Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Gamma distribution ▷ Relationship to standard gamma distribution

Theorem: Let $X$ be a random variable following a gamma distribution with shape $a$ and rate $b$:

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

Then, the quantity $Y = b X$ will have a standard gamma distribution with shape $a$ and rate $1$:

\[\label{eq:Y-snorm} Y = b X \sim \mathrm{Gam}(a,1) \; .\]

Proof: Note that $Y$ is a function of $X$

\[\label{eq:Y-X} Y = g(X) = b X\]

with the inverse function

\[\label{eq:X-Y} X = g^{-1}(Y) = \frac{1}{b} Y \; .\]

Because $b$ is positive, $g(X)$ is strictly increasing and we can calculate the cumulative distribution function of a strictly increasing function as

\[\label{eq:cdf-sifct} F_Y(y) = \left\{ \begin{array}{rl} 0 \; , & \text{if} \; y < \mathrm{min}(\mathcal{Y}) \\ F_X(g^{-1}(y)) \; , & \text{if} \; y \in \mathcal{Y} \\ 1 \; , & \text{if} \; y > \mathrm{max}(\mathcal{Y}) \; . \end{array} \right.\]

The cumulative distribution function of the gamma-distributed $X$ is

\[\label{eq:gam-cdf} F_X(x) = \int_{-\infty}^{x} \frac{b^a}{\Gamma(a)} t^{a-1} \exp[-b t] \, \mathrm{d}t \; .\]

Applying \eqref{eq:cdf-sifct} to \eqref{eq:gam-cdf}, we have:

\[\label{eq:Y-cdf-s1} \begin{split} F_Y(y) &\overset{\eqref{eq:cdf-sifct}}{=} F_X(g^{-1}(y)) \\ &\overset{\eqref{eq:gam-cdf}}{=} \int_{-\infty}^{y/b} \frac{b^a}{\Gamma(a)} t^{a-1} \exp[-b t] \, \mathrm{d}t \; . \end{split}\]

Substituting $s = b t$, such that $t = s/b$, we obtain

\[\label{eq:Z-cdf-s2} \begin{split} F_Y(y) &= \int_{-b \infty}^{b (y/b)} \frac{b^a}{\Gamma(a)} \left(\frac{s}{b}\right)^{a-1} \exp\left[-b \left(\frac{s}{b}\right)\right] \, \mathrm{d}\left(\frac{s}{b}\right) \\ &= \int_{-\infty}^{y} \frac{b^a}{\Gamma(a)} \, \frac{1}{b^{a-1} \, b} \, s^{a-1} \exp[-s] \, \mathrm{d}s \\ &= \int_{-\infty}^{y} \frac{1}{\Gamma(a)} s^{a-1} \exp[-s] \, \mathrm{d}s \end{split}\]

which is the cumulative distribution function of the standard gamma distribution.

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Metadata: ID: P112 | shortcut: gam-sgam | author: JoramSoch | date: 2020-05-26, 23:14.