(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%)

(b)Suppose that \( f \) is a function whose derivative is \( f'(x)=x/(1+x^2) \). Use the result in part (a) to show that \( |f(b)-f(a)| \le \frac{1}{2}|b-a| \) for any \( a \) and \( b \). (5%)

114年 - 114 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144147

113年 - 113 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144169

112年 - 112 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144168

111年 - 111 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#145446

110年 - 110 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146105

110年 - 110 國立臺灣大學_碩士班招生考試_應用數學科學研究所數理統計組:機率統計#102196

109年 - 109 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#145663

108年 - 108 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146113

107年 - 107 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146100

105年 - 105 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146121