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:

    “When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
    —Edward T. Hall (b. 1914)

    “We should stop looking to law to provide the final answer.... Law cannot save us from ourselves.... We have to go out and try to accomplish our goals and resolve disagreements by doing what we think is right. That energy and resourcefulness, not millions of legal cubicles, is what was great about America. Let judgment and personal conviction be important again.”
    —Philip K. Howard, U.S. lawyer. The Death of Common Sense: How Law Is Suffocating America, pp. 186-87, Random House (1994)

    “The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
    —George Steiner (b. 1929)