3.11 Functions of random variables

A function of random variables is a transformation (function) of \(\mathbb{R}^n\) into \(\mathbb{R}^n\), denoted \(g:\mathbb{R}^n \rightarrow \mathbb{R}^n\).

Example 3.46 Let the random variable be \(X \sim \mathcal{U}(-1,1)\) and the function \(Y=X^2\). It is possible to obtain the pdf \(g(y)\). Initially consider the graphs of \(f(x)\) by \(x\) and \(y\) by \(x\). From the graphs it is possible to verify that \(x \in (-1,1)\), \(f(x)=\frac{1}{2}\) and \(y \in (0,1)\). The limits \(-\sqrt{y}\) and \(\sqrt{y}\) are obtained during Step 1 below.

Step 1

\[\begin{align*} G(y) &= Pr(Y < y) \\ &= Pr(X^2 < y) \\ &= Pr(-\sqrt{y} < X < \sqrt{y}) \\ &= \int_{-\sqrt{y}}^{\sqrt{y}} \frac{1}{2} dx \\ &= \frac{1}{2} [\sqrt{y} - (-\sqrt{y})] \\ G(y) &= \sqrt{y} \end{align*}\]

Checking
\(G(0) = \sqrt{0} = 0\)
\(G(1) = \sqrt{1} = 1\)

Step 2

\[\begin{align*} g(y) &= G'(y) \\ &= \frac{1}{2} y^{\frac{1}{2}-1} \\ &= \frac{1}{2 \sqrt{y}}\\ g(y) &= \frac{\sqrt{y}}{2y} \end{align*}\]

Checking
\(\int_{0}^{1} \frac{1}{2} y^{-\frac{1}{2}} dy = \frac{1}{2}[2y^{\frac{1}{2}}] \Big|_0^1 = 1\)

Finally, it is possible to graph \(g(y)\) by \(y\), verifying \(g(y) \in (1/2,\infty)\).