Linear functions of a random variable

In this article, we consider linear functions of a random variable. That is, given a random variable $X$ , we consider functions of the form $Y = aX+b$ .

First, let us consider discrete random variables. Given a discrete random variable $X$ , we see how the probability mass function and the cumulative distribution function of $Y = aX+b$ are related to those of $X$ .

As is usual, let $p_X(x)$ and $F_X(x)$ denote the probability mass function and the cumulative distribution function respectively of $X$ . Then, we have:

$\begin{aligned} p_Y(x) & = P\{Y = x\} \\ & = P\{aX+b = x\} \\ & = P\left\{X = \frac{x-b}{a}\right\} \\ & = p_X\left(\frac{x-b}{a}\right) \end{aligned}$

Thus, the probability mass function of $X$ is horizontally stretched by a factor $a$ and then right-shifted by $b$ to give the probability mass function of $aX+b$ . Note that if $a < 0$ , a horizontal stretch by a factor of $a$ implies a horizontal stretch by $|a|$ and a reflection about the $Y$ -axis.

To determine $F_Y(x)$ , let us first consider the case of $a > 0$ .

$\begin{aligned} F_Y(x) & = P\{Y \le x\} \\ & = P\{aX+b \le x\} \\ & = P\left\{X \le \frac{x-b}{a}\right\} \\ & = F_X\left(\frac{x-b}{a}\right) \end{aligned}$

Thus, when $a > 0$ , the cumulative distribution function of $X$ is horizontally stretched by a factor $a$ and then right-shifted by $b$ to give the cumulative distribution function of $aX+b$ .

Let us now consider the case of $a < 0$ .

$\begin{aligned} F_Y(x) & = P\{Y \le x\} \\ & = P\{aX+b \le x\} \\ & = P\left\{X \ge \frac{x-b}{a}\right\} \\ & = P\left\{X = \frac{x-b}{a}\right\} + P\left\{X > \frac{x-b}{a}\right\} \\ & = p_X\left(\frac{x-b}{a}\right) + 1 - P\left\{X \le \frac{x-b}{a}\right\} \\ & = p_Y(x) + 1 - F_X\left(\frac{x-b}{a}\right) \end{aligned}$

Now, let us turn to continuous random variables. Given a continuous random variable $X$ , we see how the probability density function and the cumulative distribution function of $Y = aX+b$ are related to those of X.

As is usual, let $f_X(x)$ and $F_X(x)$ denote the probability density function and the cumulative distribution function respectively of $X$ .

For small $\epsilon > 0$ , we know that $P\{x\le X \le x+\epsilon\} \approx \epsilon f_X(x)$ .

For the case of $a > 0$ , we have:

$\begin{aligned} \epsilon f_Y(x) & \approx P\{x \le Y \le x+\epsilon\} \\ & = P\{x \le aX+b \le x+\epsilon\} \\ & = P\left\{\frac{x-b}{a}\le X \le \frac{x+\epsilon - b}{a}\right\} \\ & = P\left\{\frac{x-b}{a}\le X \le \frac{x - b}{a}+\frac{\epsilon}{a}\right\} \\ & \approx \frac{\epsilon}{a}f_X\left(\frac{x-b}{a}\right) \end{aligned}$

Thus, for $a > 0$ , we have $f_Y(x) = \frac{1}{a}f_X\left(\frac{x-b}{a}\right)$ .

For the case of $a < 0$ , we have:

$\begin{aligned} \epsilon f_Y(x) & \approx P\{x \le Y \le x+\epsilon\} \\ & = P\{x \le aX+b \le x+\epsilon\} \\ & = P\left\{\frac{x+\epsilon-b}{a}\le X \le \frac{x - b}{a}\right\} \\ & = P\left\{\frac{x-b}{a}+\frac{\epsilon}{a}\le X \le \frac{x - b}{a}\right\} \\ & \approx -\frac{\epsilon}{a}f_X\left(\frac{x-b}{a}\right) \end{aligned}$

Thus, for $a < 0$ , we have $f_Y(x) = -\frac{1}{a}f_X\left(\frac{x-b}{a}\right)$ .

We combine the two cases of $a > 0$ and $a < 0$ and say that for any $a \ne 0$ , $f_Y(x) = \frac{1}{|a|}f_X\left(\frac{x-b}{a}\right)$ .

Thus, the probability density function of $X$ is horizontally stretched by a factor $a$ , right-shifted by $b$ and then, vertically shrunk by a factor $|a|$ to give the probability density function of $aX+b$ . Note that, if $a < 0$ , a horizontal stretch by a factor of $a$ implies a horizontal stretch by $|a|$ and a reflection about the $Y$ -axis.

To determine $F_Y(x)$ , we first consider the case of $a > 0$ .

$\begin{aligned} F_Y(x) & = \int\limits_{-\infty}^{x}f_Y(t)\,dt \\ & = \int\limits_{-\infty}^{x}\frac{1}{a}f_X\left(\frac{t-b}{a}\right)\,dt \\ & = \frac{1}{a}\int\limits_{-\infty}^{\frac{x-b}{a}}f_X(u)a\,du \\ & = \int\limits_{-\infty}^{\frac{x-b}{a}}f_X(u)\,du \\ & = F_X\left(\frac{x-b}{a}\right) \end{aligned}$

Thus, when $a > 0$ , the cumulative distribution function of $X$ is horizontally stretched by a factor $a$ and then right-shifted by $b$ to give the cumulative distribution function of $aX+b$ .

Now, let us consider the case of $a < 0$ .

$\begin{aligned} F_Y(x) & = \int\limits_{-\infty}^{x}f_Y(t)\,dt \\ & = \int\limits_{-\infty}^{x}-\frac{1}{a}f_X\left(\frac{t-b}{a}\right)\,dt \\ & = -\frac{1}{a}\int\limits_{\infty}^{\frac{x-b}{a}}f_X(u)a\,du \\ & = \int\limits_{\frac{x-b}{a}}^{\infty}f_X(u)\,du \\ & = 1 - \int\limits^{\frac{x-b}{a}}_{-\infty}f_X(u)\,du \\ & = 1 - F_X\left(\frac{x-b}{a}\right) \end{aligned}$