Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability functions ▷ Probability density function of strictly decreasing function

Theorem: Let $X$ be a continuous random variable with possible outcomes $\mathcal{X}$ and let $g(x)$ be a strictly decreasing function on the support of $X$. Then, the probability density function of $Y = g(X)$ is given by

\[\label{eq:pdf-sdfct} f_Y(y) = \left\{ \begin{array}{rl} -f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; , & \text{if} \; y \in \mathcal{Y} \\ 0 \; , & \text{if} \; y \notin \mathcal{Y} \end{array} \right.\]

where $g^{-1}(y)$ is the inverse function of $g(x)$ and $\mathcal{Y}$ is the set of possible outcomes of $Y$:

\[\label{eq:Y-range} \mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace \; .\]

Proof: The cumulative distribution function of a strictly decreasing function is

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

Note that for continuous random variables, the probability of point events is

\[\label{eq:pdf-cont} \mathrm{Pr}(X = a) = \int_a^a f_X(x) \, \mathrm{d}x = 0 \; .\]

Because the probability density function is the first derivative of the cumulative distribution function

\[\label{eq:pdf-cdf} f_X(x) = \frac{\mathrm{d}F_X(x)}{\mathrm{d}x} \; ,\]

the probability density function of $Y$ can be derived as follows:

1) If $y$ does not belong to the support of $Y$, $F_Y(y)$ is constant, such that

\[\label{eq:pdf-sdfct-p1} f_Y(y) = 0, \quad \text{if} \quad y \notin \mathcal{Y} \; .\]

2) If $y$ belongs to the support of $Y$, then $f_Y(y)$ can be derived using the chain rule:

\[\label{eq:pdf-sdfct-p2} \begin{split} f_Y(y) &\overset{\eqref{eq:pdf-cdf}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y) \\ &\overset{\eqref{eq:cdf-sdfct}}{=} \frac{\mathrm{d}}{\mathrm{d}y} \left[ 1 - F_X(g^{-1}(y)) + \mathrm{Pr}(X = g^{-1}(y)) \right] \\ &\overset{\eqref{eq:pdf-cont}}{=} \frac{\mathrm{d}}{\mathrm{d}y} \left[ 1 - F_X(g^{-1}(y)) \right] \\ &= -\frac{\mathrm{d}}{\mathrm{d}y} F_X(g^{-1}(y)) \\ &= - f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; . \end{split}\]

Taking together \eqref{eq:pdf-sdfct-p1} and \eqref{eq:pdf-sdfct-p2}, eventually proves \eqref{eq:pdf-sdfct}.

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Metadata: ID: P188 | shortcut: pdf-sdfct | author: JoramSoch | date: 2020-11-06, 05:30.