Question

1 versus H:λ 2. Find a 6. Consider Neyman-Pearson Lemma. Consider testing Ho:λ suitable number k so that this lemma can be applied. Do you see any change in k if we replace 1 and 2 above by 4 and 51 our X is still Poisson from number 5; choose any meaningful alpha for number 6 and do the problem.) (Question 5. For the random variable X following a Poisson distribution with mean 2 Consider testing Ho: λ 1 versus H1: λ > 1. Build up a suitable test and calculate its power.) Flust Idefine the entšcal region for the test Then descibe the test procedure and finaly find the Xgiven ne Fatse Camscänner s -[a+ie Th orem 8.3.12 (Neyman-Peareon Lemma) Conaider testing Ho:0 versus H: -6,, tahere the pdf or pmf oomesponding to eia f(xe), 0,1, using a teat with rejection region R that satisfies and for some k 2 0, and (8.3.2) Then a. (Sufficiency) Any test that satisfies (8.3.1) and (8.3.2) is a UMP level a test. Necessity) f there ezists a test satisfying (8.3.1) and (8.3.2) with k > 0, then every UMP level a test is a size a test (satisfies (8.3.2) and every UMP level a test satisfies (8.3.1) ezcept perhaps on a set A satisfying Poo (X E A) Po, (X e A) 0. Proof We will prove the theorem for the case that f(x Bo) and f(x(0) are pdís of continuous random variables. The proof for discrete random variables can be accom- ed by replacing integrals with sums. (See Exercise 8.21.)

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