Monte Carlo methods use randomness to solve problems that might be deterministic in principle.

We can estimate some difficult or impossible mathematical problems by Monte Carlo methods.

$($a$)$ Let f$($x$)=$ e

$2$x$$x

$2$

$.$ We want to estimate the definite integral I$1=$

R $\backslash$infty

$0$

f$($x$)$dx$.$ Consider

an environment where only exponential random variables exp$(1)$ can be generate.

$1.$ Suppose we would like to use control variate approach to estimate I$1$ with the help of

g$($x$)=$ x$.$ Specify the $\backslash$lambda

$$ you choose. $[5$ points$]$

$2.$ Write down an algorithm to simulation I$1$ by control variate approach. $[5$ points$]$

solution: