Question

Consider the memoryless binary channel shown in the figure below:

If the input are equiprobable with $P(S_0=P(S_1)=0.5$, then the conditional entropy $H\left(Y/X\right)$ is equal to _____ bits/symbol

• A. 0.985

Answer 1

The correct option is A 0.985
$\begin{array}{cc}P\left(Y|X\right)& =\left[\begin{array}{cc}0.4& 0.6\\ 0.5& 0.5\end{array}\right]\\ P\left(X\right)& =\left[\begin{array}{cc}0.5& 0.5\end{array}\right]\\ P\left[\right(X;Y\left)\right]& =\left[P\right(X){]}_d\left[P\left(\frac{Y}{X}\right)\right]\\ & =\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]\left[\begin{array}{cc}0.4& 0.6\\ 0.5& 0.5\end{array}\right]\\ & =\left[\begin{array}{cc}0.2& 0.3\\ 0.25& 0.25\end{array}\right]\\ \therefore H\left(Y|X\right)& =\sum _{i=0}^1\sum _{j=0}^1-P(x_i;y_j){\log }_{2}P\left(\frac{y_j}{x_i}\right)\\ & =-0.2{\log }_2\left(0.4\right)-0.3{\log }_2\left(0.6\right)-0.25{\log }_2\left(0.5\right)-0.25{\log }_2\left(0.5\right)\\ & =0.985bits/symbol\end{array}$