MATH 235A Probability Theory
Lecture Notes, Fall 2009
Part III: The central limit theorem
Dan Romik
Department of Mathematics, UC Davis
Draft (December 4, 2009)
Lecture 10: Stirlings formula and the de Moivre-Laplace
theorem
We start with a motivating exa
MATH 235A Probability Theory
Lecture Notes, Fall 2009
Part II: Laws of large numbers
Dan Romik
Department of Mathematics, UC Davis
Draft version of 11/2/2009 (minor typos corrected 11/5/09)
Lecture 7: Expected values
7.1
Construction of the expectation op
MATH 235A Probability Theory
Lecture Notes, Fall 2009
Part I: Foundations
Dan Romik
Department of Mathematics, UC Davis
Draft version of 10/14/2009
Lecture 1: Introduction
1.1
What is probability theory?
In this course well learn about probability theory.
Homework Set No. 9 Probability Theory (235A), Fall 2009
Posted: 11/24/09 Due: Friday, 12/4/09 (Note extended due date!)
1. Compute the characteristic functions for the following distributions.
(a) Poisson distribution: X Poisson().
(b) Geometric distribut
Homework Set No. 8 Probability Theory (235A), Fall 2009
Posted: 11/17/09 Due: 11/24/09
1. (a) Prove that if X, (Xn ) are random variables such that Xn X in probability
n=1
then Xn = X .
(b) Prove that if Xn = c where c R is a constant, then Xn c in probab
Homework Set No. 7 Probability Theory (235A), Fall 2009
Posted: 11/10/09 Due: 11/17/09
1. (a) Read, in Durretts book (p. 63 in the 3rd edition) or on Wikipedia, the statement
and proof of Kroneckers lemma.
(b) Deduce from this lemma, using results we lear
Homework Set No. 6 Probability Theory (235A), Fall 2009
Posted: 11/3/09 Due: 11/10/09
1. Let f : [0, 1] R be a continuous function. Prove that
1
1
1
.
0
0
f
0
x1 + x 2 + . . . + xn
n
dx1 dx2 . . . dxn f (1/2).
n
2. A bowl contains n spaghetti noodles arra
Homework Set No. 5 Probability Theory (235A), Fall 2009
Posted: 10/27/09 Due: 11/3/09
1. Prove that if X is a random variable that is independent of itself, then there is a
constant c R such that P(X = c) = 1.
2. (a) If X 0 is a nonnegative r.v. with dist
Homework Set No. 4 Probability Theory (235A), Fall 2009
Posted: 10/20/09 Due: 10/27/09
1. If P, Q are two probability measures on a measurable space (, F ), we say that P is
absolutely continuous with respect to Q, and denote this P < Q, if for any A F ,
Homework Set No. 3 Probability Theory (235A), Fall 2009
Posted: 10/13/09 Due: 10/20/09
1. Let X be an exponential r.v. with parameter , i.e., FX (x) = (1 ex )1[0,) (x). Dene
random variables
Y
=
X := supcfw_n Z : n x
Z = cfw_X := X X
(the integer part of
Homework Set No. 2 Probability Theory (235A), Fall 2009
Posted: 10/6/09 Due: 10/13/09
1. (a) Let X be a random variable with distribution function FX and piecewise continuous
density function fX . Let [a, b] R be an interval (possibly innite) such that
P(
Homework Set No. 1 Probability Theory (235A), Fall 2009
Due: 10/6/09
1. (a) If (, F , P) is a probability space and A, B F are events such that P(B ) = 0, the
conditional probability of A given B is denoted P(A|B ) and dened by
P(A|B ) =
P(A B )
.
P(B )
P
MATH/STAT 235A Probability Theory
Lecture Notes, Fall 2011
Dan Romik
Department of Mathematics, UC Davis
November 11, 2011
Contents
Chapter 1: Introduction
5
1.1
What is probability theory? . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The a
Homework Set No. 7 Probability Theory (235A), Fall 2011
Due: 11/15/11
1. Prove that if F and (Fn ) are distribution functions, F is continuous, and Fn (t)
n=1
F (t) as n for any t R, then the convergence is uniform in t.
2. Let (x) = (2 )1/2 ex
2 /2
be t
Solutions to Homework Set No. 6 Probability Theory (235A), Fall 2011
1. Let f : [0, 1] R be a continuous function. Prove that
1
1
1
f
.
0
0
0
x1 + x 2 + . . . + xn
n
dx1 dx2 . . . dxn f (1/2).
n
Solution. Let X1 , X2 , . . . be independent uniformly distr
Homework Set No. 6 Probability Theory (235A), Fall 2011
Due: 11/08/11
1. Let f : [0, 1] R be a continuous function. Prove that
1
1
1
.
0
0
f
0
x1 + x 2 + . . . + xn
n
dx1 dx2 . . . dxn f (1/2).
n
Hint: Interpret the left-hand side as an expected value; us
Solutions to Homework Set No. 5 Probability Theory (235A), Fall 2011
Solution to 1.
(a)
ydF (y ) = E(X ).
dxdF (y ) =
0
y =x
y
dF (y )dx =
0
0
P(X x)dx =
0
0
Here, dF (x) means f (x)dx if X has a density. Otherwise, the integral represents a
Lebesgue-Stie
Homework Set No. 5 Probability Theory (235A), Fall 2011
Due: Tuesday 11/01/11 at discussion section
1. (a) If X 0 is a nonnegative r.v. with distribution function F , show that
P(X x) dx.
E(X ) =
0
(b) Prove that if X1 , X2 , . . . , is a sequence of inde
Solutions to Homework Set No. 4 Probability Theory (235A), Fall 2011
1. A function : (a, b) R is called convex if for any x, y (a, b) and [0, 1] we
have
(x + (1 )y ) (x) + (1 )(y ).
(a) Prove that an equivalent condition for to be convex is that for any x
Homework Set No. 4 Probability Theory (235A), Fall 2011
Due: 10/25/11 at discussion section
1. A function : (a, b) R is called convex if for any x, y (a, b) and [0, 1] we
have
(x + (1 )y ) (x) + (1 )(y ).
(a) Prove that an equivalent condition for to be c
Solutions to Homework Set No. 3 Probability Theory (235A), Fall 2011
1. Let X be an exponential r.v. with parameter , i.e., FX (x) = (1 ex )1[0,) (x). Dene
random variables
Y
=
X := supcfw_n Z : n x
Z = cfw_X := X X
(the integer part of X ),
(the fractio
Homework Set No. 3 Probability Theory (235A), Fall 2011
Due: Tuesday 10/18/11 at discussion section
1. Let X be an exponential r.v. with parameter , i.e., FX (x) = (1 ex )1[0,) (x). Dene
random variables
Y
=
X := supcfw_n Z : n x
Z = cfw_X := X X
(the in
Solutions to Homework Set No. 2 Probability Theory (235A), Fall 2011
1. (a) Let X be a random variable with distribution function FX and piecewise continuous
density function fX . Let [a, b] R be an interval (possibly innite) such that
P(X [a, b]) = 1,
an
Homework Set No. 2 Probability Theory (235A), Fall 2011
Posted: 10/3/11 Due: 10/11/11
1. (a) Let X be a random variable with distribution function FX and piecewise continuous
density function fX . Let [a, b] R be an interval (possibly innite) such that
P(
Solutions to Homework Set No. 1 Probability Theory (235A), Fall 2011
1. (a) If (, F , P) is a probability space and A, B F are events such that P(B ) = 0, the
conditional probability of A given B is denoted P(A|B ) and dened by
P(A B )
.
P(B )
Prove the t
Homework Set No. 1 Probability Theory (235A), Fall 2011
Due: 10/4/11 at discussion section
1. (a) If (, F , P) is a probability space and A, B F are events such that P(B ) = 0, the
conditional probability of A given B is denoted P(A|B ) and dened by
P(A|B