Let $X_1$, $X_2$ be two i.i.d. random variables and $X_1$ is uniformly distributed (discrete) on the set $\{1,2,3\}.$ Show that:
$$E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$$
Can someone give me a hint how to start?
Thanks in advance.
2026-04-01 19:09:49.1775070589
$X_1$, $X_2$ i.i.d RVs, $X_1$ is uniformly distributed. Show $E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$
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$$ 2E\left(\frac{X_1}{X_1+X_2}\right) = E\left(\frac{X_1}{X_1+X_2}\right) + E\left(\frac{X_2}{X_1+X_2}\right) = E\left(\frac{X_1+X_2}{X_1+X_2}\right) = 1 \\ \implies 2E\left(\frac{X_1}{X_1+X_2}\right) = 1 \iff E\left(\frac{X_1}{X_1+X_2}\right) = \frac{1}{2} $$
The proof uses the linearity of expectation.