110年 - 110 國立臺北大學_碩士班一般入學考試試題_金融與合作經營學系：統計學#146912

(a) (5%) Compute \( P(X_1=1, X_2=2, X_3=1) \).

(b) (5%) Determine \( P(X_1+X_2+X_3=4) \).

(c) (5%) If Y equals the maximum of \( X_1 \), \( X_2 \), \( X_3 \), find \( P(Y \le 2) = P(X_1 \le 2) P(X_2 \le 2) P(X_3 \le 2) \). (Please show the process of calculation and round to four decimal places)

(a) (5%) Find the method of moments estimator for \( \theta \).

(b) \( (5\%) \) Is your estimator in part (a) an unbiased estimator for \( \theta \)

(c) (5%) Given the following \( n = 5 \) observations of \( X \), give a point estimate of \( \theta \): 6.6, 7.7, 6.9, 8.3, 7.5.

(Please show the process of calculation)

3. (10%) A random variable X has a Poisson distribution with λ > 0. Given X, a random variable Y is N(μ, X).

(a) (5%) What is the E(XY)?
(b) (5%) What is the \( \operatorname{Var}(\mathbf{Y}) \)? (HINT: \( \operatorname{Var}(\mathbf{Y}) = \operatorname{E}(\mathbf{Y}^2) - \operatorname{E}(\mathbf{Y})^2 \)

(Please show the process of calculation)

4. \( (10\%) \) \( X{1}, X{2}, \ldots, X{n} \) are the independent and identically distributed random variables and follow the distribution with mean \( \mu \) and variance \( \sigma^2 \). Let \( \bar{X} = \sum{i=0}^{n} X_i / n \). Please show that \( \bar{X} \) is a consistency estimator of \( \mu \).

5. (10%) A test of epidemic disease has the following results: to those patients who have been infected, the correct rate (showing positive) is \( 90\% \); however, to the healthy people, there still exists \( 1\% \) error rate (showing positive). Assume the prevalence rate is \( 0.1\% \) (that is, there are 10 persons really infected in 10000 sample). If a man's test reports positive, what is the probability he really infected?

6. (10%) A random variable \( X \) follows lognormal distribution, \( \log X \sim N(\mu, \sigma^2) \). Derive the probability density function of \( X \).

7. (15%) \( X_1, X_2, \dots, X_{64} \) follow identical Possion(1) distribution independently. Please calculate the probability \( P(X_1+X_2+\dots X_{64}>62) \). 【Hint: by Central Limit Theorem】

8. (15%) \( X \sim N(0,1) \) and \( Y \sim N(0,1) \), \( X \) and \( Y \) are independent. Please derive the probability density functions of \( W = X / Y \).