By Raymond W. Yeung (auth.)

ISBN-10: 1441986081

ISBN-13: 9781441986085

ISBN-10: 1461346452

ISBN-13: 9781461346456

A First direction in details thought is an updated creation to info conception. as well as the classical subject matters mentioned, it presents the 1st finished remedy of the idea of I-Measure, community coding conception, Shannon and non-Shannon kind info inequalities, and a relation among entropy and staff concept. ITIP, a software program package deal for proving details inequalities, can be integrated. With numerous examples, illustrations, and unique difficulties, this booklet is great as a textbook or reference e-book for a senior or graduate point direction at the topic, in addition to a reference for researchers in comparable fields.

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**Sample text**

We will determine the largest c which satisfies D(pllq) ~ cd2(p, q). a) Let A = {x : p(x) ~ q(x)} , p = {p(A) , 1 - p(A)} , and ij {q(A), 1- q(A)} . Show that D(pllq) ~ D(pllij) and d(p,q) = d(p, ij). b) Show that toward determining the largest value of c, we only have to consider the case when X is binary. c) By virtue of b), it suffices to determine the largest c such that p I-p q 1- q p log - + (1 - p) log - - - 4c(p - q)2 ~ 0 for all 0 ~ p , q ~ 1, with the convention that 0 log % = 0 for b ~ 0 and a log IT = 00 for a > O.

At the beginning of Chapter 2, we mentioned that the entropy H(X) measures the amount of information contained in a random variable X. In this chapter, we substantiate this claim by exploring the role of entropy in the context of zero-error data compression. 41 R. W. 1 A FIRST COURSE IN INFORMATION THEORY THE ENTROPY BOUND In this section, we establish that H(X) is a fundamental lower bound on the expected length of the number of symbols needed to describe the outcome of a random variable X with zero error.

1'1 - 1 elements. From the last theorem, we have = x,Y = 1) :S log(lXI where this upper bound does not depend on x. 171) which completes the proof. 0 Very often, we only need the following simplified version when we apply Fane's inequality. The proof is omitted. 1'1 . 32 A FIRST COURSE IN INFORMATION THEORY Fano's inequality has the following implication. If the alphabet X is finite. as Pe ~ O. 159) tends to 0, which implies H(XIX) also tends to O. However. this is not necessarily the case if X is infinite, which is shown in the next example.

### A First Course in Information Theory by Raymond W. Yeung (auth.)

by Michael

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