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.

Read more about this topic:  Incomplete Gamma Function

Famous quotes containing the words functions, random and/or variables:

    “The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
    —Ralph Waldo Emerson (1803–1882)

    “Assemble, first, all casual bits and scraps
    That may shake down into a world perhaps;
    People this world, by chance created so,
    With random persons whom you do not know—”
    —Robert Graves (1895–1985)

    “Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
    —Paul Valéry (1871–1945)