School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #12 Solutions
Problem 1
11.1)
a)Mn=1/n
=1/n[xn+(n-1)Mn-1]=1/n xn+(1-1/n)Mn-1
Clearly if Mn-1 is given then Mn depends only on
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #8 Solutions
Problem
9.2)
(a) is
If =1:
If =2:
.
.
If =6:
1
the output of fair die
Xn:.111
Xn:.222
Xn:.666
(b)P[Xn=1]=pcfw_k=1
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #10 Solutions
Problem 1
10.1)
a)SX(f)=F[g(
=AT
Table in Appendix B
b) SX(f)=g( )
Rx()=Aw
Problem 2
10.5)Rx()=
a)from Fourier t
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #11 Solutions
Problem 1
10.27)
a)Syx(f)=H(f)Sx(f)=
Rxy=
[Syx(f)]=
b)Sy(f)=
Ry()=
>0
Sx(f)=
[Sy(f)]=
c)Ry(0)=
Problem 2
10.34)R
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #9 Solutions
Problem 1
9.37)
a)pcfw_T1<T2=
=
b)f1(t)=
dt=
=2/3
f2(t)=
F1(t)=1F2(t)=1fmin(t)=f1(t)+f2(t)-f1(t)F2(t)-F1(t)f2(t)=
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #6 Solutions
Problem 1
5.48)
(a) P[ <1/2, |y|<1/2]= P[
<1/2] P[|y|<1/2]= P[x< ] P[y<1/2]=
(1/2)(
(b) P[4x<1,y<0]= P[x<1/4] P[y
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #2 Solutions
Problem 1
2.26) Identities of this type are shown by application of the
axioms. We begin by treating (A B) as a s
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #1 Solutions
Problem 1
2.2) The output of this experiment consists of a pair of numbers
(x,y)where x=number of dots in first t
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #5 Solutions
Problem 1
5.20)
(b) FX(x)= Fxy(x,)= 1-
x>1
0
otherwise
Fy(y)= Fxy(,y)= 1-
y>1
0
otherwise
(c)Pcfw_x<3,y<=5)=Fxy(3
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #3 Solutions
Problem 1
4.12)
(b) P[x<=-1]=.5
P[x=-1]=.5
P[x<.5]=P[x<=.5]-p[x=.5]=
- 0= .75
P[-.5<x<.5]=P[x<=.5]-P[x=.5]-P[x<=-
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #7 Solutions
Problem 1
7.1)
(a)E[x+y+z]=E[x]+E[y]+E[z]=0
VAR[x+y+z]=VAR(x)+VAR(y)+VAR(z)+2COV(x,y)+2COV(x,z)+2COV(y,z)=
1+1+1+