# 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.

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

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.

• I know the answer to the question. But I need to learn the method behind calculating it. Not just having it computed. Thanks for the title edit though, that is correct :) – user100002 Dec 3 '15 at 12:20
• Ah. But the method will usually be just to add Poisson probabilities (and having a computer do it). Are you looking for the Poisson probability mass function? That is $$\frac{\lambda^k}{k!}e^{-\lambda}$$ for parameter $\lambda$ and outcome $k$ - you'd add this up for your $\lambda=40$ and $k=0, \dots, 50$. Or are you in fact interested in the numerics? I think I may not be understanding your question completely... – Stephan Kolassa Dec 3 '15 at 12:30
• Well in this case, it's an exercise we are supposed to do with pen and paper and a simple calculator. So using advanced computing like ppois functions isn't an option. It's still possible to add them together manually of course, but that doesn't seem right with such large numbers. So I just assumed there was another way to calculate it? The actual problem. Is a sample of 100 within a time interval, where 40% is the average success rate, and I need to figure out the probability of 50% or more. So the data should be correct to do it? Unless I'm missing something... – user100002 Dec 3 '15 at 12:37
• I wouldn't use a Poisson distribution for a success rate. The Poisson is discrete and unbounded. You'd rather use a beta distribution, which is bounded to the interval $[0,1]$. Or, if you explicitly have 100 trials (and not only success rates), you may want to look at the binomial distribution, which may be enlightening. – Stephan Kolassa Dec 3 '15 at 12:42
• Are you allowed to use another distribution method when you are given a time interval? It states specifically that you normally get 100 contacts within a week. It doesn't just say 100 trials. It makes it pretty clear that it's Poisson distributed? – user100002 Dec 3 '15 at 13:14