101年 - 101 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#146329

1.Let \( (X, Y) \) be distributed with joint density

\[ f(x, y) = \begin{cases} [1 + xy(x² + y²)]/4, & \text{if } |x|<1, |y|<1, \\ 0, & \text{otherwise.} \end{cases} \]
Are \( X \) and \( Y \) independent? If not, find the covariance between \( X \) and \( Y \). (15%)

2. Let \( X \sim B(n, p) \), as \( n \to \infty \), \( p \to 0 \) and \( np \to \lambda \in (0, \infty) \). Then show that

\[ \binom{n}{x} p^x (1-p)^{n-x} \xrightarrow{n \to \infty} e^{-\lambda} \frac{\lambda^x}{x!} . \quad (10\%) \]

3. Let \( X \) be a random variable such that \( E(X^{2k}) = \frac{(2k)!}{2^k k!} \), \( E(X^{2k+1}) = 0 \), \( k = 0, 1, 2, \dots \). Find the moment generating function (m.g.f.) of \( X \) in terms of the moments given and the probability distribution function (p.d.f.) of \( X \). (10%)

4. Suppose that certain events occur in time intervals of length \( t \) according to Poisson distribution with parameter \( \lambda t \). Then show that the waiting time between any two such successive occurrences of events is an r.v. \( T \) which has negative exponential distribution with parameter \( \lambda \), by showing that \( P(T > t) = e^{-\lambda t} \), \( t > 0 \). (15%)

二、統計
1. Let \( X_1, X_2, \dots, X_n \) be i.i.d. r.v.'s from \( U(\theta_1, \theta_2) \). If \( \theta = (\theta_1, \theta_2) \) is unknown, then find the sufficient for. (7%)

2. If \( \mathcal{F} \) is the family of all Poisson p.d.f., \( \theta \in (0, \infty) \). Is \( \mathcal{F} \) complete? (8%)

3. Let \( X_1, X_2, \dots, X_n \) be i.i.d. r.v.'s from \( B(1, \theta) \), \( \theta \in (0, 1) \). Show a UMVUE of \( \theta \) and the Cramer-Rao bound. (10%)

4.If \( X_1, X_2, \dots, X_n \) are i.i.d. r.v.'s from binomial distribution \( B(m, \theta) \), \( \theta \in (0, 1) \), show that \( \overline{X}/m \) is the MLE of \( \theta \). (7%)

5.The lifetime of an automobile battery is described by an r.v. \( X \) having normal distribution with unknown mean \( \mu \) hour and known s.d. \( \sigma=150 \) hour. Let \( X_1, X_2, \cdots, X_{25} \) be independent r.v.'s distributed as \( X \) and suppose that \( \bar{x}=1725 \) hour. Test the hypothesis \( H_0: \mu=1800 \) versus \( H_1: \mu<1800 \) at level of significance \( \alpha=0.01 \) and make a conclusion. (10%)

6. Let \( X_1, X_2, \cdots, X_n \) be i.i.d. r.v.'s with (finite) unknown mean \( \mu \) and (finite) known variance \( \sigma^2 \), and suppose that \( n \) is large. \( \overline{X}_n \) is the sample mean of the \( X \)'s. Use the C.L.T. to construct a 95% confidence interval for \( \mu \) when \( n=100 \) and \( \sigma=1 \). (8%)

Table: Area \( \Phi(x) \) under the standard normal curve to the left of \( x \)