Probability density function of a strictly increasing function of a continuous random variable

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability density function ▷ Probability density function of strictly increasing function

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

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

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 increasing function is

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

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

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

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

$\label{eq:pdf-sifct-p2} \begin{split} f_Y(y) &\overset{\eqref{eq:pdf-cdf}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y) \\ &\overset{\eqref{eq:cdf-sifct}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_X(g^{-1}(y)) \\ &= f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; . \end{split}$

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

∎

Sources:

Taboga, Marco (2017): "Functions of random variables and their distribution"; in: Lectures on probability and mathematical statistics, retrieved on 2020-10-29; URL: https://www.statlect.com/fundamentals-of-probability/functions-of-random-variables-and-their-distribution#hid4.

Metadata: ID: P185 | shortcut: pdf-sifct | author: JoramSoch | date: 2020-10-29, 06:21.