Strong law of large numbers
Last updated:10-14-2003
1
Borel-Cantelli Lemmas If An is a sequence of subsets of we let
lim sup An = lim An = cfw_ that are in innetly many An n=m
m
(the limit exists since the sequence is decreasing in m) and let
lim inf An
Notes on Memoryless Random Variables
Math 468 / 568 Spring 2010
klin@math.arizona.edu
A random variable X is memoryless if for all numbers a and b in its range, we have
P (X > a + b|X > b) = P (X > a) .
(1)
(We are implicitly assuming that whenever a and
Solutions to Homework Set No. 5 Probability Theory (235A), Fall 2013
1. If X 0 is a nonnegative r.v. with distribution function F , show that
P(X x) dx.
E(X ) =
0
Solution.
P(X x)dx =
0
0
y =x
y
dF (y )dx =
dxdF (y ) =
0
0
ydF (y ) = E(X ).
0
Here, dF (x)
Solutions to Homework Set No. 3 Probability Theory (235A), Fall 2013
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. 6 Probability Theory (235A), Fall 2013
Due: 11/11/13 at discussion section
1. Let X and (Xn ) be random variables dened on a probability space (, F , P ). Show
n=1
P
that if Xn X (convergence in probability) then there is a subsequence (X
MATH 235B Probability Theory
Lecture Notes, Winter 2011
Dan Romik
Department of Mathematics, UC Davis
March 15, 2012
Contents
Chapter 1: A motivating example for martingales
4
Chapter 2: Conditional expectations
7
2.1
Elementary conditional expectations .
10/1/13
UC Davis M ath: Blank syllabus
Department Syllabus
MAT 235A: Probability Theory
When taught: Fall, every year (alternating years, taught by Dept of Statistics)
Suggested text: Probability- Theory and Examples, by Rick Durrett ($70, ISBN: 053442441
MATH/STAT 235A Probability Theory
Lecture Notes, Fall 2013
Dan Romik
Department of Mathematics, UC Davis
October 16, 2013
Contents
Chapter 1: Introduction
6
1.1
What is probability theory? . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
The al
MATH/STAT 235A Probability Theory
Lecture Notes, Fall 2013
Dan Romik
Department of Mathematics, UC Davis
June 26, 2013
Contents
Chapter 1: Introduction
5
1.1
What is probability theory? . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The algeb
Homework Set No. 5 Probability Theory (235A), Fall 2013
Due: 11/4/13 at discussion section
To get full credit, solve problems 13 and at least one of problems 46.
1. If X 0 is a nonnegative r.v. with distribution function F , show that
P(X x) dx.
E(X ) =
0
Solutions to Homework Set No. 4 Probability Theory (235A), Fall 2013
1. Compute E(X ) and Var(X ) when X is a random variable having each of the following
distributions:
1. X Binomial(n, p).
2. X Poisson().
3. X Geom(p).
4. X U cfw_1, 2, . . . , n (the di
Homework Set No. 4 Probability Theory (235A), Fall 2013
Due: 10/28/13 at discussion section
1. Compute E(X ) and Var(X ) when X is a random variable having each of the following
distributions:
1. X Binomial(n, p).
2. X Poisson().
3. X Geom(p).
4. X U cfw_
Solutions to Homework Set No. 3 Probability Theory (235A), Fall 2013
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. 2 Probability Theory (235A), Fall 2013
Due: 10/14/13 at discussion section
1. (Transformation of a random variable) Let X be a random variable with distribution function FX and piecewise continuous density function fX . Let [a, b] R be an
Homework Set No. 3 Probability Theory (235A), Fall 2013
Due: 10/21/13 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 integer pa
Solutions to Homework Set No. 2 Probability Theory (235A), Fall 2013
1. (Transformation of a random variable) Let X be a random variable with distribution function FX and piecewise continuous density function fX . Let [a, b] R be an interval
(possibly inn
Solutions to Homework Set No. 1 Probability Theory (235A), Fall 2013
1. (The total probability formula) 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
Homework Set No. 1 Probability Theory (235A), Fall 2013
Due: 10/7/11 at discussion section
1. (The total probability formula) 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 denote
MAT235: Discussion 4
1
Order statistics
Let X1 , , Xn be independent independent identically distributed variables with a common density
function f . Let X(1) X(n) denote the order statistics. Then the density of X(n) can be
obtained as:
P (X(n) x) = P (m
MAT235: Discussion 3
1
Convolution
Joint density:
P (X, Y ) A) =
fX,Y (x, y )dxdy.
A
Convolution of X, Y is
fX +Y (z ) =
f (x, z x)dx.
If X and Y are independent, then
fX (x)fY (z x)dx =
fX +Y (z ) =
fX (z y )fY (y )dy.
Example: Find the density function
MAT235: Discussion 5
1
Simple random walk
d
d
d
Let cfw_(Sn ) be the simple symmetric random walk on Z d . That is S0 = 0, Sn = n=1 Xk where
n=0
k
X1 , X2 , are i.i.d. d-dimensional random vectors distributed uniformly on the 2d points cfw_ej : j =
1, ,
MAT235: Discussion 1
9/30/13
1
Moments of Discrete Random Variables
E (X n ) =
xn f (x) where f is the probability function of X .
Example: Find the expectation of X Bin(n, p).
Method 1: Using the denition of expectation.
n
n
E (X ) =
kf (k ) =
k=0
k
k=0
MAT235: Discussion 2
10/7/13
1
Construction of non-measurable set
For probability measure there are some properties we would like it to have. For example, we would
like the probability of the union of a nite or innite sequence of disjoints sets to be equa