Formulas for cumulative Poisson and binomial probabilities

Steve Simon

2007-02-19

I am updating some material about Poisson regression and noticed that some of the tests and confidence intervals rely on a percentile from a Chi-squared distribution or a gamma distribution. In previous work on binomial confidence intervals

and

These can be found on page 127 and page 40

The Wikipedia entries on the Poisson distribution and the binomial distribution refer to the incomplete gamma function and the regularized incomplete beta function, respectively

[Update: March 21

The left side of the equation equals

and the right side of the equation equals

You can compute this by using integration by parts. If you let

then the integral simplifies to

or

Repeat the process again to get

and again and again until you get down to

A gamma distribution with shape parameter 1 is simply an exponential distribution and this last probability works out directly to equal