Unformatted text preview: D . Is D one of the common random variables? Problem 3 Suppose the length L and width W of a rectangle are independent and each uniformly distributed over the interval [0 , 1] . Let C = 2 L + 2 W (the length of the perimeter) and A = LW (the area). Find the means, variances, and probability densities of C and A . Problem 4 Consider two random variables X and Y . For simplicity, assume that they both have zero mean. (a) Show that X and E [ X  Y ] are positively correlated. (b) Show that Cov ( Y,E [ X  Y ]) = Cov ( X,Y ) . Problem 5 Let random variables X and Y be jointly uniformly distributed on the region { ≤ x ≤ 1 , ≤ y ≤ 1 } ∪ {1 ≤ x < ,1 ≤ y < } . (a) Determine the value of f XY on this region. (b) Find f X , the marginal PDF of X . (c) Find E [ Y  X = x ] for  x  ≤ 1 . (d) What is the correlation coefﬁcient of X and Y ? (e) Are X and Y independent? (f) What is the PDF of Z = X + Y ?...
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 Spring '07
 BARD
 Variance, Probability theory, common random variables, Sujay Sanghavi

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