APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW1 due January 24.
Sec 2.1, Problem 1 Let = cfw_r : r [0, 1] be the set of rational points of [0, 1]. A
the algebra of sets each of which is a nite sum of disjoint sets A of one of the forms
cfw_r : a < r < b,

APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW3 due February 26.
P
P
Sec 2.10, Ex. 4 Let n , n , and let and be equivalent (P( = ) = 0). Show
that
Pcfw_| n n | 0, n
The triangle inequality gives
a.s.
| n n | | n | + | | + | n | = | n | + | n |,
Since |

APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW2 due February 12.
Sec 2.6, Problem 1 Prove that the expectation E of a nonnegative random variable
satises:
E = sup Es,
cfw_sS:s
where S is the set of simple random variables.
We already know that E = limn

APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW4 due April 2.
Sec 2.12, Ex. 9 Let be an integer-valued random variable and (t) be its characteristic
function. Show that:
P( = k ) =
1
2
eitk (t)dt,
k = 0, 1, 2,
Since is integer-valued, we rewrite integral

NOTES FOR APM503/MAT570, APPLIED/REAL ANALYSIS, FALL 2012
JACK SPIELBERG
Contents
Part 1. Metric spaces and continuity
1. Metric spaces
2. The topology of metric spaces
3. Sequences
4. Continuous functions
5. Bounded linear maps
6. Cauchy sequences and co

APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW5 due April 25.
Problem 10 Let X1 , X2 , be a Markov chain.
a. Prove that for any positive integer n 3, 3 k n, and i1 < < in , it holds that
P ( Xi n = x i n , , Xi k = x i k | Xi k 1 = x i k 1 , , Xi 1 = x i

APPLIED PROBABILITY AND STOCHASTIC PROCESSES APM504
SPRING 2013
DR. VLADISLAV VYSOTSKY
Abstract. This course is a classical measure-theory based introduction to probability theory including expectation, notions of convergence, weak and strong laws of larg