The dive depths of sperm whales range from a few hundred metres to as much as 2 km. Suppose dives follow a Gamma(k =

2, λ) model. We want to estimate λ.

a. Let X ∼ Gamma(2, λ). Sketch on a single plot the PDF of X for two values of λ, labelling which curve corresponds to the larger value of λ and which to the smaller value.

b. If X1, . . . , Xn ∼ Gamma(2, λ) are independent, what is the distribution of X1 + · · · + Xn?

c. Explain how to formulate the maximum likelihood estimator of λ, given observations X1, . . . , Xn. Write down the MLE and an expression for its variance.

d. Given the five observations of dive depths below, measured in km, estimate λ and provide an approximate 95%

confidence interval.

0.55 1.00 0.42 1.94 1.52