Find the expectation were X’s are independent $E(4/(X_1+X_2+X_3+X_4))$.

Question

Find the expectation where X’s are independent $E(4/(X_1+X_2+X_3+X_4))$. The $X_i$‘s are exponential with parameter $3$

Answer 1

Since the $X_i$ are $\operatorname{Exp}(\lambda=3)$ IIDs, $S:=\sum_{i=1}^4X_i\sim\Gamma(\alpha=4,\,\beta=3)$ has PDF $\frac{3^4}{6}s^3e^{-3s}$ on $[0,\,\infty)$. You want$$\int_0^\infty\frac{4\times 3^4}{6}s^2e^{-3s}ds=\frac{4\times 3^4}{6}\frac{2}{3^3}=4.$$