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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: 2020-11-06 05:30:00 |
| 9 | + |
| 10 | +title: "Probability density function of a strictly decreasing function of a continuous random variable" |
| 11 | +chapter: "General Theorems" |
| 12 | +section: "Probability theory" |
| 13 | +topic: "Probability functions" |
| 14 | +theorem: "Probability density function of strictly decreasing function" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Taboga, Marco" |
| 18 | + year: 2017 |
| 19 | + title: "Functions of random variables and their distribution" |
| 20 | + in: "Lectures on probability and mathematical statistics" |
| 21 | + pages: "retrieved on 2020-11-06" |
| 22 | + url: "https://www.statlect.com/fundamentals-of-probability/functions-of-random-variables-and-their-distribution#hid7" |
| 23 | + |
| 24 | +proof_id: "P188" |
| 25 | +shortcut: "pdf-sdfct" |
| 26 | +username: "JoramSoch" |
| 27 | +--- |
| 28 | + |
| 29 | + |
| 30 | +**Theorem:** Let $X$ be a [continuous](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$ and let $g(x)$ be a strictly decreasing function on the support of $X$. Then, the [probability density function](/D/pdf) of $Y = g(X)$ is given by |
| 31 | + |
| 32 | +$$ \label{eq:pdf-sdfct} |
| 33 | +f_Y(y) = \left\{ |
| 34 | +\begin{array}{rl} |
| 35 | +-f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; , & \text{if} \; y \in \mathcal{Y} \\ |
| 36 | +0 \; , & \text{if} \; y \notin \mathcal{Y} |
| 37 | +\end{array} |
| 38 | +\right. |
| 39 | +$$ |
| 40 | + |
| 41 | +where $g^{-1}(y)$ is the inverse function of $g(x)$ and $\mathcal{Y}$ is the set of possible outcomes of $Y$: |
| 42 | + |
| 43 | +$$ \label{eq:Y-range} |
| 44 | +\mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace \; . |
| 45 | +$$ |
| 46 | + |
| 47 | + |
| 48 | +**Proof:** The [cumulative distribution function of a strictly decreasing function](/P/cdf-sifct) is |
| 49 | + |
| 50 | +$$ \label{eq:cdf-sdfct} |
| 51 | +F_Y(y) = \left\{ |
| 52 | +\begin{array}{rl} |
| 53 | +1 \; , & \text{if} \; y > \mathrm{max}(\mathcal{Y}) \\ |
| 54 | +1 - F_X(g^{-1}(y)) + \mathrm{Pr}(X = g^{-1}(y)) \; , & \text{if} \; y \in \mathcal{Y} \\ |
| 55 | +0 \; , & \text{if} \; y < \mathrm{min}(\mathcal{Y}) |
| 56 | +\end{array} |
| 57 | +\right. |
| 58 | +$$ |
| 59 | + |
| 60 | +Note that [for continuous random variables, the probability](/D/pdf) of point events is |
| 61 | + |
| 62 | +$$ \label{eq:pdf-cont} |
| 63 | +\mathrm{Pr}(X = a) = \int_a^a f_X(x) \, \mathrm{d}x = 0 \; . |
| 64 | +$$ |
| 65 | + |
| 66 | +Because the [probability density function is the first derivative of the cumulative distribution function](/P/pdf-cdf) |
| 67 | + |
| 68 | +$$ \label{eq:pdf-cdf} |
| 69 | +f_X(x) = \frac{\mathrm{d}F_X(x)}{\mathrm{d}x} \; , |
| 70 | +$$ |
| 71 | + |
| 72 | +the [probability density function](/D/pdf) of $Y$ can be derived as follows: |
| 73 | + |
| 74 | +1) If $y$ does not belong to the support of $Y$, $F_Y(y)$ is constant, such that |
| 75 | + |
| 76 | +$$ \label{eq:pdf-sdfct-p1} |
| 77 | +f_Y(y) = 0, \quad \text{if} \quad y \notin \mathcal{Y} \; . |
| 78 | +$$ |
| 79 | + |
| 80 | +2) If $y$ belongs to the support of $Y$, then $f_Y(y)$ can be derived using the chain rule: |
| 81 | + |
| 82 | +$$ \label{eq:pdf-sdfct-p2} |
| 83 | +\begin{split} |
| 84 | +f_Y(y) &\overset{\eqref{eq:pdf-cdf}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y) \\ |
| 85 | +&\overset{\eqref{eq:cdf-sdfct}}{=} \frac{\mathrm{d}}{\mathrm{d}y} \left[ 1 - F_X(g^{-1}(y)) + \mathrm{Pr}(X = g^{-1}(y)) \right] \\ |
| 86 | +&\overset{\eqref{eq:pdf-cont}}{=} \frac{\mathrm{d}}{\mathrm{d}y} \left[ 1 - F_X(g^{-1}(y)) \right] \\ |
| 87 | +&= -\frac{\mathrm{d}}{\mathrm{d}y} F_X(g^{-1}(y)) \\ |
| 88 | +&= - f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; . |
| 89 | +\end{split} |
| 90 | +$$ |
| 91 | + |
| 92 | +Taking together \eqref{eq:pdf-sdfct-p1} and \eqref{eq:pdf-sdfct-p2}, eventually proves \eqref{eq:pdf-sdfct}. |
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