Gamma Distribution

The gamma distribution is a continuous probability distribution defined on the positive real numbers, characterized by a shape parameter and a rate or scale parameter, whose density is built directly from the gamma function, which supplies the normalizing constant that makes the total probability equal to one. It generalizes several simpler and more specific distributions: with a shape parameter equal to one, the gamma distribution reduces exactly to the exponential distribution, used to model the waiting time until a single random event, while summing several independent exponential waiting times, or taking an integer shape parameter more generally, produces the Erlang distribution, first studied by the Danish engineer Agner Krarup Erlang in the early twentieth century in his foundational work on telephone traffic. Because it can take on a wide range of shapes, from sharply peaked near zero to broadly spread out, the gamma distribution is widely used to model waiting times, insurance claim sizes, rainfall amounts, and other positive quantities whose variability does not follow the symmetric bell shape of the normal distribution. It also plays a central role in Bayesian statistics, where the gamma distribution is the conjugate prior most commonly paired with data assumed to follow a Poisson or exponential distribution, meaning that combining a gamma prior with such data produces another gamma distribution as the updated result.