Finding very high cumulative Poisson probabilities? I have to find the probability of a Poisson distributed problem where the average rate of success is 40 and I need to find the probability of something being > 50.
Normally I'd just either add the probabilities together manually or look at Poisson probability tables.
But adding 50 probability calculations together manually takes a while. And the tables only go up to 15.
So how do you find the probability when you are dealing with "large" numbers.
 A: There's a direct connection between the Poisson and the chi-squared distributions.
See for example, the first few lines of Johnson (1959) [1] which gives a direct derivation (before extending what was then already a known result). This fact can also be seen in the Wikipedia article on the Poisson distribution.
If I have this right, given $Y\sim \text{Pois}(\lambda)$ then for $X\sim$ chi-squared with $\nu=2y$, we have $P(Y\geq y) = P(X<2\lambda)$.

If the Poisson mean is not too small and a reasonably close approximation suffices, the Anscombe transformation can often be useful.
[1] N. L. Johnson (1959),
"On an Extension of the Connexion Between Poisson and χ2 Distributions,"
Biometrika, Vol. 46, No. 3/4 (Dec.), pp. 352-363 
A: I wouldn't necessarily call this "very high" probabilities. I'd reserve that label for situations where we have to think about numerical accuracy, which doesn't really come into play here yet.
The solution in your case: install R and type:
> 1-ppois(50,40,lower.tail=TRUE)
[1] 0.05262805

Look at ?ppois for the help page.
