# 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 $$y$$.

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.