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 you
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.
(
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]
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 -e
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
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
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 prob
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 poin