Problem based on Poisson distribution

Question

It is required to obtain the following:

$\lim_{n\rightarrow \infty}e^{-n}\sum_{j=n}^{4n}\frac{n^j}{j!}$

The only thing that I understand here is that the expression inside the limit is a pmf of Poisson distribution but not able to calculate this.

I made a substitution $x=j-n$ also but that didn't work either.

Any help?

Thanks

Answer 1

Let $X_n$ have Poisson distribution with parameter $n$. Let $Y_n=\frac {X_n -n} {\sqrt n}$. It is known that $Y_n$ converges to standard normal distribution. The given probability is $P\{n \leq X_n \leq 4n\} =P\{0 \leq Y_n \leq 3\sqrt n\}$ so the limit is $\frac 1 2$.