What is the expected maximal value of $n$ draw from a Poisson distribution?

Question

Let's draw $n$ values from a Poisson distribution with rate $r$. What is the expected maximal value?

Numerically, this can be estimated with this very simple R code

n = 300
r = 0.7

largest = 0
nbReps = 5e4
for (i in 1:nbReps)
{
    largest = largest + max(rpois(n,r))
}
print(largest / nbReps)

For $n=300$ and $r=0.7$, the answer seems to be around $4.062$

Answer 1

Let $M_n = \max(X_1,\ldots,X_n)$, where $X_i $ are iid to $X \sim \mathrm{Poisson}(r)$. Then $$\mathbb{P}(M_n \leq m) = [\mathbb{P}(X \leq m)]^n = \left( e^{-r}\sum_{k=0}^m \frac{r^k}{k!}\right)^n$$ So $$ \mathbb{P}(M_n = m) = \left( e^{-r}\sum_{k=0}^m \frac{r^k}{k!}\right)^n - \left( e^{-r}\sum_{k=0}^{m-1} \frac{r^k}{k!}\right)^n$$ $$ \mathbb{E}[M_n] = \sum_{m=0}^\infty m\mathbb{P}(M_n = m) = e^{-nr}\sum_{m=0}^\infty m\left[ \left( \sum_{k=0}^m \frac{r^k}{k!}\right)^n - \left( \sum_{k=0}^{m-1} \frac{r^k}{k!}\right)^n \right]$$ Not sure if this can be simplified further.