A non-negative continuous probability distribution indexed by two strictly positive parameters.

If a variable $X$ follows a Gamma distribution with shape parameter $k > 0$ and scale parameter $\theta$, then it is a continuous random variable with probability density function:

$$ p(x) = \frac{1}{\theta^k}\frac{1}{\Gamma(k)}x^{k-1}e^{-\frac{x}{\theta}} $$

It follows that $\mathbb{E}(X) = k \theta$ and ${\rm var}(X) = k \theta^2$.

In some texts, the Gamma distribution is parameterized by the rate parameter $\beta$, instead of the scale parameter $\theta$, of which it is the reciprocal, in other words: $\beta = 1/\theta$.

The exponential distribution ($k=1$) and the $\chi^2$ distribution (with $\nu$ degrees of freedom, $k=\frac{\nu}{2}$, $\theta = 2$) are both special cases of the Gamma distribution.

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