Section 9.1 3. Let X, Y, and Z br three independent random variables such that E(X) = E(Y) = E(Z) = 0, and Var(X) = Var(Y) = Var(Z) = 1. Calculate E[(X2(Y + 5Z)2]. 5. A company puts five different types of prizes into their cereal boxes, one in each box and in equal proportions. If a customer decides to collect all five prizes, what is the expected number of the boxes of cereals that he or she could buy? 7. A cultural society is arranging a party for its members. The cost of a band to play music, the amount that the caterer will charge, the rent of a hall to give the party, and other expenses (in dollaes) are uniform random variables over the intervals (1300, 1800), (1800, 2000), (800, 1200), and (400, 700), respectively. If the number of party guests is a random integer from (150, 200], what is the least amount that the society should charge each participant to have no loss, on average? Section 9.2 1. For random variables X, Y, and Z prove that (a) Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z). (b) Cov(X, Y + Z) = Cov(x, y) + Cov(X, Z). 3. Prove that Var(X - Y) = Var(X) + Var(Y) - 2Cov(X, Y). 5. In n independent Bernoulli trials, each with probability of success p, let X be the number of successes and Y the number of failures. Calculate E(XY) and Cov(X, Y). 9. Let X and Y be independent random variables with expected vaules and μ1 and μ2, and variances σ12 and σ22, respectively. Show that Var(XY) = σ12σ22 + μ12σ22 + μ22σ12. 11. Find the variance of sum of n randomly and independently selected point from the interval (0, 1). Section 9.3 1. Let X and Y be jointly distributed, with ρ(X, Y) = 1/2, σX = 2, σY = 3. Find Var(2X - 4Y + 3). 3. A stick of length 1 is broken into two pieces at a random point. Find the correlation coefficient and the covariance of these pieces. 5. Is it possible that for some random variables X and Y, ρ(X, Y) = 3, σX = 2 , and σY = 3? -- ╰▲▲╯ ┌ ┌ ┌ ^ ^ ∣ ∣ 〥 ∣ o o▕ ◥ ☆╭╮☆╭╮ ╭┐╭─┐╭─╮∣╭╮ ˙ ★ ˙ ★ ◥︺／ ★ ╮ ∣ ∣∣ ∣ ∣∣ ∣☆ ├☆ ~想飞就飞~ │ \ ╰ └─╯└─╯╭ ∣☆─╰╰─╯└╰╯ ★ ˙ ★ ★ ＝＝ ＝＝ ╰─╯ orexis --

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