三、(10%) The moment-generating function of X is M(t) = (0.25 + 0.75e^t)^16. What are the p.d.f. of X as well as the expectation E(X) and variance Var(X)? (Please show the details)

四、(10%) Let \( X_1, X_2, ..., X_n \) be a random sample from \( N(\mu, \sigma^2) \) with known mean \( \mu \). Moreover, Let \( \chi^2_{p(n)} \) denote the \( 100p^{\text{th}} \) percentile of \( \chi^2(n) \), the Chi-Square distribution with \( n \) degrees of freedom, where \( 0 \le p \le 1 \). Namely, \( P(Y \le \chi^2_{p(n)}) = p \), where \( Y \sim \chi^2(n) \) Describe how you would construct a \( 100(1-\alpha)\% \) confidence interval for the unknown variance \( \sigma^2 \), where \( 0 \le \alpha \le 1 \).

三、(10%) The moment-generating function of X is \( M(t) = (0.25 + 0.75e^{t})^{16} \). What are the p.d.f. of X as well as the expectation E(X) and variance Var(X)? (Please show the details)

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