[StATS]: Formulas for cumulative Poisson and binomial probabilities (created 20070219).
I am updating some material about Poisson regression and noticed that some of the tests and confidence intervals rely on a percentile from a Chisquared distribution or a gamma distribution. In previous work on binomial confidence intervals
 I had noticed the use of a beta distribution and an F distribution. It seems odd to apply percentiles from continuous distributions for confidence intervals involving counting
 but the formulas do indeed work. There are well known relationships for the cumulative distributions of the Poisson and binomial distributions that lead to these formulas.
and
These can be found on page 127 and page 40

respectively of

Statistical Distributions Second Edition. Merran Evans

Nicholas Hastings

Brian Peacock (1993) New York: John Wiley & Sons. [BookFinder4U link]](http://www.bookfinder4u.com/detail/0471559512.html)
The Wikipedia entries on the Poisson distribution and the binomial distribution refer to the incomplete gamma function and the regularized incomplete beta function, respectively
 and this is
 I suspect
 another way of deriving the above relationships.
[Update: March 21
 2007] The relationship between the Poisson and the Chisquared random variable is fairly easy to show if you recognize the relationship between the Chisquared distribution and the Gamma distribution. The first equation above can be rewritten as
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
This page was written by Steve Simon while working at Children’s Mercy Hospital. Although I do not hold the copyright for this material
 I am reproducing it here as a service
 as it is no longer available on the Children’s Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Poisson regression or Category: Probability concepts.
Probability concepts](../category/ProbabilityConcepts.html). regression](../category/PoissonRegression.html) or [Category: for pages similar to this one at [Category: Poisson with general help resources. You can also browse Children’s Mercy Hospital website. Need more information? I have a page reproducing it here as a service
 as it is no longer available on the Hospital. Although I do not hold the copyright for this material
 I am This page was written by Steve Simon while working at Children’s Mercy