A nonstationary respectively: random sequence X[n] has the following mean and autocovariance sequences, Ux[n]=cos(tn); Kx[n+l, n] = Kx[l] = {1,2,3,2,1} 1 Note that the up-arrow “↑” in Kx[l] above indicates the l = 0 position and that the covariance values not given are zero. This random sequence is processed through a linear system described by the difference equation y[n] =2x[n] +x[n+1]+x[n-1], -

3. Let X1, X2,..., be a sequence of independent and identically distributed random variables. Let N be Poisson distributed with mean u and is independent of the X,'s. Define N W =EX;. i=1 We define W = 0 if N = 0. (a) Suppose each X; is normally distributed with mean 0 and variance 1. Work out the moment generating function for W given N. [4 marks] (b) Show that the moment generating function of W is given by Mw (t) = exp(ue"2 - ), t eR. [5 marks] (c) Calculate the mean and variance of W. [5 marks] (d) Now consider Z = NX1. Find the mean and variance of Z. [6 marks]

Suppose there are three possible states of nature, and the class-conditional PDFS are the Cauchy distributions 1 p(rw;) = 1 i = 1,2,3. (1) 2 1+ bị Let ai = -2, a2 = 0, az = 2, bị = b2 = 1, b3 = 0.5. Let P(wi) = P(w2) = 0.4, P(w3) = 0.2. (a) Find the decision boundaries and the decision regions for the Bayesian decision (b) Calculate the probability of error for the classification done according to this

7. Prove or disprove that the following function is a cumulative distribution function (cdf). If the function is a cdf, is the random variable continuous or discrete? Why? (Note: This is a special case of a logistic distribution.) Fx(x)= 1 1+ e-z -∞<<∞.