Utah State University
ECE 6010
Stochastic Processes
Homework # 8
Due Friday October 31, 2003
1. Suppose cfw_Xt , t 0 is a homogeneous Poisson process with parameter . Dene a
random variable as the time of the rst occurrence of an event. Find the p.d.f. an
Utah State University
ECE 6010
Stochastic Processes
Homework # 7
Due Friday Nov. 5, 2004
1. Suppose X t , t 0 is a homogeneous Poisson process with parameter . Dene a random
variable as the time of the rst occurrence of an event. Find the p.d.f. and the m
Utah State University
ECE 6010
Stochastic Processes
Homework # 6
Due Friday October 28, 2005
1. Suppose cfw_Xn ) is a sequence of independent r.v.s each of which is uniformly disn=1
tributed on the interval (0, 1). Dene a sequence of r.v.s cfw_Zn by Zn =
Utah State University
ECE 6010
Stochastic Processes
Homework # 5
Due Friday October 8, 2005
1. Box Muller: Let X1 U (0, 1) and X2 U (0, 1) (independent). Let
Y1 =
2 ln X1 cos 2X2
Y2 =
2 ln X1 sin 2X2
Show that
Y1 N (0, 1)
Y2 N (0, 1).
2. If X and Y are in
Utah State University
ECE 6010
Stochastic Processes
Homework # 4
Due Friday Oct. 7, 2005
1. Suppose X N (, ).
(a) Show that E [X] = and cov(X, X) = .
(b) Show that AX + b N (A + b, AAT ).
(c) Suppose > 0 and write = CC T . Show that C 1 (X ) N (0, I ).
2.
Utah State University
ECE 6010
Stochastic Processes
Homework # 3
Due Friday Sept. 23, 2005
1. Suppose X and Y are the indicator functions of events A and B , respectively. Find
(X, Y ), and show that X and Y are independent if and only if (X, Y ) = 0.
2.
Utah State University
ECE 6010
Stochastic Processes
Homework # 2
Due Friday Sept. 16, 2005
1. Suppose X is a r.v. with c.d.f. FX . Prove the following:
(a) FX is nondecreasing.
(b) lima FX (a) = 1.
(c) lima FX (a) = 0.
(d) FX is right continuous.
(e) P (a
Utah State University
ECE 6010
Stochastic Processes
Homework # 1
Due Friday September 9.
Reading
G&S, Chapter 1
Exercises
1. Create a list of all the stochastic processes you can think of that might occur in the real world
(not just examples from the text
ECE 6010
Lecture 10 Markov Processes
Basic concepts
A Markov process cfw_Xt one such that
P (Xtk+1 = xk+1 |X (tk ) = xk , X (tk1 ) = xk1 , . . . , X (t1 ) = x1 ) = P (Xtk+1 = xk+1 |Xtk = xk )
(for a discrete random process) or
f (xtk+1 |Xtk = xk . . . ,
ECE 6010
Lecture 9 Linear Minimum Mean-Square Error Filtering
Background
Recall that for random variable X and Y with nite variance, the MSE E [(X h(Y ) 2 ] is
minimized by h(Y ) = E [X |Y ]. That is, the best estimate of X using a measured value of
Y is
ECE 6010
Lecture 8 Random Processes in Linear Systems
Continuous time systems
Recall: A signal Xt through a linear system produces an output
h(t, s)Xs ds.
Yt =
The system is causal if h(t, s) = 0 for t < s. In this case,
t
Yt =
h(t, s)Xs ds.
The system
ECE 6010
Lecture 7 Analytical Properties of Random Processes
Let Xt be a function of time, and let h(t) be the impulse response of a (continuoustime) linear time invariant system. If Xt is the input to the system, then the output is
Yt = Xt h(t) =
h(t )X
ECE 6010
Lecture 5 Sequences and Limit Theorems
Convergent sequences of real numbers and functions
Denition 1 Let x 1 , x 2 , . . . be a sequence of real numbers. This sequence converges to a
point x R if for every > 0 there is an N Z such that x n x < fo
ECE 6010
Lecture 4 Change of Variables
Reading from G&S: Section 4.7, 4.8, 4.9, 4.10, 4.11
Changing variables: One dimension
A simply invertible function
Let Y = g (X ), where X is a continuous r.v. and g is a one-to-one, onto, measurable
function. Then
F
ECE 6010
Lecture 3 Random Vectors
Grimmet & Stirzaker: Section 4.9
Random Vectors
Random vectors are an extension of the bivariate random variables.
n r.v.s X1 , X2 , . . . , Xn dene a measurable mapping from an underlying sample space
(, F ) to (Rn , B n
pq~k&y~uwveqeGqFPt
o n m l i f y pd re r |thvte d he uy r r
yqstqe$eeysr@f )8qk jh D#xp)8q~k&eUesegev
r pv |ehh yt yv ct rte e xv r p n m l i i u o n m l i f ddet yh y r rth e x
ere n
db m k 9G ~uvqzeqcfw_FeFm k 9 syecsyq&tqet
pd re r |thvte d he uy r
uhz h
$gte
q ~ w heww h j g u vw j e s se vh u wqwh de qh d h
Gm 3tm rty9if9fh E pcfw_&ipGg rvhfutiiFfryG9if
hqw u vw u vw h swh dehq u v e v sw v q e vwq h hqw u v hqh
tTRikRiiuxGtt iw iuii f h if5tw fir
Y R R R & j j
~ ki Gk i
w e h e x e v
Utah State University
ECE 6010
Stochastic Processes
Homework #11
Due Friday Dec 10, 2004
These problems come from the Leon-Garcia text.
1. Let Mn denote the sequence of sample means from an i.i.d. random process Xn :
Mn =
X1 + x 2 + + X n
n
(a) Is Mn a Ma
ECE 6010
Programming Assignment #2
System and Autoregressive Identication
1
Introduction
System identication is the means by which systems are modeled mathematically based on measured data.
It is often a precursor to other engineering tasks, such as contr
Utah State University
ECE 6010
Stochastic Processes
Programming Exercise # 1
Due Friday September 9.
Introduction
This exercise will provide an opportunity to do some calculations and plots with actual data. The
intent is to make some of the abstract conc
Utah State University
ECE 6010
Stochastic Processes
Homework # 11 Solutions
1. Let Mn denote the sequence of sample means from an iid random process Xn :
Mn =
X1 + X 2 + + X n
.
n
(a) Is Mn a Markov process?
n
=
1
n
=
Mn
1
1
Xn + (1 )Mn1
n
n
Xi =
i=1
1
[X
Utah State University
ECE 6010
Stochastic Processes
Homework # 8 Solutions
1. Suppose cfw_Xt , t 0 is a Wiener process. Dene a process cfw_Y t , t 0 by Yt = Xt+D Xt for a
xed positive number D .
(a) Find the mean and autocorrelation functions of cfw_Y t .
Utah State University
ECE 6010
Stochastic Processes
Homework # 9 Solutions
1. Suppose cfw_Xt , t R is a ramdom process with power spectral density
SX ( ) =
1
.
(1 + 2 )2
Find the autocorrelation function of Xt .
RX ( )
= F 1 cfw_SX ( ) = F 1
=
=
=
=
=
=
1
Utah State University
ECE 6010
Stochastic Processes
Homework # 8 Solutions
1. Suppose cfw_Xt , t 0 is a Wiener process. Dene a process cfw_Y t , t 0 by Yt = Xt+D Xt for a
xed positive number D .
(a) Find the mean and autocorrelation functions of cfw_Y t .
Utah State University
ECE 6010
Stochastic Processes
Homework # 7 Solutions
1. Suppose cfw_Xt , t 0 is a homogeneous Poisson process with parameter . Dene a random variable as the time of the rst occurrence of an event. Find the p.d.f. and the mean of .
p.
Utah State University
ECE 6010
Stochastic Processes
Homework # 6 Solutions
1. Suppose cfw_Xn ) is a sequence of independent r.v.s each of which is uniformly disn=1
tributed on the interval (0, 1). Dene a sequence of r.v.s cfw_Zn by Zn = n(1 Yn ), where
Y
Utah State University
ECE 6010
Stochastic Processes
Homework # 5 Solutions
1. Let X1 U (0, 1) and X2 U (0, 1) (independent). Let Y1 =
Show that Y1 N (0, 1) and Y2 N (0, 1)
2 ln X1 cos(2X2 ) and Y2 = 2 ln X1 sin(2X2 ).
Here we have,
Y12 + Y22 = 2 ln X1 [(c
Utah State University
ECE 6010
Stochastic Processes
Homework # 4 Solutions
1. Suppose X N (, ).
(a) Show that E [X] = and cov(X, X) = .
Using characteristic functions:
1
X (u) = exp[iuT uT u]
2
Taking the gradient with respect to u we have
1
X (u)
= exp[i
ECE 6010 Stochastic Processes Homework #3
Problems from Grimmet & Stirzaker:
1. Prob 2.7.4
a)
1
3
3
1
1
P( < x ) = F( ) F( ) = .
2
2
2
2
2
b)
c)
1
P (1 < X < 2) = F (2) F (1) = .
2
1
P (Y X ) = P (X 2 X ) = P (X 1) = F (1) = .
2
d)
e)
1
3
P (X 2Y ) = P (X