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Unformatted text preview: 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 ln2) > 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 ....
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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