hw4 - EE 278 Statistical Signal Processing Homework#4 Due...

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EE 278 October 14, 2009 Statistical Signal Processing Handout #6 Homework #4 Due: Wednesday October 21 1. Two envelopes. A fixed amount a is placed in one envelope and an amount 5 a is placed in the other. One of the envelopes is opened (each envelope is equally probable), and the amount X is observed to be in it. Let Y be the (unobserved) amount in the other envelope. a. Find E parenleftBig Y X parenrightBig . b. Find E parenleftBig X Y parenrightBig . c. Find E( Y ) E( X ) . 2. Schwarz inequality. a. Prove the following inequality, which is known as the Schwarz inequality . ( E( XY ) ) 2 E( X 2 ) E( Y 2 ) . Hint: Use the fact that E ( ( X + aY ) 2 ) 0 for every real number a . b. Prove that equality holds if and only if either Y = cX or X = cY for some constant c . c. Use the Schwarz inequality to show that the correlation coefficient ρ X,Y satisfies | ρ X,Y | ≤ 1 . d. Show that E ( ( X + Y ) 2 ) ( radicalbig E( X 2 ) + radicalbig E( Y 2 ) ) 2 . This is called the triangle inequality . 3. Jensen’s inequality. (Bonus) A function g ( x ) is said to be convex on an interval ( a, b ) if for every x 1 , x 2 in ( a, b ) and for every λ satisfying 0 λ 1, g ( λx 1 + (1 - λ ) x 2 ) λg ( x 1 ) + (1 - λ ) g ( x 2 ) . Further, g ( x ) is said to be strictly convex if equality holds only for λ = 0 and λ = 1. It can be shown that if g ( x ) is twice differentiable, then it is convex iff g primeprime ( x ) 0 for all x in ( a, b ) and strictly convex iff g primeprime ( x ) > 0 for all x in ( a, b ). See Figure 1. a. Show that if g ( x ) is convex on ( a, b ) and X ∈ X ⊂ ( a, b ) is a discrete random variable, then E ( g ( X ) ) g ( E( X ) ) . This is called Jensen’s inequality. (Jensen’s inequality holds for continuous random variables as well, which can be shown by “discretizing” the random variable and using a limiting argument.) Hint: use induction on the number of
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