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Please, check my work. Is it correct that if $X_1,X_2,\ldots,X_n$ are independent Poisson random variables, each with a parameter $\lambda$, then $$ E\left( e^{-\,\frac{X_1+X_2+\ldots+X_n}{n}}\right)=e^{-n\lambda(e^{-\,\frac{1}{n}}+1)}? $$

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$$E\left(e^{-\frac{X_1+\cdots+X_n}{n}}\right)=E\left(\prod_{i=1}^ne^{-\frac{X_i}{n}}\right)=E\left(e^{-\frac{X_1}{n}}\right)^n$$ as the $X_i$'s are all independent. $E(e^{-X_1/n})=\sum_{k=0}^{\infty}e^{-k/n}e^{-\lambda}\frac{\lambda^k}{k!}=e^{-\lambda}e^{\lambda e^{-1/n}}=e^{\lambda(1-e^{-1/n})}$ $$E\left(e^{-\frac{X_1+\cdots+X_n}{n}}\right)=e^{n\lambda(1-e^{-1/n})}$$

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