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mtsample-sol - EE 278 Statistical Signal Processing Sample...

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EE 278 Saturday, July 18, 2009 Statistical Signal Processing Handout #9 Sample Midterm Solutions 1. (20 points) Inequalities. a. (2 points) None . Since P( A | B ) = P( B | A )P( A ) P ( B ) and P( A ) can be larger or less than P( B ), no equality or inequality holds in general. b. (2 points) . Since p X ( x ) = Σ y p X,Y ( x, y ) and p X,Y 0, p X ( x ) p X,Y ( x, y ). c. (2 points) None . Here f X ( x ) = integraltext −∞ f X,Y ( x, y ) dy , so we cannot in general say how f X ( x ) relates to f X,Y ( x, y ). If f X ( x ) > 0, then f Y | X ( y | x ) = f X,Y ( x, y ) /f X ( x ) can be greater or less than 1. d. (3 points) =. Since expectation is linear, E parenleftbigg X 1 X 1 + X 2 parenrightbigg + E parenleftbigg X 2 X 1 + X 2 parenrightbigg = E parenleftbigg X 1 + X 2 X 1 + X 2 parenrightbigg = E(1) = 1 . e. (3 points) . Let g ( x ) = x log 2 x . Then g ( x ) is convex since g ′′ ( x ) = 1 / ( x ln 2) > 0 for x > 0. Now, by Jensen’s inequality and E ( X ) = 2, E( X log 2 X ) E( X ) log 2 E( X )) = 2 log 2 2 = 2 . f. (4 points) . Consider P { ( XY ) 2 > 16 } ≤ E (( XY ) 2 ) 16 radicalbig E ( X 4 ) E ( Y 4 ) 16 = 1 8 , where the first inequality follows from the Markov inequality and the second inequality follows from the Schwarz inequality.
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