Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Gamma distribution ▷ Median

Theorem: Let $X$ be a random variable following a gamma distribution:

\[\label{eq:gam} X \sim \mathrm{Gam}(a, b) \; .\]

Then, the median of $X$ is

\[\label{eq:gam-med} \mathrm{median}(X) = \frac{1}{b} \gamma_a^{-1}\left( \frac{\Gamma(a)}{2} \right)\]

where $\Gamma(x)$ is the gamma function and $\gamma_s^{-1}(y)$ is the inverse function of the lower incomplete gamma function $\gamma(s,x)$ in which the first argument is $s$.

Proof: The median is the value at which the cumulative distribution function is $1/2$:

\[\label{eq:median} F_X(\mathrm{median}(X)) = \frac{1}{2} \; .\]

The cumulative distribution function of the gamma distribution is

\[\label{eq:gam-cdf} F_X(x) = \frac{\gamma(a,bx)}{\Gamma(a)}\]

where the lower incomplete gamma function $\gamma(s,x)$ is given by

\[\label{eq:low-inc-gam-fct} \gamma(s,x) = \int_0^x t^{s-1} \exp(-t) \, \mathrm{d}t \; .\]

Thus, the inverse CDF, i.e. the quantile function, is

\[\label{eq:gam-cdf-inv} x = \frac{1}{b} \gamma_a^{-1}(p \Gamma(a))\]

where $\gamma_a^{-1}(y)$ is the inverse function of $\gamma(a,x)$. Setting $p = 1/2$, we obtain:

\[\label{eq:gam-med-qed} \mathrm{median}(X) = \frac{1}{b} \gamma_a^{-1}\left( \frac{\Gamma(a)}{2} \right) \; .\]
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Metadata: ID: P518 | shortcut: gam-med | author: JoramSoch | date: 2025-10-24, 13:32.