Gamma distribution

The gamma distribution is a continuous probability distribution on positive values, parameterized by shape and scale. It generalizes the exponential and chi-squared distributions.

Gamma function: A function extending factorials to non-integer arguments, defined for positive real inputs by an improper integral. It normalizes the gamma density through the factor Γ(shape).
Poisson process: A counting process with independent increments and a constant event rate. Waiting for a fixed number of Poisson events produces a gamma-distributed time.
Chi-squared distribution: A distribution of the sum of squares of independent standard normal random variables. It is gamma with shape equal to half the degrees of freedom and scale two.
Rate parameterization: A convention that describes an exponential-family distribution using the reciprocal of its scale parameter. Some gamma formulas use rate instead of scale, changing the density’s written form.
Shape parameter: A parameter that controls a distribution’s form, including its skewness and behavior near a boundary. In the gamma distribution, shape determines whether the density peaks away from zero.
Erlang distribution: A gamma distribution with positive integer shape, used for sums of exponential waiting times. It models time until a specified number of events in a Poisson process.
Beta distribution: A continuous distribution on the interval from zero to one, controlled by two positive shape parameters. Normalizing two independent gamma variables produces a beta-distributed proportion.
Exponential family: A broad class of probability distributions whose densities have a particular factorized form in parameters and data. The gamma distribution belongs to this family, enabling convenient inference and sufficient statistics.
Scale parameter: A positive parameter that stretches a probability distribution along its measurement axis. Gamma scale changes the distribution’s spread and mean without changing its shape.
Bayesian statistics: A statistical framework that updates probability distributions for unknown quantities using observed data. Gamma distributions serve as conjugate priors for rates in Poisson and exponential models.

Linked from 7 pages

Method of momentsRelated: Its mean and variance provide two equations for estimating shape and scale.