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:

    “Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
    —Friedrich Nietzsche (1844–1900)

    “Man always made, and still makes, grotesque blunders in selecting and measuring forces, taken at random from the heap, but he never made a mistake in the value he set on the whole, which he symbolized as unity and worshipped as God. To this day, his attitude towards it has never changed, though science can no longer give to force a name.”
    —Henry Brooks Adams (1838–1918)

    “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)