Homework 5 - drawn by first choosing its distance D from...

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FALL 2011 Instructor: Sujay Sanghavi [email protected] Homework 5 Due: October 13th in class Problem 1 Let Y be a random variable with probability density function (PDF) f Y ( y ) = 1 + y, - 1 y 0 , y, 0 < y 1 , 0 , otherwise . Find (a) P ( | Y | < 0 . 5) . (b) P ( Y > 0 | Y < 0 . 5) . (c) E [ Y ] . Problem 2 The random variable X has the PDF f X ( x ) = { cx - 2 , 1 x 2 0 , otherwise (a) Determine the value of c . (b) Find the CDF of X . (c) Let A be the event { X > 1 . 5 } . Calculate P ( A ) and the conditional PDF of X given that event A has occurred. Problem 3 Let X be a random variable with PDF f X ( x ) = ± 2 x/ 5 if 2 < x 3 , 0 otherwise . and let f ( X ) = X 2 . Calculate E [ f ( X )] and Var [ f ( X )] . Problem 4 A circle has radius r . Any chord of the circle is at distance at most r from the center. A random chord is
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Unformatted text preview: drawn by first choosing its distance D from the center uniformly from the interval [0 ,r ] , and then choosing any chord at that distance from the center. Find the PDF of L , the length of the chord. Draw a figure to illustrate. Problem 5 Let the random variables X and Y be jointly uniformly distributed over the region shown. (a) Determine the value of f X,Y on the region shown. (b) Find f X , the mean and variance of X . (c) Find and sketch E [ Y | X = x ] as a function of x . Be sure to specify which range of x this conditional expectation is well defined for....
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This note was uploaded on 02/20/2012 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas.

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Homework 5 - drawn by first choosing its distance D from...

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