Find the density function of $Y=X^{2}$

Question

Let X have density function

\begin{align} f(x) = \begin{cases}e^{-x} & x > 0\\0 & otherwise\end{cases} \end{align}

Find the density function of $Y=X^{2}$

Answer 1

$X$ is exponential with parameter $1$, so $F_X(t)=1-e^{-t}$

$Y=X^2$ takes positive values, so for $t>0$:

$F_Y(t)=P(X^2\le t) = P(X\le\sqrt t) = F_X(\sqrt t)=1-e^{-\sqrt t}$

$f_Y(t)=\frac{d}{dt}F_Y(t)=\frac{1}{2\sqrt t}e^{-\sqrt t}$ when $t>0$ and $0$ otherwise.