 Gamma distribution

The Gamma distribution with shape $a>0$ and rate $r>0$ has probability density functionThe gamma function normalizes its integral to one. Its mean is $a/r$ and its variance is $a/r^2$, obtained by integrating powers of $x$. At $a=1$ it is the exponential distribution; integer $a$ gives the waiting time to the $a$th event of a rate-$r$ Poisson process. Independent gamma variables with the same rate add their shapes. Some sources use scale $1/r$ instead of rate, so the parameter convention must accompany the formula.

The Gamma distribution is a continuous probability distribution defined by two parameters: shape (often denoted as \( k \) or \( \alpha \)) and scale (denoted as \( \theta \) or \( \beta \)). It is widely used in various fields, including statistics, finance, and engineering, due to its ability to model waiting times and processes that are characterized by events that occur independently at a constant average rate.