(1) \( (5\%) \) Find the moment generating function of \( \mathbf{W}_{\mathrm{n}} \)

(2) \( (5\%) \) Find the limiting distribution of \( \mathbf{W}_{\mathrm{n}} \)

(1) \( (5\%) \) Show that the maximum likelihood estimator of \( \theta \) is \( \hat{\theta} = -\left(\frac{1}{n}\right)\sum{i=1}^{n}\ln(Xi) \).

(2) \( (5\%) \) Show \( \operatorname{E}(\hat{\theta}) = \theta \) and thus \( \hat{\theta} \) is an unbiased estimator of \( \theta \).

5、(12%) An opposition party asserts that there being 64% citizens do not support a new policy launched by the government. If their assertion is near true, how many numbers of persons have to be surveyed if they want the estimation error less than 0.04 under confidence level 95%? On the other hand, if the real ratio is not known, how many numbers of persons have to be surveyed? 【Z₀.₉₇₅=1.96 · Z₀.₉₅=1.645】

6、(12%) A random variable Y follows Uniform(-1,3). Derive the probability density function of \( \mathrm{W} = \mathrm{Y}^2 \).

7. (12%) A test has probability \( q \) of success. \( X \) represents the random variable of trial times that first success happens. Please derive \( E(X) = 1 / q \).

8. (14%) P represents the probability of adverse reaction not happening for a vaccination. P is a random variable with density functions:$$\left\{ \begin{array}{ll} 12p^2(1-p) & 0 \le p \le 1 \\ 0 & otherwise \end{array} \right.$$Let n be the number of persons who have been injected. Y is the number of persons that adverse reaction does not happen after vaccination. Please calculate E(Y) if n=10.

(1)(10%) Show that the maximum likelihood estimator of \( \theta \) is \( \hat{\theta} = -(1 / n)\sum{i = 1}^{n}\ln (Xi) \)

(2)(5%) Show the \( \operatorname{E}(\hat{\theta}) = \theta \) and thus \( \hat{\theta} \) is an unbiased estimator of \( \theta \).

2. (10%) Let \( X \) have an exponential distribution with mean \( \theta > 0 \). Show that \( P(X > x + y | X > x) = P(X > y) \).

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