Suppose that X1, . . . , Xn form a random sample from the exponential distribution with

Question:

Suppose that X1, . . . , Xn form a random sample from the exponential with unknown parameter β, and suppose that it is desired to test the following hypotheses:
H0: β ≥ 1/2,
H1: β < 1/2.
Show that at every level of significance α0 (0 < α0 < 1), there exists a UMP test that specifies rejecting H0 when Xn ≥ c, for some constant c. Distribution
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