Consider a distribution with pdf f(x; theta) = (theta)x^(theta - 1) if 0 < x < 1 and zero otherwise.
a) Based on a random sample of size n = 1, find the most powerful test of Ho: theta = 1 against Ha: theta = 2 with alpha = 0.05.
b) Compute the power of the in a) for the alternative theta = 2.
c) Derive the most powerful test for the hypotheses of a) based on a random sample of size n.