Question

Consider a random process X(t) defined by X(t) - Ycoset, 0st where o is a constant 1. and Y is a uniform random variable over (0,1) (a) Classify X(t) (b) Sketch a few (at least three) typical sample function of X(t) (c) Determine the pdfs of X(t) at t 0, /4o, /2, o. (d) EX() (e) Find the autocorrelation function Rx(t,s) of X(t) (f) Find the autocovariance function Rx(t,s) of X(t)

Answer 1

A Tandom pro Cess XC) s defined by X CE-YCaSut 470 ue Camstant (o,1) y uniform random vamable aves classty Xt Stationany Sense Stationany pa ces will be process. But it cuill not be Y t) au Wide X CE) Let usTT Xt) I At t o -% Xt)-0 X)--I 3u/2 At 七 wt 21T At t ! -1F uat ut wt functfon of tme fs Calleof whfch Ps a A 1andom vaniable nandom dcers

C) Det es m? ne Pobabilfty denstty functt on.s of x) at teo, 12 aus l 4 Co f(y,)= Probability densfty function CoS cst XEE ay Co O fCy,0)= Qt t 0 Co r f Cog fs 4 4 at t-T Cos T fC9, CoS at t ' ao CoG T T Cog fus IL fC, Tao) at t (t

which gives the rando mean Parocess fiy2 X Ct)=YCog cast E Cxu .dy y Cog uot Gog cot Hese Hence Because here teme mean is Pt is not a Auto orrelation independent of Stationay function mot wide Sense pocess of X omu E L XCt)XCt2) ACF 1 Sa0 = Cos ue t2 t e [ y t Cog uo a Cog uo 2 3 0 RxCtiyt,) =Co9 wt, Coswte

f) functi on Auto covarian ce CyCt,t)ltt)-yt) t) CxCti,t) Cos wts where xlti - Ex(t CoS wta Ctsta CoS wt Co5 wta 4 Co5 wt CoS wt2 3 Cog wti CoS wt 1a

Mean functien ytt gives as the expected Vaue of x at time t' Note:- K Cti,t) and C, Ct,t) givas ntranation about how XCt) and Xlta) X Telated are are uneraelated tf Cxlt,t) XCt and XLt - 0 Cernel ated posttively they are +Ve thay megatve ly orarel ated are =-Vo Similaoi ty of Rx Ctit)gives nformattom XC and x (t abeut