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Unformatted text preview: EE 278 Monday, July 13, 2009 Statistical Signal Processing Handout #6 Homework #3 Due Monday, July 20, 2009 1. Schwarz inequality. a. Prove the following inequality: ( E ( XY )) 2 ≤ E ( X 2 ) E ( Y 2 ) . Hint: Use the fact that E (( X + aY ) 2 ) ≥ 0 for any real number a . b. Prove that equality holds if and only if Y = cX for some constant c , and find c in terms of the second moments of X and Y . c. Use the Schwarz inequality to show that the correlation coefficient | ρ X,Y | ≤ 1 . d. Show that E ( ( X + Y ) 2 ) ≤ p E ( X 2 ) + p E ( Y 2 ) 2 . This is called the triangle inequality . 2. Jensen Inequality. Definition: A function g ( x ) is said to be convex (see Figure 1) over an interval ( a, b ) if for every x 1 , x 2 ∈ ( a, b ) and 0 ≤ λ ≤ 1, g ( λx 1 + (1- λ ) x 2 ) ≤ λg ( x 1 ) + (1- λ ) g ( x 2 ) . A function g is said to be strictly convex if equality holds only if λ = 0 or λ = 1. Moreover, if the function is twice differentiable, then it is convex iff g 00 ( x ) ≥ 0 for all x ∈ ( a, b ) (and strictly convex iff strict inequality holds for all x ∈ ( a, b )). a. Show that if g is a convex function over ( a, b ) and X ∈ X ⊂ ( a, b ) is a discrete random variable , then E ( g ( X )) ≥ g ( E ( X )) . (Hint: Use induction on the number of x points with nonzero probability, i.e., such that p X ( x ) > 0.) Note: This is called the Jensen inequality and it holds for continuous random variables as well (this can be shown by “discretizing” the random variable and using a limiting argument. You dont need to provide a proof of this.). b. Find the inequality relationship ( ≤ or ≥ ) between : i. E ( e 2 X ) and e E (2 X ) . ii. E (ln X ) and ln( E ( X )), for X ≥ 0. iii. ( E ( X 2 )) 6 and E ( X 12 )....
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- Spring '11
- Signal Processing