P5.9 Let X, be an i.i.d. random sequence with marginal pdf fn(x) =e , x20. Let Y, be another i.i.d. random sequence, independent of X,, with marginal pdf fn(y) = 2e ZY, y20. Define a random sequence U, by the difference equation Un= Xn+ Xn-1+ Yn The random sequence Un can be thought of as the result of passing Xn through a first order moving average filter and then adding noise. Determine: (a) the mean pu(n) of Un, (b) the auto-correlation Ru(m, n), and (c) the marginal density f,(u) of Un at a fixed n. Answers: : (a) , (b) Ru(m, m-0) = Ru(() = 25 9 -+-6(0)+6(6+1)+6(6-1), (c)2e *[(x-1) +e *Ju(x). 4