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

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

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

Then, the cumulative distribution function of $X$ is

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

where $\Gamma(x)$ is the gamma function and $\gamma(s,x)$ is the lower incomplete gamma function.

Proof: The probability density function of the gamma distribution is:

\[\label{eq:gam-pdf} f_X(x) = \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x] \; .\]

Thus, the cumulative distribution function is:

\[\label{eq:gam-cdf-s1} \begin{split} F_X(x) &= \int_{0}^{x} \mathrm{Gam}(z; a, b) \, \mathrm{d}z \\ &= \int_{0}^{x} \frac{b^a}{\Gamma(a)} z^{a-1} \exp[-b z] \, \mathrm{d}z \\ &= \frac{b^a}{\Gamma(a)} \int_{0}^{x} z^{a-1} \exp[-b z] \, \mathrm{d}z \; . \end{split}\]

Substituting $t = b z$, i.e. $z = t/b$, this becomes:

\[\label{eq:gam-cdf-s2} \begin{split} F_X(x) &= \frac{b^a}{\Gamma(a)} \int_{b \cdot 0}^{b x} \left(\frac{t}{b}\right)^{a-1} \exp\left[-b \left(\frac{t}{b}\right)\right] \, \mathrm{d}\left(\frac{t}{b}\right) \\ &= \frac{b^a}{\Gamma(a)} \cdot \frac{1}{b^{a-1}} \cdot \frac{1}{b} \int_{0}^{b x} t^{a-1} \exp[-t] \, \mathrm{d}t \\ &= \frac{1}{\Gamma(a)} \int_{0}^{b x} t^{a-1} \exp[-t] \, \mathrm{d}t \; . \end{split}\]

With the definition of the lower incomplete gamma function

\[\label{eq:low-inc-gam-fct} \gamma(s,x) = \int_{0}^{x} t^{s-1} \exp[-t] \, \mathrm{d}t \; ,\]

we arrive at the final result given by equation \eqref{eq:gam-cdf}:

\[\label{eq:gam-cdf-qed} F_X(x) = \frac{\gamma(a,bx)}{\Gamma(a)} \; .\]
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Metadata: ID: P178 | shortcut: gam-cdf | author: JoramSoch | date: 2020-10-15, 12:34.