Let t, 0 be a real-valued continuous-time Random Process with the following properties

Problem 4: Let X t, t 2 0 be a real-valued continuous-time Random Process with the following properties: (1) X, E {1, 1} for all t >_> 0, (2) X 0 = 1 with probability one, (3) t,- ,i = 1 ,2, be the random switching times for X , that is ithi = 1 then tht = -1, where At, = Ck tk_1 , k = 1, 2, is iid sequence with exponential distribution with parameter/l > U, Atk ~ IlaAt, t 2 0. (a) Let T], = :11 At,- , k = 1, 2, , determine the density function and cumulative distribution mction of T], (b) Let N,, t 2 0 denote the number of random switch points of the random process X, in the time interval (0, t] with N0 = 0. (i) SHOW that' Probabtttty(N, _ k)- _ Ea w\" for k = 1, 2, (ii) Compute: E[N,],E[(N, E[N,] D2] Hui, 2] 1: (iii) SHOW that Probability((N,,, N ,) _ k) _ \"it: e\" for k = 1,2, (iv) SHOW that: If (S, t) and (11,17) are disjoint intervals then (Nt NS) and (Nv N u) are independent. (v) Compute: ProbabilityOf, = 1), Probabilityi', = 1), E[X,], and SHOW that RXCt + t, t) = E[X,+,X,] = trim, for 00