Let $X_1,...,X_n$ be positive-integer-valued random variables with the same distribution. Then:

$$E\left[\frac{X_i}{X_1+...+X_n}\right]=E\left[\frac{X_j}{X_1+...+X_n}\right]$$ $\forall 1\le i,j\le n, i\ne j$.

I know that it must be true but I don't see why this is implied by $E[X_i]=E[X_j]$. Can anyone prove it? Thanks in advance!

  • $\begingroup$ The result is basically saying chose any random variable then the expectation of that over the sun will be $E$. So it doesn't matter which random variable you choose. $\endgroup$ – Karn Watcharasupat Dec 12 '17 at 17:11

Note that $$ \frac{X_i}{X_1+...+X_n}\stackrel{d}{=}\frac{X_j}{X_1+...+X_n} $$ since the $X_k$ are identically distributed. The result follows.

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