# hw04 - (b What value of p maximizes your answer in(a...

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ORIE 360/560 Fall 2007 Assignment 4 Due Tuesday, Sept. 25 at 2:00 pm Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. Suppose X has pdf f ( x ) = a + bx 2 , for 0 x 1 . Let c = EX . Find a and b in terms of c . Problem 2. The random vector ( X,Y ) has joint pdf f ( x,y ) = 6 7 ± x 2 + xy 2 ² for 0 x 1 , 0 y 2 . Find Cov( X,Y ). Problem 3. Let X U (0 ,a ) and let Y = min( X,a/ 2), for some a > 0. (a) Find the cdf F Y of Y . (b) Find EY . Problem 4. Abdul and Bertha are squash players. They agree to play a best-of-ﬁve set. That is, they play until Abdul or Bertha has won three games, then they stop. Suppose Abdul wins each game with probability p (0 , 1), and suppose that the outcomes of the various games are independent. (a) Find the expected number of games played.
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Unformatted text preview: (b) What value of p maximizes your answer in (a)? Problem 5. Suppose X and Y are random variables satisfying EX = EY and EX 2 = EY 2 < ∞ . Show that Cov( X-Y,X + Y ) = 0. The following problems are optional for students enrolled in 360, required for students enrolled in 560. Problem 6. Prove that if EX = Var X = 0, then P [ X = 0] = 1. Hint: ﬁrst, let a > and prove P [ | X | ≥ a ] = 0 . Then, explain why this fact means P [ X = 0] = 1 . Problem 7. Let X 1 ,...,X n be independent random variables uniformly distributed on [0 , 1]. Let Y = max( X 1 ,...,X n ). (a) Find the cdf F Y of Y . (b) Find EY ....
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