$X$ 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!
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 $n.$