I have to calculate the limit of the sequence $E((\frac{X_n-n}{\sqrt{n}})^+)$, where $X_n$ is Poi(n) and $\xi^+$ means "Maximum of $\xi$ and $0$". I get $e^2/\sqrt{2\pi}$ and I am not sure that this is correct. My calculation is straightforward, using Stirling formula at the end. It is rather long so I don't really want to place it here. Perhaps somebody has a better idea? Thanks in advance!
2026-03-29 19:44:46.1774813486
Expected value for random variables based on Poisson distribution
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Hint: Let $(Y_n)$ be i.i.d. with $Poss(1)$ distribution. The $X_n$ has same distribution as $Y_1+Y_2+...+Y_n$. Now just apply CLT to find the limit.
You will need the following: If $Z_n \to Z$ in distribution and $EZ_n^{2}$ is bounded the $EZ_n \to $EZ$.