Untitled - Problem 3(45 points Let X and Y be two jointly...

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Unformatted text preview: Problem 3 (45 points) Let X and Y be two jointly continuous random variables with pdf __ C D<x+y£1,fl<r—y£1 fx.y(r,y}—{O otherwise where C is a positive constant. Sketch the region where fxy(1‘,y) is nonzero. (5 points) Now, in each question given below and on the next page‘ only one of the given four answers is correct. In each case. circle the letter corresponding to the correct answer. Here each correct choice counts +4 points. whereas an incorrect choice receives —1 point. You. can use the space provided at the bottom of each page as well as the facing pages for scratch work. (i) The value of 0 in the expregion for fxyflfizy) is equal to: a. l b. 2 c. :1 11. none of these (it) The expected value of X‘ E[X], is a. 0 b. 0.5 c. 1 1:]. none of these (iii) The probability P(X g 0.5,}’ S 0.25] is given as: a. 7/16 1:. 1/16 c. 3/16 6. none of these (iv) The correlation coefficient of X, Y is given as: a. 0 b. 1 c. —0.5 «.1. none ofthese (v) The conditional expectation of X given ‘1’. EIX|YL is given as: a, 1 b. 0.5 c. V (1. none of these (vi) The random variables X and Y are a. uncorrelated but dependent b. independent c. positively correlated d. none ofthese (vii) Consider the random variables Z = X + Y and 11/ = X — Y. The random variables Z and 'W are a. uncorrelated but dependent b. independent 1:. positively correlated d. none of these (viii) The variance of E[X|Y] is a. O b. 1 c. 0.5 :1. none ofthese (ix) Consider the random variable U = X —— Y — 1. The probability P(U > X) is a. 0 b. [1.5 c. l (1. none of these (x) Consider the random variable U = X —— Y — l. The variance of U is a. 1 b. (1.5 c. 1/12 d. none of these ...
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