/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q.9.17 Show that for any discrete rando... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Q.9.17 (page 413)

Show that for any discrete random variable Xand functionf

H(f(X))鈮�H(X)

Short Answer

Expert verified

The given statement is proved below.

Step by step solution

01

Given Information

We have to prove that

H(f(X))鈮�H(X)

02

simplify

Consider two random variables, XandfX. Using the formula about the entropy of two random variables, we have that

HX,fX=HX+HXfX

Observe that knowing X, random variable fXbecomes absolutely deterministic, i.e. there is no more uncertainty about its value. Hence HXfX=0which implies that HX,fX=HX. On the other hand, we can also write

HX,fX=HfX+HfXX鈮�HfX

which finally implies that

HfX鈮�HX

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

On any given day, Buffy is either cheerful (c), so-so (s), or gloomy (g). If she is cheerful today, then she will be c, s, or g tomorrow with respective probabilities .7, .2, and .1. If she is so-so today, then she will be c, s, or g tomorrow with respective probabilities .4, .3, and .3. If she is gloomy today, then Buffy will be c, s, or g tomorrow with probabilities .2, .4, and .4. What proportion of time is Buffy cheerful?

A pair of fair dice is rolled. Let

X=1ifthesumofthediceis60otherwise

and let Y equal the value of the first die. Compute (a) H(Y), (b) HY(X), and (c) H(X, Y).

Suppose that 3 white and 3 black balls are distributed in two urns in such a way that each urn contains 3 balls. We say that the system is in state i if the first urn contains i white balls, i = 0, 1, 2, 3. At each stage, 1 ball is drawn from each urn and the ball drawn from the first urn is placed in the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth stage, and compute the transition probabilities of the Markov chain {Xn, n 脷 0}.

asdsadasdsadsad

Prove that if X can take on any of n possible values with respective probabilities P1, ... ,Pn, then H(X) is maximized when Pi = 1/n, i = 1, ... , n. What is H(X) equal to in this case?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.