It is given that $X_1,X_2,X_3 \sim^{\text{i.i.d}} R(0,1)$. Find $E(\frac{X_1+X_2}{X_1+X_2+X_3})$. Now, I have tried this for a while and I somehow feel that if I can show that $\frac{X_1+X_2}{X_1+X_2+X_3} $ and $X_1+X_2+X_3$ are independent, then I can write $E(\frac{X_1+X_2}{X_1+X_2+X_3})=\frac{E(X_1+X_2)}{E(X_1+X_2+X_3)}$, after which it is easy. Please help!
2026-02-22 22:55:15.1771800915
$X_1,X_2,X_3 \sim^{\text{i.i.d}} R(0,1)$. Find $E(\frac{X_1+X_2}{X_1+X_2+X_3})$
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Hint: $$\mathsf E(\tfrac{X_1+X_2}{X_1+X_2+X_3}) ~{=\mathsf E(\tfrac{X_1}{X_1+X_2+X_3})+\mathsf E(\tfrac{X_2}{X_1+X_2+X_3})\\= 1-\mathsf E(\tfrac{X_3}{X_1+X_2+X_3})}$$
Now, $X_1,X_2,X_3$ are identically and independently distributed so...