Stat 310A/Math 230A Theory of Probability
Homework 1 Solutions
Andrea Montanari
Due on 9/30/2010
Option 1: Exercises on measure spaces
Exercise [1.1.4]
1. A and B \ A are disjoint with B = A (B \ A) so P(A) + P(B \ A) = P(B ) and rearranging gives the
des
Stat 310A/Math 230A Theory of Probability
Homework 7
Andrea Montanari
Due on November 18, 2010
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same proble
Stat 310A/Math 230A Theory of Probability
Homework 1
Andrea Montanari
Due on October 3, 2012
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) for
each problem. Staple sheets referring to the same problem
Stat 310A/Math 230A Theory of Probability
Homework 1
Andrea Montanari
Due on September 30, 2010
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) for
each problem. Staple sheets referring to the same prob
Stat 310A/Math 230A Theory of Probability
Midterm Solutions
Andrea Montanari
November 1, 2010
The midterm was long! This will be taken into account in the grading. We will assign points proportionally to the number of questions answered (e.g. Problem 1 co
Stat 310A/Math 230A Theory of Probability
Homework 3
Andrea Montanari
Due on 10/14/2009
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same problem, and
Stat 310B: Problem Set 2
Due in class on Tuesday, January 31
1. Let Xn be a martingale with X0 = 0 and E(Xn2 ) < . Show that for
any > 0,
E(Xn2 )
P ( max Xm )
.
1mn
E(Xn2 ) + 2
Hint: Use the fact that (Xn + c)2 is a submartingale and optimize
over c.
2.
Stat 310B: Problem Set 1
Due Tuesday, January 24
1. On the unit interval with Lebesgue measure, let X() = and for an arbitrary positive
integer n let Y () = n[n], where [x] denotes the largest integer x. What is E(X|Y )?
Explain your answer.
2. Suppose E(
STATS310A: Problem Set 4
Solve the five problems:
- Billingsley, chapter 13: 6, 91 , 10
- Billingsley, chapter 14: 5, 8
Also, the following additional problem: let F1 (x), F2 (x) be distribution functions on the line, R.
Define FU (x, y) = mincfw_F1 (x),
Stats 310A: HW3
Due 10-17-2016 in class
Reading
Billingsley Probability and Measure 3rd edition: sections 10 and 11.
Book Problems
Billingsley: 10.1, 10.2, 10.6, 11.1, 11.2.
Additional Problem
With the notation as in class, let mn = max1in si . Prove that
STATS 310: Homework 8
Problem 1
Total variation between two measures and , denoted as dT V (, ), is defined as follows:
dT V (, ) = k kT V = sup | (A) (A)|
AF
Prove the following properties of dT V :
Show that dT V (, ) = dT V (, ), dT V (, ) dT V (, ) +
Stat 310A/Math 230A Theory of Probability
Practice Final Solutions
Andrea Montanari
December 2, 2010
Problem 1
b
Let = cfw_0, 1N be the space of innite binary sequences = (1 , 2 , 3 , . . . ), and, for a b, write a for
the vector (a , a+1 , . . . , b ). L
Stat 310A/Math 230A Theory of Probability
Practice Midterm Solutions
Andrea Montanari
October 28, 2010
Problem 1
Consider the measurable space (, F ), with: = cfw_0, 1N the set of (innite) binary sequences =
(1 , 2 , 3 , . . . ); F the -algebra generated
Stat 310A/Math 230A Theory of Probability
Homework 2
Andrea Montanari
Due on October 7, 2010
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same problem,
Stat 310A/Math 230A Theory of Probability
Homework 2 Solutions
Andrea Montanari
Due on 10/7/2010
Exercises on measurable functions and Lebesgue integration
Exercise [1.2.14]
The same method works for all four parts.
1. Since B = (cfw_(, ] : R), it follows
Stat 310A/Math 230A Theory of Probability
Homework 3 Solutions
Andrea Montanari
Due on 10/14/2010
Exercises on inequalities and convergence
Exercise [1.3.21]
(a). Cauchy-Schwarz implies
2
(EY I(Y >a) ) EY 2 P(Y > a)
For EY > a 0 the left hand side is larg
Stat 310A/Math 230A Theory of Probability
Homework 4
Andrea Montanari
Due on October 21, 2010
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same problem
Stat 310A/Math 230A Theory of Probability
Homework 4 Solutions
Andrea Montanari
Due on October 25, 2010
Exercises on independent random variables and product measures
Exercise [1.4.18]
We will use the following fact repeatedly: the probability that X is d
Stat 310A/Math 230A Theory of Probability
Homework 5 Solutions
Andrea Montanari
Due on November 4, 2010
Exercises on the law of large numbers and Borel-Cantelli
Exercise [2.1.5]
Let > 0 and pick K = K ( ) nite such that if k K then r(k ) . Applying the Ca
Stat 310A/Math 230A Theory of Probability
Homework 6
Andrea Montanari
Due on November 11, 2010
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same proble
Stat 310A/Math 230A Theory of Probability
Homework 6 Solutions
Andrea Montanari
Due on November 11, 2010
Exercises on the law of large numbers and central limit theorem
Exercise [2.3.13]
Clearly, |Xn | = |Xn1 |Un |, resulting with
n
log |Xn | =
log |Uk |
Stat 310A/Math 230A Theory of Probability
Homework 7 Solutions
Andrea Montanari
Due on November 18, 2010
Exercises on weak convergence of distributions
Exercise [3.2.8]
p
D
(a). It is easy to see that Yn c if and only if Yn c (for example, this is a conse
Stat 310A/Math 230A Theory of Probability
Homework 8
Andrea Montanari
Due on December 2, 2010
Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per
each problem. Staple sheets referring to the same problem
Stat 310A/Math 230A Theory of Probability
Homework 8 Solutions
Andrea Montanari
Due on December 2, 2010
Exercises on characteristic functions
Exercise [3.3.10]
1. Denoting by X () the ch.f. of X , since X and X are i.i.d., the ch.f. of X is X () = X ().
H
Stat 310A/Math 230A Theory of Probability
Practice Final
Andrea Montanari
November 30, 2010
Solutions should be complete and concisely written. Please, mark clearly the beginning and end of each
problem. You have 3 hours but you are not required to solve
Stat 310A/Math 230A Theory of Probability
Practice Midterm
Andrea Montanari
October 23, 2010
This is a practice midtetm: rules described beow will be the same as for the real Midterm. Solutions wil be
posted.
Solutions should be complete and concisely wri