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| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Joram Soch" |
| 6 | +affiliation: "BCCN Berlin" |
| 7 | +e_mail: "joram.soch@bccn-berlin.de" |
| 8 | +date: 2023-11-24 13:08:34 |
| 9 | + |
| 10 | +title: "Scaling of a random variable following the gamma distribution" |
| 11 | +chapter: "Probability Distributions" |
| 12 | +section: "Univariate continuous distributions" |
| 13 | +topic: "Gamma distribution" |
| 14 | +theorem: "Scaling of a gamma random variable" |
| 15 | + |
| 16 | +sources: |
| 17 | + |
| 18 | +proof_id: "P426" |
| 19 | +shortcut: "gam-scal" |
| 20 | +username: "JoramSoch" |
| 21 | +--- |
| 22 | + |
| 23 | + |
| 24 | +**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [gamma distribution](/D/gam) with shape $a$ and rate $b$: |
| 25 | + |
| 26 | +$$ \label{eq:gam} |
| 27 | +X \sim \mathrm{Gam}(a,b) \; . |
| 28 | +$$ |
| 29 | + |
| 30 | +Then, the quantity $Y = c X$ will also be gamma-distributed with shape $a$ and rate $b/c$: |
| 31 | + |
| 32 | +$$ \label{eq:gam-scal} |
| 33 | +Y = b X \sim \mathrm{Gam}\left( a, \frac{b}{c} \right) \; . |
| 34 | +$$ |
| 35 | + |
| 36 | + |
| 37 | +**Proof:** Note that $Y$ is a function of $X$ |
| 38 | + |
| 39 | +$$ \label{eq:Y-X} |
| 40 | +Y = g(X) = c X |
| 41 | +$$ |
| 42 | + |
| 43 | +with the inverse function |
| 44 | + |
| 45 | +$$ \label{eq:X-Y} |
| 46 | +X = g^{-1}(Y) = \frac{1}{c} Y \; . |
| 47 | +$$ |
| 48 | + |
| 49 | +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 |
| 50 | + |
| 51 | +$$ \label{eq:pdf-sifct} |
| 52 | +f_Y(y) = \left\{ |
| 53 | +\begin{array}{rl} |
| 54 | +f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; , & \text{if} \; y \in \mathcal{Y} \\ |
| 55 | +0 \; , & \text{if} \; y \notin \mathcal{Y} |
| 56 | +\end{array} |
| 57 | +\right. |
| 58 | +$$ |
| 59 | + |
| 60 | +The [probability density function of the gamma-distributed](/P/gam-pdf) $X$ is |
| 61 | + |
| 62 | +$$ \label{eq:gam-pdf} |
| 63 | +f_X(x) = \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x] \; . |
| 64 | +$$ |
| 65 | + |
| 66 | +Applying \eqref{eq:pdf-sifct} to \eqref{eq:gam-pdf}, we have: |
| 67 | + |
| 68 | +$$ \label{eq:Y-pdf} |
| 69 | +\begin{split} |
| 70 | +f_Y(y) &= \frac{b^a}{\Gamma(a)} [g^{-1}(y)]^{a-1} \exp[-b g^{-1}(y)] \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \\ |
| 71 | +&= \frac{b^a}{\Gamma(a)} \left( \frac{1}{c} y \right)^{a-1} \exp\left[-b \left( \frac{1}{c} y \right) \right] \, \frac{\mathrm{d}\left( \frac{1}{c} y \right)}{\mathrm{d}y} \\ |
| 72 | +&= \frac{b^a}{\Gamma(a)} \left( \frac{1}{c} \right)^{a} \left( \frac{1}{c} \right)^{-1} y^{a-1} \exp\left[- \frac{b}{c} y \right] \, \frac{1}{c} \\ |
| 73 | +&= \frac{(b/a)^a}{\Gamma(a)} y^{a-1} \exp\left[- \frac{b}{c} y \right] |
| 74 | +\end{split} |
| 75 | +$$ |
| 76 | + |
| 77 | +which is the [probability density function](/D/pdf) of a [gamma distribution](/D/gam) with shape $a$ and rate $b/c$. |
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