A) Fit the rate parameter of an exponential distribution from a simulated data set (use set.seed(123); x=rexp(200,rate=pi);) using an MLE based optimization using optimize() and the dexp() function in R

B) Next, repeat the above parameter estimation but here find an analytical expression for the MLE rb by differentiating the negative log likelihood function for the scenario above (i.e., for an exponential distribution with rate r) for a given data set (x1, . . . , xN ). Recall f(x) = r exp(r x), for x 0.

C) Use the boot package in R to calculate 95% confidence intervals for the estimate of r used above (for example, see https://www.statmethods. net/advstats/bootstrapping.html).