $m$ and $n$

$E[\frac{X_1+...+X_m}{X_1+...+X_n}]=\frac{m}{n}$ ?

## marked as duplicate by NCh, GNUSupporter 8964民主女神 地下教會, Community♦Feb 15 '18 at 2:19

• Hint. Assuming that $Y_i = \frac{X_1}{X_1+\cdots+X_n}$ is integrable, you can check that $Y_i$’s have all the same distribution and sums up to $1$. – Sangchul Lee Feb 15 '18 at 1:05
• The condition $E[X_i]>0$ does not guarantee that the sum in the denominator cannot be zero. And in this case the fraction is undefined. – NCh Feb 15 '18 at 1:19
• @YOUSEFY Please find an expectation of r.v. $X$ with the following pmf: $P(X=0)=1/2=P(X=1)$. – NCh Feb 15 '18 at 14:17