ESE 520 Probability and Stochastic Processes
Instructor: Vladimir Kurenok
EXAM 2- Practice
Name (Please print):
Total: 70 points (each problem is 10 points worth)
Instructions:
1. You must show all work to completely justify your answers in order to recei

ESE 520 Probability and Stochastic Processes
Lecture 15
Introduction to stochastic processes: general theory and facts.
Defintion 1. A stochastic process (Xt )tI will be a family of random
variables Xt defined on a probability space (, F, P ):
Xt : (, F,

ESE 520 Probability and Stochastic Processes
Lecture 12
Conditional expectation of one variable with respect to another. Application to least squares estimation
Let X, Y : (, F, P ) (IR, B(IR) be two random variables (dsicrete or
continuous). As it is wel

ESE 520 Probability and Stochastic Processes
Lecture 13
Convergence of random variables. The law of large numbers.
Given a sequence of random variables
Xn : (, F, P ) (IR, B(IR), n = 1, 2, .
it is natural to study the convergence of cfw_Xn , n = 1, 2, .
T

ESE 520 Probability and Stochastic Processes
Lecture 20
White noise continued.
To be able to work with white noise process (eventhough we cannot prove
the existence of it by Kolmogorovs theorem since we do not have the
corresponding family of FFDs we can

ESE 520 Probability and Stochastic Processes
Lecture 9
Expectation (continued). Covariance and correlation.
Some more examples.
Example 1. Let X be a random variable with N (, 2 ) probability distribution. Then V ar(X) = 2 .
Indeed:
Z
V ar(X) =
(x )2
Z
1

ESE 520 Probability and Stochastic Processes
Lecture 1
Introduction. Probability space (, F, P ).
What is Probability as a part of mathematics? Probability is dealing with
randomness as such and, in particular, help us to regorously understand
how to cal

ESE 520 Probability and Stochastic Processes
Lecture 21
White noise representations of signals.
Assume that (Xt ), t IR is a WSS process with mean (t) and covariance
function R( ).
Let us consider a deterministic dynamical system given by the following
OD

ESE 520 Probability and Stochastic Processes
Lecture 17
The Poisson process continued.
What can be said about the relation of a Poisson process (Nt ), t 0 and
the sequence of random variables cfw_Tk , k = 1, 2, . from the point of view
of their probabilit

ESE 520 Probability and Stochastic Processes
Lecture 11
Characteristic and moment-generating functions
Definition 1. For a random variable X its characteristic function X is
defined as
X (t) := E[eitX ] = E[cos tX + i sin tX].
(1)
Since |eitx | = 1, the c

ESE 520 Probability and Stochastic Processes
Lecture 10
Covariance matrix. Bivariate normal distribution
Now we extend the notion of the covariance to a random vector of any
dimension n, n = 1, 2, .
Definition 1. Let X = (X1 , X2 , ., Xn ) be a random vec

ESE 520 Probability and Stochastic Processes
Lecture 7
Functions of jointly continuous random vectors. Conditional densities.
Let X = (X1 , ., Xn ) : IRn be a random vector, n 6= 1.
Consider a transformation : IRn IRn where = (1 , ., n ) and i :
IRn IR. I

ESE 520 Probability and Stochastic Processes
Lecture 3
One-dimensional random variables and their probability distributions:
concrete examples.
A. Discrete random variables/discrete probability distributions.
From the previous discussion we can summarize

ESE 520 Probability and Stochastic Processes
Lecture 8
Expectation and its properties.
Let X : IR be a random variable.
Formally, the expected value of X (or expectation) is defined as
Z
EX =
X(w)dP (w)
(1)
where the integral in (1) is calculated as a cor

ESE 520 Probability and Stochastic Processes
Lecture 3
Probability measures on Borel sets. Random variables and their probability distributions: general theory.
Let = IR (or any set from IR) and let us look at some ways how we can
construct a probability

ESE 520 Probability and Stochastic Processes
Lecture 2
Probability space (, F, P ): continued.
Example 1. Let be any non-empty set (not necessarily a set of numbers)
and let w0 be a fixed point. For any subset A from , we define
1, w0 A
(A) :=
0, w0
/ A.

ESE 520 Probability and Stochastic Processes
Lecture 18
Gaussian processes/Brownian motion.
Definition 1. A stochastic process (Xt ), t I is said to be Gaussian if all
its finite-dimensional distributions are Gaussian.
We already know that any Gaussian ve

ESE 520 Probability and Stochastic Processes
Lecture 6
Random vectors and their probability distributions. Multidimensional
cdfs and pdfs.
Under a random vector we understand a map
X = (X1 , X2 , ., Xn ) : IRn
where each Xi : IR, i = 1, 2, ., n is random

ESE 520 Probability and Stochastic Processes
Lecture 22
Markov processes.
Definition 1. A stochastic process (Xt ), t I is called a n-dimensional
Markov process (n 1) if for any finite set of times 0 t1 t2 .
tm s t, any Borel set B IRn , and any points x

ESE 520 Probability and Stochastic Processes
Instructor: Vladimir Kurenok
EXAM 1
Name (Please print):
crimp/cfw_Z
Total: 70 points (each problem is 10 points worth)
Instructions:
1. You must Show all work to completely justify your answers in order to rec

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May 11, 2016
ESE 520 Probability and Stochastic Processes
Instructor: Vladimir Kurenok
Optional exam
Name (Please print):
Total: 70 points (each problem is 10 points worth)
Instructions:
1. You must show all work to completely justify your answers in or

March 3, 2016
ESE 520 Probability and Stochastic Processes
Instructor: Vladimir Kurenok
EXAM 1
Name (Please print):
Total: 70 points (each probiem is 10 points worth)
Instructions:
1. You must show all work to completely justify your answers in order

April 28, 2016
ESE 520 Probability and Stochastic Processes
Instructor: Vladimir Kurenok
EXAM 2
Name (Please print):
Total: 70 points (each problem is 10 points worth)
Instructions:
1. You must show all work to completely justify your answers in order t