Define a random process X(t) as follows: (1) X(t) assumes only one of two possible levels 1 or -1 at any time, (2) X(t) switches back and forth between its two levels randomly with time, (3) the number of level transitions in any time interval is a Poisson random variable, that is, the probability of exactly k transitions, when the average rate of transitions is λ, is given by [(t)/k!] exp(-), (4) transitions occurring in any time interval are sta- tistically independent of transitions in any other interval, and (5) the levels at the start of any interval are equally probable. X(t) is usually called the random telegraph process. It is an example of a discrete random process. lation function of the precess. (d) Discuss the stationarity of X(t).

2. (5 ts) Suppose X is a discrete random variable with probability mass function (pmf) f(x) = c(x - 1)2 for r= -4,-2,2,4, | where c is a constant. (a) Find c. (b) Find P(-2 < X < 5).

Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).

1. (a) lind autocovariance function and autocorrelation function of Poisson Process. (b) If X(1) = A cos ct+ Bsin et and Y) and B are uncorrelated random variables with mean 0 and variance 1. Examine whether {X(1)} and {Y(!)} are jointly widen -sense stationary or not. Bcoset - Asin ct. where c is a constant, A

3. Suppose X is a discrete random variable with pmf defined as p(x) = log10 ( for %3D x = {1,2,3,...9} Prove that p(x) is a legitimate pmf.