Incomplete Gamma Function - Regularized Gamma Functions and Poisson Random Variables

Regularized Gamma Functions and Poisson Random Variables

Two related functions are the regularized Gamma functions:

is the cumulative distribution function for Gamma random variables with shape parameter and scale parameter 1.

When is an integer, is the cumulative distribution function for Poisson random variables: If is a random variable then

Pr(X<s) = \sum_{i<s} e^{-\lambda} \frac{\lambda^i}{i!} = \frac{\Gamma(s,\lambda)}{\Gamma(s)} = Q(s,\lambda).

This formula can be derived by repeated integration by parts.

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