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

科目：研究所、轉學考(插大)◆統計學 | 年份：110年 | 選擇題數：0 | 申論題數：12

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所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (0)

申論題 (12)

1. (15%) Let X₁, X₂, X₃ denote a random sample of size n = 3 from a distribution with the p.d.f.

$$f(x) = \binom{2}{3} \binom{3}{2}^{x-1}, x=1, 2, 3, \dots.$$

That is, X₁, X₂, X₃ are mutually independent and each has the geometric distribution.

(a) (5%) Compute \( \mathrm{P}(\mathrm{X}1 = 1, \mathrm{X}2 = 2, \mathrm{X}_3 = 1) \).
(b) \((5\%)\) Determine \(\mathrm{P}(\mathrm{X}1 + \mathrm{X}2 + \mathrm{X}_3 = 4)\).
(c) (5%) If Y equals the maximum of \( X{1}, X{2}, X{3} \), find \( P(Y \leq 2) = P(X{1} \leq 2)P(X{2} \leq 2)P(X{3} \leq 2) \).

(Please show the process of calculation and round to four decimal places)
2. (15%) Let X₁, X₂, ..., Xₙ be a random sample of size n from uniform distribution on interval (θ-1, θ+1).

(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%) \( X{1}, X{2}, \ldots, X{64} \), follow identical Possion(1) distribution independently. Please calculate the probability \( P(X{1} + X{2} + \ldots + X{64} > 62) \). [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 \( W = X / Y \).