Consider the definite integral

$I={\displaystyle \underset{0}{\overset{}{}}}{x}^2{e}^{-2x}dx.$

$($a$.)$ Explain how you would estimate the integral above by interpreting it

as the expectation of a Gamma distributed random variable with the

following density function:

where $>0$ are constant parameters. You should clearly state the

random variable and the Monte Carlo estimator you would use for I.

$($b$.)$ Calculate the mean and variance of the Monte Carlo estimator of $I$

and use the Central Limit Theorem to give an asymptotic $95\%$ confi$-$

dence interval for the Monte Carlo estimate of I.

$($c$.)$ Apart from the one obtain in part $($a$.),$ suggest another interpretation

of I as the expectation of a function of a random variable. You should

clearly state the function, the new random variable and the Monte

Carlo estimator you would use for I.