Question

[CLU: Let g(t) be the rectangular pulse shown in Fig. Q3. The random process X(t) is defined as X(t)=Ag(t), where A assumes the values +1 with equal probability. Figure Q3 (a) Find the pmf of X(t). (b) Find uxo). (c) Find the joint pmf of X(t) and X(t+d). (d) Find Cx{1.1+d), d > 0.

Answer 1

From given figure that 9(t) 2 retangular Pulre and it lies outfide of the St) cleare interval [o,] 1(x(4)=0)%=D1 P(c0)=9(0)=1)= P (>ct) = -9(4)=-1) - 2. b) malt)= 1KP[A(t) = 1] +(-1)0[nC4) =-1] %3D mi(t)= 12 +-1)* 을 te0s는 2- 2. 2.

mx(t)=0 %3D to find x (t +d) we have X(t) and PlaCe) = E1,x(t +d) = 0]= e 2. P(xct)=0,x(t+d) = 0] =1- 3. For equation O condition tE l0,1]t +dE[o,]. f& equation ☺ condition te (0,], t +d ¢ (o11) FR equation Condition will be tfl0,1], t+d¢[o,1]. will be will be

) cx(e,t+a)= EL^ce)x (t+d)] -mx(t).mx(t+d) CP+7)x:(77*]7 = =P(act)}=1,a(t+d) =1] +P[xC+)=-1,A(t+d) = -1 abready in from ore solued the Vales that %3D 2. 2- 2. that means 17t€[0,1],t tdEfe,!] 0-7 Romaining coses this is valid