hwset3 - EE 278 Monday, July 13, 2009 Statistical Signal...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )....
View Full Document

Page1 / 5

hwset3 - EE 278 Monday, July 13, 2009 Statistical Signal...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online