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hw05 - EX and Var X Problem 4 Suppose n balls are...

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ORIE 360/560 Fall 2007 Assignment 5 Due Friday, Oct. 5 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. This problem extends the investment example from class. Suppose you are investing \$20 , 000 in two mutual funds. Let X be the return of fund 1 and Y the return of fund 2. Suppose EX = EY = μ , but the returns have different variances, Var X = σ 2 X and Var Y = σ 2 Y . Suppose further that the correlation of X and Y is ρ . Find the amount of money a that you should invest in fund 1 (so that \$20 , 000 - \$ a is invested in fund 2) in order to minimize the resulting variance. Problem 2. A continuous random vector ( X, Y ) has joint pdf f X,Y ( x, y ) = ( 3 x if 0 x 1 and - (1 - x ) y (1 - x ) , 0 otherwise. (a) Find Cov( X, Y ). (b) Are X and Y independent? (c) Find the ρ | X | , | Y | , the correlation of | X | and | Y | . Problem 3. A continuous random variable has density f X ( x ) = λ 1 - e - λ e - λx , for 0 x 1 , where λ > 0. (a) Compute the moment generating function φ X of X . (b) Use your answer to part (a) to compute
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Unformatted text preview: EX and Var X . Problem 4. Suppose n balls are distributed at random into r boxes. Let X i = 1 if the i th box is empty, X i = 0 otherwise. (a) Compute EX i . (b) Compute EX i X j , for i 6 = j . (c) Let S be the number of empty boxes. Find ES and Var S . Problem 5. Suppose ( X,Y ) is uniformly distributed on the 2-dimensional unit disc D 2 = { ( x,y ) : x 2 + y 2 ≤ 1 } . Find . .. (a) . ..the marginal cdf F X of X . (b) . ..the conditional cdf of Y given X = 1 / 2. The following problem is optional for students enrolled in 360, required for students enrolled in 560. Problem 6. Suppose X is a continuous random variable with pdf f , and suppose that X is bounded. That is, there exists a constant a > 0 such that P [-a ≤ X ≤ a ] = 1. Prove that the moment generating function φ X of X satisﬁes φ X ( t ) < ∞ for all real t . Hint: ﬁrst assume t > ....
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