#$2($Conditional Monte Carlo vs$.$ Stratified Sampling$)$ Suppose that $Y$ is a binomial random variable with parameters $n=10$ and $p=0\mathrm{.}5.$ Suppose that, conditioned on $Y=y,x$ is a normal random variable with mean $y$ and variance $4.$ We want to use simulation to efficiently estimate $=P(x5).$

$($a$)$ Design a simulation algorithm that only generates i$.$i$.$d$.$ random variables.

$($b$)$ Design a simulation algorithm that uses conditional Monte Carlo method. Clearly specify the estimator you are using. Calculate the variance of the conditional Monte Carlo estimator;

$($c$)$ Design a simulation algorithm that uses stratified sampling. Assuming that each stratum $k$ contains $n{p}_k$ samples. Clearly specify the estimator you are using. Calculate the expected variance of this stratified sampling method.

$($d$)$ If we use the optimal number of samples for part $($c$),$ will your variance result change?

$($e$)$ Compare the variance reduction results of part $($b$)$ and part $($c$)$ against part $($a$)$ and explain why one is greater than the other.

Please provide python code. thanks