91Ó°ÊÓ

A transition probability matrix is said to be doubly stochastic if $$\sum_{i=0}^{M} P_{i j}=1$$ for all states \(j=0,1, \ldots, M .\) Show that such a Markov chain is ergodic, then \(\prod_{j}=1 /(M+1), j=0,1, \ldots, M\).

A pair of fair dice is rolled. Let $$ X=\left\\{\begin{array}{ll} 1 & \text { if the sum of the dice is } 6 \\ 0 & \text { otherwise } \end{array}\right. $$ and let \(Y\) equal the value of the first die. Compute (a) \(H(Y),(\mathrm{b}) H_{Y}(X),\) and \((\mathrm{c}) H(X, Y)\).