Problem 3 Discrete-time counterpart of Poisson counting process. Let X1, X2,.. . be a sequence of i.i.d. Bernoulli (p) random variables, i.e., 1 - P. 1 =0 P(Xn = x) = r =1 otherwise. Let Yn = Ex-1 Xx denote the number of arrivals up to time n, n e {1, 2, 3,...}. Let ; denote the length of time between the (i - 1)th and the ith arrival, i.e., A; = n; - nj-1, where n; is the time index of the ith arrival. In this problem, we will establish analogies between counting processes in continuous time and in discrete time. (a) Find the distribution of the number of arrivals in interval [I, m], i.e., Ek, Xk = Ym - Yi-1. Are arrivals in nonoverlapping intervals independent? (b) Find P(1 = n) and P(2 = n2|1 = ni). Show that 1, 2, ... are i.i.d. geometrically distributed random variables. (c) Show that the geometric PMF has the following "memoryless property": for A ~ geometric(p), P(A > n+ k | A > k) is independent of k. (d) The interarrival times of the continuous-time Poisson process are exponentially distributed. Does the exponential distribution also have a similar "memoryless property"? (e) Find the mean and (auto) covariance functions of Yr. Is Y, a WSS process? (f) Find the mean and (auto)covariance functions of the continuous time Poisson counting process N(t). Is N(t) a WSS process