114年 - 114 國立臺北大學_碩士班一般入學考試試題_統計學系：數理統計#147007

(a) \( (10\%) \) Let \( Z = X + Y \). Find the probability density function of \( Z \).

(b) \( (10\%) \) Let \( W = \ln \left(\frac{Y}{X}\right) \). Find the probability density function of \( W \).

(a) \( (5\%) \) Find the probability density function of \( X \)

(b) \( (10\%) \) Find \( \operatorname{E}(\mathbf{X}) \) and \( \operatorname{Var}(\mathbf{X}) \).

(c) \( (5\%) \) Find \( \mathrm{P}(\mathrm{X} < 0.3) \)

(a) \( (5\%) \) Find the joint probability function of X and Y.

(b) \( (5\%) \) Find the marginal distribution of \( X \) and the marginal distribution of \( Y \).

(a) \( (5\%) \) Find a complete sufficient statistic of \( \theta \)

(b) \( (5\%) \) Find the uniformly minimum-variance unbiased estimator (UMVUE) of $$\tau(\theta) = P_\theta[X_1 = 1].$$

(a) \( (5\%) \) Find the method of moments estimator (MME), \( \hat{\theta} \)

(b) \( (5\%) \) Find the asymptotic variance of \( \hat{\theta} \)

(c) \( (5\%) \) Find a consistent estimator of \( \theta \). Prove your claim.

(a) \( (5\%) \) It is known that \( Y_{1}(\theta),\ldots ,Y_{n}(\theta) \) are i.i.d. with \( N(K_{1},K_{2}) \) . Find \( K_{1} \) and \( K_{2} \)

(b) \( (5\%) \) Find a pivotal quantity of \( \theta \)

(c) \( (5\%) \) Use the pivotal quantity of (b) to find a \( 100(1 - \alpha)\% \) confidence set, \( C(X) \), for \( \theta \).

(d) \( (5\%) \) Find a condition such that \( C(X) \) is a finite interval.

4. (5 %) Let f(x|θ) be the logistic location pdf $$f(x|\theta) = \frac{e^{(x-\theta)}}{(1 + e^{(x-\theta)})^2}, -\infty < x < \infty, -\infty < \theta < \infty.$$ Based on one observation, X, find the most powerful size α test of H₀: θ = 0 versus H₁: θ = 1.