Suppose that X1, , Xn form a random sample from the Poisson distribution with unknown mean Let 0 and 1 be specified values such that 1 0 0, and suppose that it is desired to test the following simple hypotheses H0 0, H1 1 a Show that the value of () () is minimized by a test procedure which rejects H0 when Xn c b Find the value of c c For 0 1 4, 1 1 2, and n 20, determine the minimum value of () () that can be attained

Question: Suppose that X1, . . . , Xn form a random sample from the Poisson distribution with unknown mean . Let 0 and 1 be

Suppose that X1, . . . , Xn form a random sample from the Poisson distribution with unknown mean λ. Let λ0 and λ1 be specified values 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 Xn > c.
b. Find the value of c.
c. For λ0 = 1/4, λ1 = 1/2, and n = 20, determine the minimum value of α(δ) + β(δ) that can be attained.

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