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Q. 9.15

Expert-verified

A First Course in Probability

Found in: Page 413

Book edition 9th

Author(s) Sheldon M. Ross

Pages 432 pages

ISBN 9780321794772

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Short Answer

A coin having probability p = 2 3 of coming up heads is flipped 6 times. Compute the entropy of the outcome of this experiment.

The entropy of the outcome of this experiment is $H (X) = 5.5098$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Information

We have given the probability of the coin $p = \frac{2}{3}$ .

Step 2: Simplify

Considering random vector $X = (X_{1}, \dots, X_{6})$ where $X_{i}$ are independent equally distributed variables with distribution which given as

$X_{i} ~ Bern (\frac{2}{3})$

Calculating the entropy of $X$ .

localid="1648135034976" $H (X) = - \sum_{x \in {(0, 1)}^{6}} p (x) log p (x)$

This sum contains $2^{6}$ summands. Then,

localid="1648135254966" $\sum_{x \in {(0, 1)}^{6}} = p (x) log p (x) = - (\begin{matrix} 6 \\ 0 \end{matrix}) {(\frac{1}{3})}^{6} log {(\frac{1}{3})}^{6} = - (\begin{matrix} 6 \\ 1 \end{matrix}) {(\frac{1}{3})}^{5} \frac{2}{3} log {(\frac{1}{3})}^{5} \frac{1}{6} = - (\begin{matrix} 6 \\ 6 \end{matrix}) {(\frac{2}{3})}^{6} log {(\frac{2}{3})}^{6}$

Hence,

$H (X) = 5.5098$

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