Q. 9.12 - A First Course in Probability

Question

Determine the entropy of the sum that is obtained when a pair of fair dice is rolled.

Step-by-Step Solution

Answer

The entropy of the sum  is $H\left(X\right)-3.27$ when a pair of fair dice is rolled.

Step 1: Given Information

We have to find the entropy of the sum that is obtained when a pair of fair dice is rolled.

Step 2: Simplify

We have$X\in \left\{2,...,12\right\}$ and the distribution of $X$ is

$X~\left(\begin{array}{ccccccccccc}2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ 1/36& 2/36& 3/36& 4/36& 5/36& 6/36& 5/36& 4/36& 3/36& 2/36& 1/36\end{array}\right)$

Now, the entropy will be

$$\begin{gathered} H\left(X\right)=-\sum _{i=2}^{12}{p}_i{\log }_2{p}_i \\ =-\underset{j=1}{\overset{5}{\sum \frac{j}{36}}}{\log }_2\frac{j}{36}-\frac{6}{36}{\log }_2\frac{6}{36} \\ =3.27 \end{gathered}$$