(1) Calculate the moment generating function (m.g.f.) of X.　(7%)

(2) Find the expectation and using m.g.f. of X.　(8%)

4. Let and be independent r.v.’s distributed as Poisson with mean and set .Show that the conditional p.d.f. of , given , is .　(10%)

二、統計部分： 1. Of the first 100 patients to undergo a new operation, 6 die. Can you calculate the95% CI for the probability of dying with this procedure? If so, calculate theinterval. If not, what information do you need? What assumptions must you make?(10%)

2. A new diet and exercise program has been advertised as remarkable way toreduce blood glucose levels in diabetic patients. Ten randomly selected diabeticpatients are put on the program, and the results after 1 month are given by thefollowing table:Do the data provide sufficient evidence to support the claim that the newprogram reduces blood glucose level in diabetic patients? Use α=0.05, and　\( t_{0.05,9} = 1.833 \)。. (15%)

3. Let be a random sample from an exponential distribution with probability densityfunctionFind the constant c for which where is the 3rd order statistic of the X’s. (10%)

(1) Construct a 95% two-sided confidence interval for . (7%)

(2) It is desired to test against , using . If population mean were really equal to45, what is the power of this test? (8%)T　able: Area under the standard normal curve to the left of x

(a) \( f(x) = \ln(\tan^{-1} x) \). (5%)

(b) \( g(x) = \csc^{-1}(e^x) \). (5%)

(a) Show that \( -1/2 \leq x/(1+x^2) \leq 1/2 \) for every value of \( x \). (5%)

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