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Unformatted text preview: EE 178 Wednesday, March 18, 2009 Probabilistic Systems Analysis Handout #24 Sample Final Solution 1. a. ≤ . This follows by the unionofevents bound: P( ∪ 100 i =1 A i ) ≤ 100 summationdisplay i =1 P( A i ) ≤ 10 2 parenleftbigg max 1 ≤ i ≤ 100 P( A i ) parenrightbigg ≤ . 01 . b. ≤ . For each Z = z , Var( X + Y  Z = z ) = Var( X  Z = z ) + 2Cov( X,Y  Z = z ) + Var( Y  Z = z ) = Var( X  Z = z ) + Var( Y  Z = z ) , by independence . Thus E[Var( X + Y  Z )] = E[Var( X  Z )] + E[Var( Y  Z )]. But since E(Var( X  Z )) ≤ Var( X ) and E(Var( Y  Z )) ≤ Var( Y ) (from the law of conditional variances), it follows that E [Var( X + Y  Z )] ≤ Var( X ) + Var( Y ) . c. ≥ . Let Z = X 2 , then E( Z 2  Y ) = E( X 4  Y ) and E( Z  Y ) = E( X 2  Y ). The result follows by the fact that the second moment is larger than or equal to the square of the mean. d. =. Consider E( X 2 Y 6 ) = E( X 2 ) E( Y 6 ) by independence = (1 + 1)(5 × 3) = 30 . e. NONE. Uncorrelation does not necessarily imply independence, which is needed for equality. f. ≥ . We use Markov inequality P { ( X + Y ) 2 < 2 } = 1 P { ( X + Y ) 2 ≥ 2 } ≥ 1 E[( X + Y ) 2 ] 2 = 1 1 2 (E( X 2 ) + 2 E( XY ) + E( Y 2 )) = 1 2 . 1 2. Clearly F Z ( z ) = 0 if z ≤ 0. From Figure 1, for z > 0, F Z ( z ) = P { Z ≤ z } = 1 P { Z > z } = 1 2 integraldisplay ∞ z integraldisplay x − z e − y e − x dy dx by symmetry = 1 2 integraldisplay ∞ z e − x (1 e − ( x − z ) ) dx = 1 2( e − z 1 2 e − z ) = 1 e − z . Thus Z ∼ Exp(1). y = x z y = x + z z z x y Figure 1: The set { Z ≤ z } . 3. Note that U and V are independent. a. We first note that, when either x < 0 or u < 0, F X,U ( x,u ) = 0, and when both x > 1 and u > 1, F X,U ( x,u ) = 1. When 0 ≤ u ≤ x ≤ 1, F X,U ( x,u ) = P { X ≤ x,U ≤ u } = P { U ≤ u } = u. When 0 ≤ x ≤ u ≤ 1, 2 F X,U ( x,u ) = P { X ≤ x,U ≤ u } = P { UV ≤ x,U ≤ u } = integraldisplay x P { V ≤ 1 } du ′ + integraldisplay u x P braceleftBig V ≤ x u ′ bracerightBig du ′ = x integraldisplay u x x...
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This note was uploaded on 10/26/2011 for the course MATH 180C taught by Professor Eggers during the Spring '09 term at Aarhus Universitet.
 Spring '09
 Eggers
 Systems Analysis

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