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32) are called the (one-step) transition probabilities of X . The distribution of X O is called the initial distribution of the Markov chain. The one-step transition probabilities and the initial distribution completely specify the distribution of X. 30) P(X0 = zo,. . , Xt = Zt) = P(X0 = 50) P(X1 = 51 I Xo = 20). P(Xt = Zt = P(X0 = 20) P(X1 = 51 I Xo = 50) ’ . P(Xt = Zt I xo = zo, . . xt-1 = 21-1) I x,-1 = X t - 1 ) . Since 8 is countable, we can arrange the one-step transition probabilities in an array.

This follows from Jensen’s inequality (if 4 is a convex function, such as - In, then IE[q5(X)]2 \$(E[X])). 57) is related to the CE in the following way: where f is the (joint) pdf of (X,Y ) and fx and fy are the (marginal) pdfs of X and Y, respectively. In other words, the mutual information can be viewed as the CE that measures 32 PRELIMINARIES the distance between the joint pdf f of X and Y and the product of their marginal pdfs fx and f y ,that is, under the assumption that the vectors X and Y are independent.

D) Calculate E[X I Y = y] for all y E ( 0 , l ) . e) Determine the expectations of X and Y . 21 Let { N t ,t 2 0) be a Poisson process with rate A = 2. 22 P(N4 = 3 I N2 = 1,N3 = 2), E“4 I N2 = 21, P(N[2,7]= 4, N[3,8] = 6), E[N[4,6]IN[1,5]= 31. 1 1. Let U1, U2, . . denote the times of success for the Bernoulli process X. a) Verify that the “intersuccess” times U 1 , U2 - U l , . . are independent and have a geometric distribution with parameter p = Ah. 24 If { N , , t 2 0 ) is a Poisson process with rate A, show that for 0 j = O , 1 , 2 , .