Consider two p.d.f.€™s f0(x) and f1(x) that are defined as follows:

and

Suppose that a single observation X is taken from a distribution for which the p.d.f. f (x) is either f0(x) or f1(x), and the following simple hypotheses are to be tested:
H0: f (x) = f0(x),
H1: f (x) = f1(x).
a. Describe a test procedure for which the value of α(δ) + 2β(δ) is a minimum.
b. Determine the minimum value of α(δ) + 2β(δ) attained by that procedure.

1 for 0 sxs1, 0 otherwise, 2x for 01 0 otherwise. f(x) =