1. Given a random sample X1 , X2 ,..., Xn from a Gamma γ(α, β) population,

(i) Determine the MLE for the parameter β when α is speci?ed.

(ii) When α is not speci?ed, there is no closed form solution for the maximum likelihood estimates for the two unknown parameters. However, show, without solving the simultaneous maximum likelihood equations, that the method of moments estimates are di?erent from the maximum likelihood estimates.

2. The average of a sample of size n = 20 from a normal population with unknown mean μ and variance σ2 = 5 was obtained as x¯ = 74.4; the sample variance was obtained as s2 = 5.6

(i) Determine an interval (x¯ -w, x¯ + w) such that the probability that the true value of μ lies in this interval is  0.90.

(ii) Repeat (i) when the desired probability is 0.95

(iii) Repeat (i) and (ii) when the population variance is not given.