Derive the maximum likelihood estimator of ?, and explain why this is a maximum.

Question (a) A random sample {X1, X2,....Xn} is drawn from the following probability distribution:

f(x; ?) = {2x/ (1-?2) for ? ? x ? 1
0 otherwise.
i. Derive the maximum likelihood estimator of ?, and explain why this is a maximum.
ii. If n = 4, such that x1 = 0.70, x2 = 0.92, x3 = 0.21 and x4 = 0.34, determine the maximum likelihood estimate of O.
iii. As n ? ?, would you expect the maximum likelihood estimate of B to get closer to 6? Briefly explain your answer.

(b) A random sample of size n = 9 drawn from a normal distribution had a sample variance of s2 = 72.76. Construct a 90% confidence interval for ?2.

(c) Let X be a random variable such that X ~ Exp(?). Suppose we wish to test the null hypothesis Ho : ? = 2 vs. H1 : ? > 2. Consider the test procedure which rejects Ho if X < 0.75.
i. Compute the probability of a Type I error, expressing your answer in terms of the exponential constant e.
ii. Determine the power function of the test, expressing your answer in terms of the exponential constant e.

(b) A random sample of size n = 9 drawn from a normal distribution had a sample variance of 62 = 72.76. Construct a 90% confidence interval for o2.

(c) Let X be a random variable such that X ". Exp(A). Suppose we wish to test the null hypothesis Ho : A = 2 vs. HI : A > 2. Consider the test procedure which rejects Ho if X < 0.75.

i. Compute the probability of a Type I error, expressing your answer in terms of the exponential constant e.

ii. Determine the power function of the test, expressing your answer in terms of the exponential constant e.