Prob18W-08.pdf - ELG 3126 RANDOM SIGNALS AND SYSTEMS Winter...

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ELG 3126 RANDOM SIGNALS AND SYSTEMS Winter 2018 ASSIGNMENT 8 (due at 11:30 AM Thursday, March 15 in class) 1. A point is chosen at random from the region on the Cartesian plane { ( x, y ) | | x | + | y | ≤ 1 } . The location of the point in Cartesian coordinates is ( X , Y ). Find the joint pdf f XY ( x, y ), and the marginal density functions f X ( x ) and f Y ( y ), and determine if X and Y are independent random variables. 2. Suppose X and N are two independent random variables, where X is a discrete random variable that is equally likely to be - 1 or 1 (and cannot be any other value) and N has the Laplacian distribution where f N ( y ) = 2 e - 4 | y | . Suppose Z = X + N . (a) What is the joint probability distribution function of X and N , F XN ( x, y )? (b) Find the conditional density function of X given N = y , f X ( x | N = y ) for all y R . (c) Find the conditional density function of N given X = x , f N ( y | X = x ) for all x R . (d) What is the conditional density function of Z given N = y , f Z ( z | N = y ) for all y R , and then show the density function of Z is f Z ( z ) = e - 4 | z - 1 | + e - 4 | z +1 | .
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  • Fall '16
  • Probability distribution, Probability theory, probability density function, density function, conditional density function

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