Let X1, . . . , Xn be an i.i.d. sample from an exponential distribution with the density...

Let X₁, . . . , X_n be an i.i.d. sample from an exponential distribution with the density function

a. Find the mle of τ .

b. What is the exact sampling distribution of the mle?

c. Use the central limit theorem to find a normal approximation to the sampling distribution.

d. Show that the mle is unbiased, and find its exact variance. (Hint: The sum of the X_i follows a gamma distribution.)

e. Is there any other unbiased estimate with smaller variance?

f. Find the form of an approximate confidence interval for τ .

g. Find the form of an exact confidence interval for τ .