Let Q be a symmetric transition probability matrix, that is qy qu for all i,j- 1,... ,N. Consider a Markov chain which, when the present state is i, generates the value of a random variable X such that P(X j) dy, and if X-j, then either moves to state j with probability b,/(b, + bi), or remains in state i otherwise, where b,, j-1,..., N, are specified positive numbers. Show that the resulting Markov chain is time reversible with stationary probabilities a Cb,, j1,.., N

2. Let Q be a symmetric transition probability matrix, that is, 53,-}- = q},- for all i, j. Consider a Markov chain which, when the present state is i, generates the value of a raindom variable X such that P{X = j} = (3,}, and if X = j, then either moves to state j with probability b}- /(b,- + bf), or remains in state 1' otherwise, where b}, j = l . . . , N, are specied positive numbers. Show that the resulting Markov chain is time reversible with limiting probabilities