1. Goal: We need to find $\operatorname{Var}(Y)$. The formula for variance is $\operatorname{Var}(Y)=E\left[Y^2\right]-(E[Y])^2$.
2. Substitute $Y=X^2$ :
$$
\operatorname{Var}\left(X^2\right)=E\left[\left(X^2\right)^2\right]-\left(E\left[X^2\right]\right)^2=E\left[X^4\right]-\left(E\left[X^2\right]\right)^2
$$
We need to find the 2nd and 4th moments of the exponential variable $X$.
3. Analyze $X$ : The p.d.f. is $f(x)=e^{-x}$. The standard form is $f(x)=\lambda e^{-\lambda x}$. By comparing these, we see the rate parameter $\lambda=1$.
4. Find $E\left[X^k\right]$ : A standard result for an exponential distribution $X \sim \operatorname{Exp}(\lambda)$ is that its $k$-th moment is:
$$
E\left[X^k\right]=\frac{k!}{\lambda^k}
$$
5. Calculate $E[Y]=E\left[X^2\right]$ : Using the formula with $k=2$ and $\lambda=1$ :
$$
E[Y]=E\left[X^2\right]=\frac{2!}{1^2}=\frac{2}{1}=2
$$
6. Calculate $E\left[Y^2\right]=E\left[X^4\right]$ : Using the formula with $k=4$ and $\lambda=1$ :
$$
E\left[Y^2\right]=E\left[X^4\right]=\frac{4!}{1^4}=\frac{24}{1}=24
$$
7. Calculate $\operatorname{Var}(Y)$ :
$$
\begin{gathered}
\operatorname{Var}(Y)=E\left[Y^2\right]-(E[Y])^2 \\\\
\operatorname{Var}(Y)=24-(2)^2 \\\\
\operatorname{Var}(Y)=24-4=20
\end{gathered}
$$
