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

1. If the joint probability density of X and Y is given by \[ f(x, y) = \begin{cases} \frac{2}{7}(x+2y), & \text{for } 0 < x < 1, 1 < y < 2; \\ 0, & \text{elsewhere}. \end{cases} \] Find the expected value of $g(x, y) = x/y^3$. (15%)

2. Suppose that X has an Poisson distribution with parameter ^λ. Find the moment-generating function of X. (10%)

(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 \( X_1, X_2, X_3 \) constitute from Bernoulli distribution with parameter \( \theta \). Show that \( Y = (X_1 + 2X_2 + 3X_3)/6 \) is not a sufficient estimator of Bernoulli parameter \( \theta \). (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%)

(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