- 101 Ne 2 5 Exercises 67 Question No 6 Are The Increments Of The Stochastic Process X T T 0 Defined In Ex Ample 2 1 (20.83 KiB) Viewed 237 times
Example 2.1.6. Let Y be a random variable having a U(0,1) distribution. We define the stochastic process {X(t), t > 0} by X(t) = evt for t 20 The first-order density function of the process can be obtained by using Propo- sition 1.2.2 (see Example 1.2.6): dIn(/t) S(q;t) = fx(o)(x) = ſy (ln(x/t)) dar if xe (t, te) 2 Next, the mean EX (C) of the process at time t > 0 is given by E[X(t) = 5" ellt. 1 dy=t(e - 1) for > 0 or, equivalently, by E[X(t) = ("..dr = dir -tette-1) for 120
Finally, we have X(t)X(+ + 8) = e't.ex (t + s) = €2V +(t + 8) It follows that Rx{t,t + 3) = E[X(t)X(+ + x)) = Ele='t( + )) = t+8)¢?,! Vsto