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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Problem Set 12: Solutions Joint Distributions of Random Variables 1. [Joint pmfs] (a) The marginal pmfs p X ( u ) and p Y ( v ) are column and row sums as shown in the table below. u → v ↓ 1 3 5 Row sum 4 1/12 1/6 1/12 1/3 3 1/6 1/12 1/12 1/31 1/12 1/6 1/12 1/3 Column sum 1/4 1/3 1/4 1/6 1 (b) The eyeball test tells us that X and Y are dependent random variables. Less optically, p X , Y (0 , 4) = 6 = p X (0) p Y (4) = 1 4 × 1 3 = 1 12 and so X and Y are not independent random variables. (c) P { X ≤ Y } = p X , Y (0 , 3) + p X , Y (0 , 4) + p X , Y (1 , 3) + p X , Y (1 , 4) + p X , Y (3 , 4) = 1 2 . P { X + Y ≤ 4 } = p X (0) + p X , Y (1 , 3) + p X , Y (1 , 1) + p X , Y (3 , 1) + p X , Y (5 , 1) = 7 12 . (d) p X  Y ( u  3) = p X , Y ( u, 3) p Y (3) = 1 / 2 , 1 / 4 , 1 / 4 respectively for u = 0 , 1 , 5. E [ X  Y = 3] = 0 × 1 2 + 1 × 1 4 + 5 × 1 4 = 3 2 . var ( X  Y = 3) = E [ X 2  Y = 3] ( E [ X  Y = 3]) 2 = 1 × 1 4 + 5 2 × 1 4 3 2 2 = 26 4 9 4 = 17 4 . 2. [Drill problem on jointly continuous random variables I] The pdf is nonzero over the shaded region in the figure shown below....
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 Spring '10
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 Probability distribution, Probability theory, probability density function

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