2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Matias Heikkilä Oct 18 '16 at 13:04
2
$\begingroup$

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
[1] 0.9874311
$\endgroup$
  • $\begingroup$ 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 $\endgroup$ – Henry Oct 18 '16 at 13:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.