1
Homework 1 Solutions
Exercise 1.1.
1. If Fi are -algebras for each i I then iI Fi is a -algebra. (recall that
this was the property that allowed us to dene (A)
2. If F and G are -algebreas, F G is not necessarily a -algebra.
Solution:
1. The intersectio
HW: 2
Course: M385C/CSE384K - Theory of Probability I
Page: 1 of 4
University of Texas at Austin
HW Assignment 2
Problem 2.1. Let (S, S , ) be a nite measure space. Show that
Hint: For the second part, a measure
space with nite (and small) S will do.
(lim
HW: 7
Course: M385C/CSE384K - Theory of Probability I
Page: 1 of 4
University of Texas at Austin
HW Assignment 7
Problem 7.1. Let f : R2 R be a bounded Borel-measurable function, and let X and Y be independent
random variables. Dene the function g : R R b
Lecture 5: Fubini-Tonelli and Radon-Nikodym
1 of 13
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 5
Theorems of Fubini-Tonelli and
Radon-Nikodym
Products of measure spaces
We have seen that it is possible to dene prod
Lecture 7: Weak Convergence
1 of 9
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 7
Weak Convergence
The denition
In addition to the modes of convergence we introduced so far (a.s.convergence, convergence in probabilit
Exam: nal
Course: M385C/CSE384K - Theory of Probability I
University of Texas at Austin
Page: 1 of 2
Name :
Course (M or CSE) :
The Final Exam
Problem 1.1. Let f : R R R be a function such that
x f ( x, y) is Borel-measurable for each y R, and
1
2
3
4
5
Lecture 4: Lebesgue spaces and inequalities
1 of 10
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 4
Lebesgue spaces and inequalities
Lebesgue spaces
We have seen how the family of all functions f L1 forms a vector
spa
Lecture 1: Measurable spaces
1 of 13
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 1
Measurable spaces
Families of Sets
Denition 1.1 (Order properties). A countable1 family cfw_ An nN of
subsets of a non-empty set S i
Lecture 3: The Lebesgue Integral
1 of 14
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 3
The Lebesgue Integral
The construction of the integral
Unless expressly specied otherwise, we pick and x a measure space
(S, S ,
Lecture 4: The Simple Random Walk
1 of 9
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 4
The Simple Random Walk
We have defined and constructed a random walk cfw_ Xn nN0 in the
previous lecture. Our next t
Lecture 5: Generating Functions
1 of 16
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 5
Generating Functions
The path-counting method used in the previous lecture only works
for computations related to the
Lecture 3: Stochastic processes and simulation
1 of 11
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 3
Stochastic processes and simulation
STOCHASTIC PROCESSES
Definition 3.1. Let T be a subset of [0, ). A
Lecture 2: Mathematica in 15 min
1 of 10
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 2
Mathematica in 15 min
Mathematica is a glorified calculator. Here is how
to use it (actually, this is just a tip of
Lecture 10: Conditional Expectation
1 of 17
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 10
Conditional Expectation
The definition and existence of conditional expectation
For events A, B with P[ B] > 0, we recall th
Lecture 1: Probability review
1 of 19
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 1
Probability review
RANDOM VARIABLES
A large chunk of probability is about random variables. Instead of
giving a precise
OPERATING SYSTEMS
THREE EASY PIECES
Remzi H. Arpaci-Dusseau
Andrea C. Arpaci-Dusseau
To Vedat S. Arpaci, a lifelong inspiration
Contents
To Everyone . . . .
To Educators . . . .
To Students . . . . .
Acknowledgments .
Final Words . . . .
References . . .
Lecture 8: Markov Chains
1 of 21
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 8
Markov Chains
THE MARKOV PROPERTY
Simply put, a stochastic process has the Markov property if its future evolution depends o
Lecture 9: Classification of States
1 of 27
Course:
M362K Intro to Stochastic Processes
Term:
Fall 2014
Instructor: Gordan Zitkovic
Lecture 9
Classification of States
There will be a lot of definitions and some theory before we get
to examples. You might
Lecture 8: Characteristic Functions
1 of 9
Course:
Theory of Probability I
Term:
Fall 2013
Instructor: Gordan Zitkovic
Lecture 8
Characteristic Functions
First properties
A characteristic function is simply the Fourier transform, in probabilistic language
HW: 1
Course: M385C/CSE384K - Theory of Probability I
Page: 1 of 4
University of Texas at Austin
HW Assignment 1
Problem 1.1. A partition of a set S is a family P of non-empty subsets
of S with the property that each S belongs to exactly one A P .
1. Show
Exam: midterm
Course: M385C/CSE384K - Theory of Probability I
Page: 1 of 1
University of Texas at Austin
The Midterm Exam
Problem 1.1. Let (S, S , ) be a measure space, let T : S S be a measurable map, and let T be the
push-forward of via T . For f L0 ()
2
Homework 2
Exercise 2.1. Let X be random variable on a probability space (, F , P). Show that if Y
is a random variable which is measurable with respect to (X ) then there exists a (Borel)
measurable function f : R R such that Y = f (X ).
Hint: one way
3
Homework 3
Exercise 3.1.
1. Suppose that X is a random variable with density function f , and P(
X ) = 1. Let g : (, ) R be a strictly increasing and dierentiable function.
Compute the density of g (X ) in terms of f and g .
2. compute the density of t
4
Homework 4
Exercise 4.1. Let X1 , X2 , . . . be i.i.d. with P(Xi = (1)k k ) = C/k 2 log k for k 2 where
C is chosen to make the sum of all probabilities one. Show that E[|Xi |] = but there is a
nite constant such that
Sn
in probability as n .
n
Exercis
5
Homework 5
Exercise 5.1. Consider the space L0 (, F , P) of measurable random variables (identied up
to a.s. equality). Denote by
d(X, Y ) = E
|X Y |
.
1 + |X Y |
Show that
1. d(X, Y ) is a metric on L0 (which means d(X, Y ) 0, d(X, Y ) = 0 if and only
6
Homework 6
Exercise 6.1. If Fn converges weakly to F and F is continuous, then supx |Fn (x) F (x)| 0.
Exercise 6.2. Let Xn , n = 1, be integer valued random variables. Show that Xn converges
weakly to X if and only if P(Xn = k ) P(X = k ) as n for each
7
Homework 7
Exercise 7.1. Show that if Xn X in probability than Xn = X . Conversely, if Xn = c
where c is a constant, then Xn c in probability.
Exercise 7.2. Let X1 , X2 , . . . be i.i.d. with characteristic function (t). Denote Sn = X1 +
+ Xn .
1. If (
8
Homework 8
Exercise 8.1.
1. (conditional variance) Dene
V ar(X |G ) = E[X 2 |G ] (E[X |G ])2 .
Show that
V ar(X ) = E[V ar(X |G )] + V ar(E[X |G ]).
2. Show that if X and Y are random variables with E[X |G ] = Y and E[X 2 ] = E[Y 2 ],
then X = Y a.s.
Ex
9
Homework 9
Exercise 9.1.
1. Let (Xn )n an adapted process and N a stopping time. Show that
XN FN
2. Let N M two stopping times and A FN . Show that the random time
L = 1A N + 1Ac M,
is a stopping time.
Exercise 9.2. Show that a set of random variables (
2
Homework 2, Solutions
Exercise 2.1. Let X be random variable on a probability space (, F , P). Show that if Y
is a random variable which is measurable with respect to (X ) then there exists a (Borel)
measurable function f : R R such that Y = f (X ).
Hin