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# solution14 - 3:25 PM HW14 Solution page 1 Probability for...

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12/5/03 3:25 PM HW14 Solution page 1 Probability for Engineering Applications SOLUTION Assignment #14 1. (1 pt) Express VAR[X+Y] in terms of E[X], E[Y], VAR[X], VAR[Y], and E[XY]. What happens if E[XY]= E[X] E[Y] ? (cf. 4.59) VAR[X+Y] = E[(X+Y) 2 ] – (E[X+Y]) 2 = E[X 2 ] +2E[XY] + E[Y 2 ] – (E[X]) 2 – 2E[X]E[Y] – (E[Y]) 2 = VAR[X] + VAR[Y] +2E[XY] – 2E[X]E[Y] VAR[X+Y] = VAR[X] +VAR[Y] only when X and Y are independent (i.e., when E[XY]=E[X]E[Y]) 2. (1 pt) Find E[X 2 Y] where X is a zero-mean, unit-variance Gaussian variable, and Y is a uniform random variable in [-2, +5], and X and Y are independent. X and Y are independent ] [ ] [ ] [ 2 2 Y E X E Y X E = X is zero-mean, unit-variance Gaussian r. v. 1 ] [ = X VAR , 0 ] [ = X E 1 ]) [ ( ] [ ] [ 2 2 = + = X E X VAR X E 5 . 1 2 )) 2 ( 5 ( ] [ = - + = Y E 5 . 1 5 . 1 1 ] [ ] [ ] [ 2 2 = × = = Y E X E Y X E 3. (2pts) Random variables X and Y have the pmf shown below. (This is the same pmf as in HW 11) Are the variables independent? Correlated? What is ρ X,Y ? (cf. 4.64) Y -1 0 +1 -1 0.10 0.05 0.10 0 0.05 0.15 0.15 X +1 0.25 0.10 0.05 Marginal PMF: 25 . 0 10 . 0 05 . 0 10 . 0 ] 1 [ 35 . 0 15 . 0 15 . 0 05 . 0 ] 0 [ 4 . 0 05 . 0 10 . 0 25 . 0 ] 1 [ = + + = - = = + + = = = + + = = X P X P X P 4 . 0 25 . 0 05 . 0 10 . 0 ] 1 [ 3 . 0 10 . 0 15 . 0 05 . 0 ] 0 [ 3 . 0 05 . 0 15 . 0 10 . 0 ] 1 [ = + + = - = = + + = = = + + = = Y P Y P Y P

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