109年 - 109 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#146959

1. (10%) Suppose \( X_1 \), \( X_2 \) are independent and both follow continuous uniform distribution U(0,1). Please find the probability \( Pr(X_1^2 + X_2^2 < 1) \).

(I) (5%) \( \text{E}(\overline{\text{X}}) \)

(2) (5%) \( \text{Var}(\overline{\text{X}}) \)

(3) (5%) \( \Pr(24.18 \leq \overline{\text{X}} \leq 24.93) \) (Hint: Use the empirical rule for normal distributions)

3. (10%) Let \( X_1, X_2, \dots, X_n \) be a random sample of size n from the distribution with p.d.f. \( f(x, \theta) = \theta x^{\theta-1} \), \( 0 < x < 1 \), \( 0 < \theta < \infty \). Assume that \( \overline{X} = \sum_{i=1}^{n} X_i / n = 0.2 \) be the sample mean. Please use \( \overline{X} \) to find the moments estimator of \( \theta \) and calculate its estimate.

(1)(5%) What is the value of \( (\overline{\mu}_2 - \overline{\mu}_1) \)?

(2) (10%) Assume \( \sigma_1^2 = \sigma_2^2 \), please find an estimate of \( \text{Var}(\overline{\mu}_2 - \overline{\mu}_1) \). (Hint: \( \text{SSE} = (n-1)S_1^2 + (m-1)S_2^2 \))

5. (10%) The probabilities of two independent events A and B: \( \mathrm{P(A)} = 0.2 \), \( \mathrm{P(B)} = 0.5 \). Please calculate \( \mathrm{P(A^c \cap B^c)} \). \( \mathrm{A^c} \) and \( \mathrm{B^c} \) represent the complement of A and B.

6. (10%) Let \( X_1, X_2 \dots X_n \) be i.i.d. exponential random variables with the p.d.f. \[ f(x|\beta) = \beta e^{-\beta x} \]. Please find the maximum likelihood estimator of \( \beta \).

(1) \( (5\%) \) Explain the meaning of "hazard function".

(2) \( (10\%) \) If \( \lambda (t) \) is a constant, that is \( \lambda (t)\equiv \lambda >0 \) , please depict the distribution of X.

8. (15%) A zoologist observes the Polar bears activities nearby the North Pole for 50 weeks. He records the number of bears he finds every week. Please test if the number of bears follows Poisson distribution under significant level α=0.05?