8. (20%) Consider all possible binary relations on a set A with 4 elements. Answer the following 5 questions. Q1: How many binary relations are there? Q2: How many binary relations are reflexive? Q3: How many binary relations are symmetric? Q4: How many binary relations are anti-symmetric? Q5: How many binary relations are reflexive and symmetric and anti-symmetric simultaneously?

一、(10%) Let \( X_1 \), \( X_2 \), and \( X_3 \) be independent and have uniform distribution U(0, 1). The joint p.d.f. is \( f(x_1, x_2, x_3) = 1 \), where \( 0 \le x_1, x_2, x_3 \le 1 \). What is the probability \( P(X_1 + X_2 + X_3 \le 1) \)? (Please show the details)

二、(10%) There are two random variables \( X \) and \( Y \). Given \( Y \), \( X \) follows the conditional distribution \( f(x|y) = e^{-y}y^x/x! \), where \( x = 0, 1, 2, ... \). Moreover, \( Y \) follows the distribution \( f(y) = \lambda e^{-y\lambda} \), where \( y > 0 \) and \( \lambda > 0 \). What are expectation \( E(X) \) and the variance \( Var(X) \)? (Please show the details)

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 \( 0 \le \alpha \le 1 \).

五、(10%) Let \( f(x; \theta) = (1/\theta^2)xe^{-x/\theta} \), where \( x > 0 \) and \( \theta > 0 \). Let \( X_1, X_2, ..., X_n \) denote a random sample of size \( n \) for this distribution. What is the maximum likelihood estimator of \( \theta \). (Please show the details)

(一)E(X - Y). (3 %)

(二)Var(X - Y). (6 %)

(三)If you have known that $X+Y \le 3$, calculate Var(X/Y). (7 %)

七、X and Y are two random variables, please derive: (12 %)$$Var(X) = E[Var(X|Y)] + Var(E[X|Y])$$

(一)$W = X^2$ (10 %)

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