EE230hw6solution - Middle East Technical University...

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Middle East Technical University Department of Electrical and Electronics Engineering EE230: Probability and Random Variables HOMEWORK 6 Solutions Derived Distributions, Covariance and Correlation, Iterated Expectations, Transforms 1) a) b) c)   . . 0 15 15 , 80 50 3900 1 , ~ ~ ~ ~ w e v r v v v f v r f r v f V R R V d) P{R>90 | V>90} = 0.5 e) P{V>100 | R>90} = P{V>100 and R>90} / P{R>90} P{V>100 and R>90} = ((80 x 105) (80 x 80 / 2) (75 x 75 / 2)) / 3900 = 0.6121 P{R>90} = P{V>100 and R>90} + (25 x 25 / 2) / 3900 = 0.6923 P{V>100 | R>90} = 0.6121 / 0.6923 = 0.8842 e   e f E ~ 15 15 30 1 r v r f R ~ 15 v 15 v 30 1 v   v f V ~ 180 50 130 1 v r 180 50 15 15 65 35 165 195 90 100
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2) a-b) Let W be W = X-Y , first we will find pdf of W, then we will obtain pdf of Z using pdf of W. For this purpose firstly we will find the joint pdf of X and W then we will integrate over X to find pdf of W. Note that by multiplication rule we know that ( ) ( ) ( ) Therefore we need ( ) which will be obtained as follows: ( ) ( ) ( ) ( ) ( ) Differentiating both sides with respect to z we obtain ( ) ( ) ( ) since we know that ( ) ( ) . Therefore
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