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lecture1solutions - ISYE 6414 Spring 2009 Solution Lecture...

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ISYE 6414 - Spring 2009 Solution Lecture 1 - Prerequisites Problem 1. Q n i =1 c · e x i = c n e ( n i =1 x i ) Problem 2. The most likely sum of two rolled six-sided dice is 7 The most likely sum of three rolled seven-sided dice is 12 Problem 3. Cov ( aX + b, Y ) = E [( aX + b - E ( aX + b )) ( Y - E ( Y ))] = = E [( aX + b - aE ( X ) - b ) ( Y - E ( Y ))] = E [( a ( X - E ( X ))) ( Y - E ( Y ))] = = E [( X - E ( X )) ( Y - E ( Y ))] = aCov ( aX + b, Y ) Problem 4. Cov n X i =1 X i , n X j =1 Y j = E n X i =1 X i - E n X i =1 X i !! n X j =1 Y j - E n X j =1 Y j = = E n X i =1 X i - n X i =1 E ( X i ) ! n X j =1 Y j - n X j =1 E ( Y j ) = = E n X i =1 ( X i - E ( X i )) n X j =1 ( Y j - E ( Y j )) = = n X i =1 E ( X i - E ( X i )) n X j =1 ( Y j - E ( Y j )) = = n X i =1 n X j =1 E [( X i - E ( X i )) ( Y j - E ( Y j ))] Problem 5. By definition we know that λ ( x ) 0; x ∈ < . Also: R -∞ λ ( x ) dx = R x 0 λ ( x ) dx = R x 0 f ( x ) dx 1 - F ( x 0 ) = 1 - F ( x 0 ) 1 - F ( x 0 ) = 1. Now suppose X f ( x ) = e - x · 1 (0 , ) . Then: P ( X < x | X x 0) = P ( x 0 < X X x ) P ( X x 0 ) = P ( x 0 X x ) P ( X x 0 ) = e - x 0 - e - x e - x 0 = 1 - e - ( x - x 0 )
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