103年 - 103 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#146098

科目： 研究所、轉學考(插大)◆機率統計 | 年份：103年 | 選擇題數：0 | 申論題數：10

試卷資訊

所屬科目： 研究所、轉學考(插大)◆機率統計

選擇題 (0)

申論題 (10)

一、機率部分：
1. If the joint probability density of X and Y is given by

Find the expected value of . (15%)
2. Suppose that X has an Poisson distribution with parameter . Find the moment-generating function of X. (10%)
3. Please state and prove the following inequality.
(a) Jensen’s inequality (8%)
(b) Chebyshev’s inequality (7%)
4. If the joint pdf of X and Y is
,
Please calculate the probability . (10%)
二、統計部分：
1. If X1, X2, X3 constitute from Bernoulli distribution with parameter θ. Show that Y=(X1+2X2+3X3)/6 is not a sufficient estimator of Bernoulli parameter　θ.　(15%)
2. If 132 of 200 male voters and 90 of 159 female voters a certain candidate running for governor of Illinois, find a 99% confidence interval for the difference between the actual proportions of male and female voters who favor the candidate. (z0.025=1.96, z0.005=2.575)　　(10%)
3. Let be random sample of size three from uniform distribution, where .
(a) Find the method of moments estimator of .　(7%)
(b) Find the MLE, , and find a constant such that .　(8%)
4. Let be a random sample from , and be a random sample from , independent of the Xs. Define . Please find the level LRT of versus . (10%)








Table: Area under the standard normal curve to the left of x