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Can anyone help me with it: Using the central limit theorem for suitable Poisson random variables, prove that $$ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$ Thanks!

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    This question has been asked and answered [countless](http://math.stackexchange.com/questions/1297553) times.2015-12-07

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Hint: A Poisson$(n)$ random variable can be represented as the sum of $n$ i.i.d. Poisson$(1)$ rv's.

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    I'll give further hints if you wish.2011-02-13
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    @Shai Covo: Can you show me the steps? Thanks!2011-02-13
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    @kira: Hint 2: Move the exponent into the sum.2011-02-13
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    @kira: Find a random variable $X$ for which it holds ${\rm P}(X=n) = \sum\nolimits_{k = 0}^n {\frac{{e^{ - n} n^k }}{{k!}}} $.2011-02-14