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Chapter 9: Q. 9.14 (page 413)

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).

Short Answer

Expert verified

a) H(Y)=2.58

b) HY(X)=0.65

c) H(X,Y)=3.12.

Step by step solution

01

Part (a) Step 1: Given Information

We have given value of X=1ifthesumofthediceis60otherwise

and Yequal the value of the first die.

We have to computeH(Y).

02

Part (a) Step 2: Simplify

We haveYcan assume each value 1,......6 with equal probabilities 12. So, the entropy of Yis

localid="1651482886925" role="math" HY−ipilogpi=−log16=log6=2.58

03

Part (b) Step 1: Given Information

We have to computeHY(X).

04

Part (b) Step 2: Simplify

For y∈1,…,5, we have that X=1given that Y=ymeans that on the first die we have obtained localid="1648144538249" y, and in order to obtain 6 in the sum, we need localid="1648144481486" 6-yon the second die,

XY=y~015/61/6

So the entropy is given by

localid="1651482832761" HY=yX=−∑i=0,1pi∣ylogpi∣y=−56log56−16log16=0.65∪

05

Part (c) Step 1: Given Information

Need to compute H(X,Y).

06

Part (c) Step 2: Explanation

From the proposition in the chapter, we have

HX,Y=HY+HYX=2.58+0.5417=3.12

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Most popular questions from this chapter

A random variable can take on any of n possible values x1, ... , xn with respective probabilities p(xi), i = 1, ... , n. We shall attempt to determine the value of X by asking a series of questions, each of which can be answered “yes” or “no.” For instance, we may ask “Is X = x1?” or “Is X equal to either x1 or x2 or x3?” and so on. What can you say about the average number of such questions that you will need to ask to determine the value of X?

Show that for any discrete random variable Xand functionf

H(f(X))≤H(X)

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