1. A random variable X has the following pdf: a) Determine the value of C. b) Find the characteristic function. c) Find the variance. fx(x) = Ce-3|x| 2. Consider a random process X(t) defined by X(t) Ycos(wt + 0) = where Y and are independent rvs and are uniformly distributed over (-A,A) and (0,), respectively. a) Find the mean of X(t). b) Find the autocorrelation function Rx(t,s) of X(t). 3. The random process X(t) with autocorrelation function Rxx(t) system with the following transfer function: 2 = e -2|T| is input to a linear H(s) = (s + 1)(s + 2) a. Find the second moment of the rv X(4). b. Find the output autocorrelation function Ryy (t). 4. A binary communication system transmits signals s(t), i = 0,1, with the a priori probabilities P(So) 2/3 and P(s) = 1/3. The receiver test statistic is = z=a, +no, where a = -1 and a = 2 and no is a zero-mean Gaussian random variable with variance 0.3. The decision is made via H > Z < Ho (1) where H and Ho are the possible hypotheses. Choosing H is equivalent to deciding that signal s1(t) was sent, and choosing Ho is equivalent to deciding that signal so(t) was sent. The above equation (1) indicates that hypothesis H is chosen if z> , and hypothesis Ho is chosen if z