Exercise $9\mathrm{.}22.$ In this exercise, you estimate the integral ${\displaystyle \underset{0}{\overset{}{}}}{x}^6{e}^{-x^{1\mathrm{.}5}}dx$ via Monte Carlo. For this exercise, you need access to a computer software package that can generate independent samples from the Unif $0,1$ distribution.

Observe that

${\displaystyle \underset{0}{\overset{}{}}}{x}^5{e}^{-x^{1\mathrm{.}5}}dx={\displaystyle \underset{0}{\overset{}{}}}{x}^6{e}^{-x^{1\mathrm{.}5}+x}{e}^{-x}dx=E[x^6{e}^{-x^{1\mathrm{.}5}+x}]$

where $xExp(1).$ Set $Y=x^6{e}^{-x^{1\mathrm{.}5}+x}$ and sample $n={10}^5$ independent copies of $Y$ in order to provide a $95\%$ confidence interval for the integral.