This is a question from my Stochastic Processes class that I'm having a hard time figuring out. Does anyone know how to solve?
Let $X_{n1},....,X_{nn}$ be independent Poisson random variables with mean $\frac 1n$ Then X = $X_{n1}+...+X_{nn}$, is a Poisson random variable with mean 1. Let $M_n$ = max{$X_{n1},...,X_{nn}$}.
Find $$\lim_{n\to\infty} (\mathbb{P}M_n > \frac 12)$$ Picture of problem from textbook linked here.
The $\frac12$ is a bit of a red herring here, in that the problem remains the same if $\frac12$ is replaced by any $c$ with $0<c<1$. What you are looking for is the probability that one or more of the $X_{nk} \geq 1$, which is $1$ minus the probability that all of the $X_{nk} < 1$.
The only value that the (discrete) Poisson distribution can take on which is less than $1$ is $0$, and the probability that any given variable is $0$ is given by $$ e^{-\frac1n}\frac{(1/n)^0}{0!} = e^{-\frac1n} $$ So the probability that all $n$ are equal to zero is $$ \prod_{k=1}^n e^{-\frac1n} = e^{-1} $$ Thus the probability that the maximum is at least 1 is $$ 1 - e^{-1} $$