Poisson Process question is attached

Problem 4: Let Xt, t 2 0 be a real-valued continuous-time Random Process with the following properties: (1) Xte {-1, 1} for all t 2 0, (2) Xo = 1 with probability one, (3) ti, i = 1,2, ... be the random switching times for Xt, that is if Xt- = 1 then X + = -1, where Atk = th - th-1, k = 1, 2, ... is iid sequence with exponential distribution with parameter 1 > 0, Atk ~ le-it, t 2 0. (a) Let The = Et=1 At; , k = 1, 2, ..., determine the density function and cumulative distribution function of Tk (b) Let Nt, t 2 0 denote the number of random switch points of the random process Xt in the time interval (0, t] with No = 0. (i) SHOW that: Probability(Nt = k) = (at)" e-at for k = 1, 2, ... (ii) Compute: E[Nt], E[(Nt - E[N.])2], E[N.2] K! (iii) SHOW that: Probability((Nt+s - Ns) = k) = (t)" k ! -e-it for k = 1, 2, ... (iv) SHOW that: If (s, t) and (u, v) are disjoint intervals then (Nt - Ns) and (Nv - Nu) are independent. (v) Compute: Probability(Xt = 1), Probability(Xt = -1), E[X ], and SHOW that Ry (t + t, t) = E[Xt+tXt] = e-It, for -0o