16.3. Two-to-One Functions

Let \(X\) have density \(f_X\). As you have seen, the random variable \(Y = X^2\) comes up frequently in calculations. Thus far, all we have needed is \(E(Y)\) which can be found by the formula for the expectation of a non-linear function of \(X\). To find the density of \(Y\), we can’t directly use the change of variable formula of the previous section because the function \(g(x) = x^2\) is not monotone. It is two-to-one because both \(\sqrt{x}\) and \(-\sqrt{x}\) have the same square.

In this section we will find the density of \(Y\) by developing a modification of the change of variable formula for the density of a monotone function of \(X\). The modification extends in a straightforward manner to other two-to-one functions and also to many-to-one functions.

16.3.1. Density of \(Y = X^2\)

If \(X\) can take both positive and negative values, we have to account for the fact that there are two mutually exclusive ways in which the event \(\{ Y \in dy \}\) can happen: either \(X\) has to be near the positive square root of \(y\) or near the negative square root of \(y\).

../../_images/03_Two_to_One_Functions_4_0.png

So the density of \(Y\) at \(y\) has two components, as follows. For \(y > 0\),

\[ f_Y(y) ~ = ~ a + b \]

where

\[ a = \frac{f_X(x_1)}{2x_1} ~~~~ \text{at } x_1 = \sqrt{y} \]

and

\[ b = \frac{f_X(x_2)}{\vert 2x_2 \vert} ~~~~ \text{at } x_2 = -\sqrt{y} \]

We have used \(g'(x) = 2x\) when \(g(x) = x^2\).

For a more formal approach, start with the cdf of \(Y\):

\[\begin{split} \begin{align*} F_Y(y) ~ &= ~ P(Y \le y) \\ &= ~ P(\vert X \vert \le \sqrt{y}) \\ &= ~ P(-\sqrt{y} \le X \le \sqrt{y}) \\ &= ~ F_X(\sqrt{y}) - F_X(-\sqrt{y}) \end{align*} \end{split}\]

Differentiate both sides to get our formula for \(f_Y(y)\); keep an eye on the two minus signs in the second term and make sure you combine them correctly.

This approach can be extended to any many-to-one function \(g\). For every \(y\), there will be one component for each value of \(x\) such that \(g(x) = y\).

16.3.2. Square of the Standard Normal

Let \(Z\) be standard normal and let \(W = Z^2\). The possible values of \(W\) are non-negative. For a possible value \(w \ge 0\), the formula we have derived says that the density of \(W\) is given by:

\[\begin{split} \begin{align*} f_W(w) ~ &= ~ \frac{f_Z(\sqrt{w})}{2\sqrt{w}} ~ + ~ \frac{f_Z(-\sqrt{w})}{2\sqrt{w}} \\ \\ &= ~ \frac{\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}w}}{2\sqrt{w}} ~ + ~ \frac{\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}w}}{2\sqrt{w}} \\ \\ &= \frac{1}{\sqrt{2\pi}} w^{-\frac{1}{2}} e^{-\frac{1}{2}w} \end{align*} \end{split}\]

By algebra, the density can be written in an equivalent form that we will use more frequently.

\[ f_W(w) ~ = ~ \frac{\frac{1}{2}^{\frac{1}{2}}}{\sqrt{\pi}} w^{\frac{1}{2} - 1} e^{-\frac{1}{2}w} \]

This is a member of the family of gamma densities that we will study later in the course. In statistics, it is called the chi squared density with one degree of freedom.