Let X ∼Beta(θ, 1). Consider testing the simple hypotheses H0 : θ = 1 vs. H1 : θ = k,
where k > 1.

(a) for k = 4, determine the test procedure that minimizes the sum of the probabilities of type I and type II errors.
(b) Determine the probability of type I error and the probability of type II error using the procedure in part (a), still using k = 4.
(c) Determine the test procedure that minimizes the sum of the probabilities of type I and type II error for k in general (k > 1).

2. Suppose that X1, . . . , Xn form a random sample from a normal distribution for which the value of the mean µ is unknown, and the standard deviation is know to be 2.

Consider testing the following simple hypotheses: H0 : µ = -1 vs. H1 : µ = 1. Determine the minimum value of α(δ) + β(δ) that can be attained for each of the following values of the sample size n:
(a) n=1.
(b) n=4.
(c) n=16.
(d) n=36.

3. Suppose that X1, . . . , Xn form a random sample from a Poisson distribution with unknown parameter λ. Let λ0 and λ1 be specified such that λ1 > λ0 > 0, and suppose that it is desired to test the following simple hypotheses:
H0 : λ = λ0,
H1 : λ = λ1.
(a) Show that the value of α(δ) +β(δ) is minimized by a test procedure which rejects H0 when X > c, and determine c.
(b) For λ0 = 1/4, λ1 = 1/2, and n = 20, determine the minimum value of α(δ) +β(δ) that can be attained.
4. Suppose that X1, . . . , Xn form a random sample from a Gamma distribution for which the value of the parameter α is unknown (α > 0), and the value of the parameter β is known. Show that the joint p.d.f. of X1, . . . , Xn has a monotone likelihood ratio in
the statistic Πn

i=1Xi.

5. Reconsider the class example in which X1, . . . , Xn are a random sample of size n from a normal distribution with mean µ and known variance σ2, and you want to test H0 : µ ≤ µ0 vs. H1 : µ > µ0. We determined the UMP test procedure for a given significance level α0, and we determined an expression for the power function of this UMP test. Determine how (if at all) the rejection region changes as:
(a) the sample size n increases
(b) the significance level α0 increases
(c) the population variance σ2 increases
(d) the hypothesized value of the mean µ0 increases

Justify all of your answers mathematically, and also explain why each makes sense intuitively.