For context, I have beem learning importance sampling and tilted density as to reduce variacne in simulating. Therefore, if these are applicable please use them. Please provide very detailed solution. Thank you!

Consider the following pseudo$-$code.

For $i$ in $1$ to $5000$ :

I. Generate $U_{i}Unif[0,1].$

II$.$ Set $x_i=-log(U_i).$

III. Evaluate $h(x_i)=\frac{x_i}{(1+x_i^2{)}^2{e}^{-x_i}}.$

Compute hat$()=\frac{1}{5000}{\displaystyle \underset{i=1}{\overset{5000}{}}}h(x_i).$

$($a$)(4$ points$)$ What is the integral that hat$()$ is estimating? Your answer should be of the

form

${\displaystyle \underset{a}{\overset{b}{}}}g(x)dx$

where $a,$binR$\{-,\}.$

$($b$)(4$ points$)$ Implement the pseudo$-$code. Can you find any outliers?

$($c$)(6$ points$)$ Revise the pseudo$-$code for a better result with the same computation

efficiency. Please provide your idea and pseudo$-$code here, and compare your results

with part $($b$).$