S-72_2410_solution_2

S-72_2410_solution_2 - S-72.2410 Information Theory Haanp...

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: S-72.2410 Information Theory Haanp aa & Linja-aho Homework 2, solutions 2009 Homework 2, solutions Deadline: November 16th, 16:00 The box for returning exercises is in the E-wing, 2nd floor corridor. 1. Let X and Y be two independent integer-valued random variables. Let X be uniformly distributed over { 1 , 2 ,..., 8 } , and let Pr { Y = k } = 2- k ,k = 1 , 2 , 3 ,... (a) Find H ( X ) (b) Find H ( Y ) (c) Find H ( X + Y,X- Y ). Solution: (a) For uniform distribution H ( X ) = 8(- 1 8 log 1 8 ) = log 8 = 3 (b) H ( Y ) =- k =1 1 2 k log 1 2 k = 1 2 k k log 2 = k =0 k ( 1 2 ) k = 1 2 (1- 1 2 ) 2 = 2 (c) Because ( X,Y ) ( X + Y,X- Y ) is a one-to-one -transformation, H ( X + Y,X- Y ) = H ( X,Y ). Because X and Y are independent, H ( X,Y ) = H ( X ) + H ( Y ) = 3 + 2 = 5. 2. A deck of n cards in order 1 , 2 ,...,n is provided. One card is removed at random then replaced at random. What is the entropy of the resulting deck? Solution: There are n 2 possible ways to shuffle the deck as described, and all have the same probability. But the solution is not that simple, because some shuffling actions result in same deck. There are basically three ways to shuffle the deck: The card is removed and put back in the same place. In this case,The card is removed and put back in the same place....
View Full Document

This note was uploaded on 10/27/2010 for the course ECE 221 taught by Professor Sd during the Spring '10 term at Huston-Tillotson.

Page1 / 4

S-72_2410_solution_2 - S-72.2410 Information Theory Haanp...

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