Consider a random variable $x$ having the density

$f(x)=e^{-(x+1)}+(e-1)e^{-ex},$xin$(0,).$

$($a$)$ Design a rejection sampling algorithm for obtaining independent draws of $x$ based only upon samples $UU(0,1).$ Comment on the efficiency of your algorithm. Implement the algorithm in $R$ and use it to double$-$check your work $($as in Question $2).$

$($b$)$ Describe a method which is more efficient than your rejection sampling algorithm, but still uses only samples $UU(0,1),$ and justify why it is more efficient.

$($c$)$ Design an importance sampling $($IS$)$ algorithm for calculating the $P(xin(2,3)).$ Compare $($empirically$)$ the variance of IS estimator to the plug$-$in estimators for the same quantity that may be based upon the samplers you created in parts $(a-b).$