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 tim
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 th
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 in
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
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
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
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
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 o
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,
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
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 t
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
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. a
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 mapp
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[email protected] )8qk jh D#xp)8q~k&eUesegev
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ere n
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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. ran
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 ofte
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 actua
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 p
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) F
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 funct
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) F
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 occur
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). Den
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
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