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$X$ is the sum of $n$ independent Poisson random variables with parameter 1. Therefore $X$ has a Poisson distribution with parameter $n$. Use the central limit theorem to show that $P(X≤n)→(1/2)$.

I was able to prove that $X$ has a Poisson distribution with parameter $n$, but I'm not sure how to use this and the central limit theorem to show the converge of the probability above.

Any help/guidance would be wonderdul!

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  • $\begingroup$ Are you sure that $P(X\le n)\to 1/2$ as $n\to\infty$? Because $P(X\le n)=P(n^{-1}\sum_{i=1}^nY_i-1\le 0)$ and $n^{-1}\sum_{i=1}^nY_i\to1$ almost surely using the strong law of large numbers, where $Y_1,\ldots,Y_n$ are iid Poisson random variables with parameter $1$. $\endgroup$ – Cm7F7Bb Mar 21 '16 at 8:20
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Central limit theorem says that the mean of the sum of any large collection of random variables with finite variance will approach a normal distribution.

For any normally distributed $Z$ with mean $\mu$, $P(Z\leq \mu) = 1/2$

As $X$ gets large it resembles a normally distributed random variable with mean $n.$

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