12.6 Consider a system whose environment changes according to a Markov chain. Specifically, Yn is the state of the environment at the beginning of the nth period, where Y 5 fYn; n $ 1g is a Markov chain with a state transition probability matrix P. At the beginning of every period, a Bernoulli experiment is performed whose outcome depends on the state Yn. Specifically, given that the process is in state Yn, the probability of success is pn and probability of failure is qn 5 1 2 pn. Thus, the outcome of the experiment, Xn, depends on the state of the environment. Assume that the conditional PMF of Xn is given by pXn ðxjYÞ 5 pn x 5 1 qn x 5 2 1  Define the random variable Kn as follows: Kn 5 0 n 5 0 X1 1 X2 1?1 Xn n $ 1  If we assume that a unit positive reward is associated with a success in the Bernoulli experiment, and a unit negative reward is associated with a failure, then Kn is the total reward at the end of the nth period. The bivariate process fðKn; YnÞ; n $ 0g is a Bernoulli-modulated Markov process. Consider the case where P is the matrix P 5 p00 p01 p10 p11   5 1 2 α α β 1 2 β   Give the state transition rate diagram of the process, assuming that pn 5 p.