(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 \( W = X / Y \).

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