final - f X | Y ( x ) . (c) [ 15 points ] Find the...

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EE 505 B Fall 2011 Final Exam Nota bene : For full credit you must show all of the steps or reasoning that you used to get the answer. 1. [ 35 points ] U is a uniformly distributed random variable with probability density function f U ( u ) = b 1 0 u 1 0 for all other values of u A new random variable is generated by transforming the random variable U with the equation X = - 1 λ ln ( U ) (a) [ 10 points ] Find f X ( x ) . For reference, a graph of the function x = g ( u ) = - 1 λ ln ( u ) is shown in Figure 1. 1 u x Figure 1: A graph of the function x = - 1 2 ln ( u ) . (b) [ 10 points ] Find Pr p - 1 λ ln ( U ) λ P . (c) [ 15 points ] If λ is now a random variable Λ with probability density function f Λ ( λ ) = b 1 0 λ 1 0 otherwise so that X = - 1 Λ ln ( U ) Find the probability density function for X , i.e., f X ( x ) . 1
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2. [ 40 points ] The random variables X and Y have a joint probability density function f XY ( x , y ) = b 2 e - x - y 0 < x < y < 0 otherwise (a) [ 15 points ] Are X and Y independent? (b) [ 10 points ] Find
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Unformatted text preview: f X | Y ( x ) . (c) [ 15 points ] Find the covariance of X and Y . Hint : The formulas or numeric values for all complicated integrals have been given to you. 3. [ 25 points ] The random variables X and Y have a joint probability density function of f XY ( x , y ) = b 2 e-x-2 y x 0 and y otherwise Find the joint probability density function of the random variables W and Z , where W = 2 X-Y Z = X-2 Y Helpful formulas For a couple of problems, the following integral will be useful. i b a xe-qx dx = e-aq ( aq + 1 )-e-bq ( bq + 1 ) q 2 Also, useful information is that a random variable, e.g., Z , with probability density function f Z ( z ) = e- z has E { Z } = 1 and E { Z 2 } = 2 2 . 2...
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final - f X | Y ( x ) . (c) [ 15 points ] Find the...

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