Gamma Distribution Quiz

The two parameterizations of the gamma distribution in common use are: 1) With a shape parameter $k$ and a scale parameter $\theta$, and 2) With a shape parameter $\alpha = k$ and an inverse scale parameter $\beta = 1 / \theta$, called a rate parameter.

In both parameterizations, the shape parameter and the scale parameters are positive real numbers.

The gamma distribution is the maximum entropy probability distribution for a random variable $X$ for which $E[X] = k\theta = \alpha/\beta$ is fixed and greater than zero, and $E[\ln(X)] = \psi(k) + \ln(\theta) = \psi(\alpha) - \ln(\beta)$ is fixed (where $\psi$ is the digamma function).

The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution.

The relationship between the shape parameter $k$ and the rate parameter $\beta$ is given by $\alpha = k$ and $\beta = 1 / \theta$.