Consider three stochastic variables X_t, Y_t and Z_t which assume the values −1 or 1 for t = 1,2,3, . . .. We assume that Z can swap freely between −1 and 1 with constant probability μ_Z at each time step.

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We assume that when Z(t 1) = 1, at time 7 the two variables X and Y swap independently of each other between -1 and 1 with probability x and y. If Z(1-1) = -1, the variables X and Y are unable to change at time t. Assume that at time t = 1 X, Y and Y assume the values -1 and 1 with equal probability equal to (a) Show that Prob{X, = a} = for a (-1,1} and t = 2,3,.... (b) Show that the covariance and the mutual information between X and Y are zero, i.e. (XY) (X) (Y) = 0 and I(X, Y) = 0. = (c) Compute the conditioned probabilities P(x = a|x-1 = b) for x X, Y, Z and a,b (-1,1} and show for example that P(X = a| X-1 = b) = { | - y P(Z1-1=1)=1-x if a=b xP(Z=1)=x (d) Show that the transfer entropy between Z and X is given by if a b Tz_x=log2+ [(1 ux)log(1 ux) (2 ux)log(2 x)]. (9.92) (9.93) Plot Tz x. Can you think of an explanation of why Tzx is an increasing function of .. Now simulate the three processes X, Y and Z with the slight change that X, and Y, can swap only when Z-5 = 1. (f) Study the transfer entropy between Z- and X, and Y, and plot the transfer entropy as a function of the time lagt. (You can check your computations against Sec. 7 in [359].)