Ph.D. Preliminary Examination
Stat 501 & 502
August 20, 2008
[1 ] Let R be the set of real numbers. Let C be the set of all single-element subsets of
R.
(a) What is the eld generated by C ? Verify your answer.
(b) What is the -eld generated by C ? Verify
Ph.D. Preliminary Examination
Stat 501 & 502
May 15, 2009
[1 ] Let = N, the set of natural numbers. Dene
A = cfw_A N : A or Ac is nite.
(a) Show that A is a eld.
(b) Is A a -eld? Verify your answer.
(c) Dene the set function P by
0,
1,
P (E ) =
if E is ni
Ph.D. Preliminary Examination
Stat 501 & 502
May 14, 2010
1. A -eld is called countably generated if it is generated by a countable class C .
(a) Let B (0, 1]) be the Borel -eld on the interval (0, 1]. Show that it is countably
generated.
(b) Let G = cfw_
PhD Prelim Exam 2011
Section I: Stat 501
1. Let X and Y be random variables dened on the same probability space (, B , P ).
Show that
sup |P [X A] P [Y A]| P [X = Y ]
AB(R)
where B (R) is the Borel -eld on the real line R.
2. Suppose cfw_An n1 are indepen
Stat 501
Prelim-Spring 2012
Problem 1. Suppose and are two sets. X : is a map
with domian and range . Then X determines a function
X 1 : P ( ) P ()
dened by:
X 1 (A ) = cfw_ : X ( ) A .
Now show that if C is a class of subsets of then
X 1 ( (C ) = (X 1 (C
PH.D. Preliminary Examination
July 2, 2007
1. Let be the set of natural numbers and let =cfw_ A : A is finite or A c is finite .
Define P( A) on to be 0 or 1 according as A is finite or not. Show that is a field
but not a -field and that P is finitely add
Stat502
Preliminary Examination
Spring 2006
[1] Let cfw_Xn : n 1 be a sequence of independent random variables such that for n 1,
Pcfw_Xn = n =
1
= 1 Pcfw_Xn = 0.
n
(a) Discuss its convergence in probability, in distribution, in Lp , p > 0, and almost
eve
Stat 501 Probability Theory I
Fall 2013
Midterm 1 (due on Oct. 11, 2013)
Total points: 20
Problem 1. (4 points) If A is the eld generated by C , then A and
C generate the same -eld.
Problem 2. (4 points) Let A (C ), where C is an uncountable
collection of