3. Let $X \sim \text{Poisson}(\lambda)$, and conditional on $X = x, Y | X = x \sim \text{Binomial}(x, p)$. (10%)

(1) Compute \(\operatorname{E}(Y)\).

(2) Compute \(\operatorname{Var}(Y)\).

4. Let $X1, \ldots, X{64}$ be i.i.d. random variables with $\operatorname{E}(Xi) = 5$ and $\operatorname{Var}(Xi) = 4$. Using the Central Limit Theorem, compute the approximation of $P(\sum{i=1}^{64} Xi > 346.32)$. (10%)

5. A manufacturer claims that the defect rate of a product is $p$. A random sample of $n = 400$ products is taken, among which 36 are defective. (10%) (1) At significance level \(a = 0.05\), test \(\mathrm{H}0: p = 0.1\) vs. \(\mathrm{H}1: p \neq 0.1\).

(2) Construct a \(95\%\) confidence interval for \(p\).

II. 1. The recruitment channels for a company are as follows: based on past experience, 50% of applicants learned of the openings through job banks, 10% through social media ads, 30% through employee referrals, and 10% through campus recruiting. The one-year turnover rates (the probability of an employee resigning within a year) are as follows: 10% for those from job banks, 20% for those from social media ads, 5% for those from employee referrals, and 30% for those from campus recruiting. If an employee is still with the company after one year, what is the probability that he (she) originally learned about the recruitment information through a job bank? (7%)

2. The annual returns of two ETFs, ETF A and ETF B, both follow a normal distribution, and are independent. ETF A $\sim N[\mu = 20\%, \sigma^2 = (24\%)^2]$; ETF B $\sim N[\mu = 15\%, \sigma^2 = (18\%)^2]$ Calculate the probability that the actual annual return of ETF B is higher than the actual annual return of ETF A. (6%)

3. A random sample of 100 citizens is selected from a city to investigate the satisfaction rate with municipal services. The decision rule is as follows: if between 55 and 65 people (inclusive) express approval, it is concluded that the satisfaction rate is 60%. Please calculate: (1) The probability of a Type I error. (6%)

(2) The power of the test when the true proportion is \( p = 0.5 \). (6%)

4. Regarding survey responses, people often hesitate to provide an affirmative answer. Suppose that in a population, 80% would truthfully answer "No" to a question, while 20% would truthfully answer "Yes." However, 70% of the individuals lie about their response. Let $Y$ be the number of people who must be surveyed until the first affirmative ("Yes") response is obtained. Derive the probability distribution of $Y$. (7%)

5. In electronics, current is denoted as $I$, resistance as $R$, and power as $W = I^2 \times R$. Suppose $I$ follows a uniform distribution over the interval $(0, 1)$, and the probability density function of $R$ is $f(r) = \begin{cases} 2r, & 0 \le r \le 1 \\ 0, & \text{otherwise} \end{cases}$ Given that $I$ and $R$ are independent, please find: (1) The probability density function of \(W\). (12%)

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