四、Two independent random variables X and Y. Y follows standard normal distribution, and X is uniform between [0,1]. $Z = X + Y$. Please derive the detail process that $cov(X, Z) = var(X)$. (15%)

五、Suppose $X_1, X_2, \ldots, X_n$ are independent random variables and follow Poisson distribution with mean $\lambda$. Please use moment generating function to show that $\overline{X} = \sum X_i / n$ converges in probability to $\lambda$. (Hint: The distribution of $\overline{X}$ degenerates to a constant random variable.) (10%)

六、Suppose $X_1, X_2, \ldots, X_{10}$ are independent random variables and follow N(0, 1). Let $S = X_1+X_2+X_3+X_4+X_5$ and $W = X_6^2+X_7^2+X_8^2+X_9^2+X_{10}^2$. What is the probability $P(S/\sqrt{W} > 4.0324)$? (Rounded to third place.) (10%)

七、An urn contains six balls, one marked WIN and five marked LOSE. You and another player take turns selecting a ball from the urn, one at a time. The first person to select the WIN ball is the winner. If you draw first, find the probability that you will win (a) When the sampling is done with replacement. (10%)

(b) When the sampling is done without replacement. (5%)

八、Bowl B₁ contains 2 white chips, bowl B₂ contains 2 red chips, bowl B₃ contains 2 white and 2 red chips, and bowl B₄ contains 3 white chips and 1 red chip. The probabilities of selecting bowl B₁, B₂, B₃, or B₄ are 1/2, 1/4, 1/8, and 1/8, respectively. A bowl is selected using these probabilities, and a chip is then drawn random. Find (a) \(\mathrm{P}(\mathrm{W})\) , the probability of drawing a white chip. \((10\%)\)

(b) \(\mathrm{P(B_1|W)}\), the conditional probability that bowl \(\mathrm{B}_1\) had been selected, given that a white chip was drawn. (5%)

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