added proof "gam-scal"

Author: JoramSoch

P/gam-scal.md

@@ -0,0 +1,77 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2023-11-24 13:08:34
+
+title: "Scaling of a random variable following the gamma distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Gamma distribution"
+theorem: "Scaling of a gamma random variable"
+
+sources:
+
+proof_id: "P426"
+shortcut: "gam-scal"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [gamma distribution](/D/gam) with shape $a$ and rate $b$:
+
+$$ \label{eq:gam}
+X \sim \mathrm{Gam}(a,b) \; .
+$$
+
+Then, the quantity $Y = c X$ will also be gamma-distributed with shape $a$ and rate $b/c$:
+
+$$ \label{eq:gam-scal}
+Y = b X \sim \mathrm{Gam}\left( a, \frac{b}{c} \right) \; .
+$$
+
+
+**Proof:**  Note that $Y$ is a function of $X$
+
+$$ \label{eq:Y-X}
+Y = g(X) = c X
+$$
+
+with the inverse function
+
+$$ \label{eq:X-Y}
+X = g^{-1}(Y) = \frac{1}{c} Y \; .
+$$
+
+Because the parameters of a gamma distribution [are positive](/D/gam), $c$ must also be positive. Thus, $g(X)$ is strictly increasing and we can calculate the [probability density function of a strictly increasing function](/P/pdf-sifct) as
+
+$$ \label{eq:pdf-sifct}
+f_Y(y) = \left\{
+\begin{array}{rl}
+f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; , & \text{if} \; y \in \mathcal{Y} \\
+0 \; , & \text{if} \; y \notin \mathcal{Y}
+\end{array}
+\right.
+$$
+
+The [probability density function of the gamma-distributed](/P/gam-pdf) $X$ is
+
+$$ \label{eq:gam-pdf}
+f_X(x) = \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x] \; .
+$$
+
+Applying \eqref{eq:pdf-sifct} to \eqref{eq:gam-pdf}, we have:
+
+$$ \label{eq:Y-pdf}
+\begin{split}
+f_Y(y) &= \frac{b^a}{\Gamma(a)} [g^{-1}(y)]^{a-1} \exp[-b g^{-1}(y)] \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \\
+&= \frac{b^a}{\Gamma(a)} \left( \frac{1}{c} y \right)^{a-1} \exp\left[-b \left( \frac{1}{c} y \right) \right] \, \frac{\mathrm{d}\left( \frac{1}{c} y \right)}{\mathrm{d}y} \\
+&= \frac{b^a}{\Gamma(a)} \left( \frac{1}{c} \right)^{a} \left( \frac{1}{c} \right)^{-1} y^{a-1} \exp\left[- \frac{b}{c} y \right] \, \frac{1}{c} \\
+&= \frac{(b/a)^a}{\Gamma(a)} y^{a-1} \exp\left[- \frac{b}{c} y \right]
+\end{split}
+$$
+
+which is the [probability density function](/D/pdf) of a [gamma distribution](/D/gam) with shape $a$ and rate $b/c$.