# Approximating Maximum of $n$ iid Random Variables

I have $n$ i.i.d random variables from $X_1$ to $X_{50}$ each of these variables is a Poisson R.V with rate $\lambda=8$. I am interested to estimate the following probability $$P\left(\max(X_i) \lt 20\right)\space \text{i ranges from 1 to 50}$$
Obviously we can calculate the exact probability of $P(X_i < 20)$ using Poisson distribution and raise it to the power 50 to calculate the probability that each of $X_i$ assumes value less than 20. But is there a way to approximate this probability? More specifically, can the central limit theorem come into play? So far, what I have studied about central limit theorem is that the sum of $n$ i.i.d random variables is a Normal random variable. Is that true if we are interested in the max of $n$ i.i.d variables? Or is there an other way of approximation that might come in handy?

• CLT doesn't really say anything about extreme order statistics. There is something called Fisher-Tippett-Gnedenko theorem which is about something called "intermediate order statistics". You might want to have a look at de Haan's book "introduction to extreme value theory" for this sort of things. – Matias Heikkilä Oct 18 '16 at 13:04

The harder part is actually finding $P(X_i \lt 20)$ when $\lambda=8$. This can also be written as $P(X_i \le 19)$ and is about $0.9997471$
You then need this to be true for all $50$ independent cases so that the maximum is also less than $20$, and so you are looking for about $0.9997471^{50}$ which is about $0.9874311$. In R:
> ppois(19, lambda=8)^50

• A Gaussian approximation to the Poisson would not give a good result here: for example using something like pnorm(19.5, mean=8, sd=sqrt(8))^50 gives 0.9988043 – Henry Oct 18 '16 at 13:22