Inverse Gamma distribution

Story

If $Y$ is Gamma distributed, then $1 / Y$ is Inverse Gamma distributed.

The number of arrivals, $α$ , and the rate of arrivals, $β$ .

The Inverse Gamma distribution is supported on the set of positive real numbers.

\begin{array}{r} f (y; α, β) = \frac{1}{Γ (α)} \frac{β^{α}}{y^{α + 1}} e^{- β / y}, \end{array}

where $Γ (α)$ is the gamma function.

\begin{array}{r} F (y; α, β) = Q (α, β / x), \end{array}

Mean: $\frac{β}{α - 1}$ for $α > 1$ ; for $α \leq 1$ , the mean is undefined.

Variance: $\frac{β^{2}}{(α - 1)^{2} (α - 2)}$ for $α > 2$ ; for $α \leq 2$ , the variance is undefined.

Package	Syntax
NumPy	`1 / rng.gamma(alpha, 1/beta)`
SciPy	`scipy.stats.invgamma(alpha, loc=0, scale=beta)`
Distributions.jl	`InverseGamma(alpha, beta)`
Stan	`inv_gamma(alpha, beta)`

The Inverse Gamma distribution is useful as a prior for positive parameters. It imparts a quite heavy tail and keeps probability further from zero than the Gamma distribution.
NumPy module does not have a function to sample directly from the Inverse Gamma distribution, but it can be achieved by sampling out of a Gamma distribution and then taking the inverser, as shown in the NumPy usage above.