Statistics 451 Midterm October 11, 2013
This exam has 3 problems and is worth 40 points.
Instructor: Michael Kozdron
You have 50 minutes to complete this exam. Please read all instructions carefully, and
check your answers. Show all work neatly and in ord
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 25, 2013
Lecture #10: Continuity of Probability
Recall that last class we proved the following theorem.
Theorem 10.1. Consider the real numbers R with the Borel -algebra B , and let P be a
probabi
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 27, 2013
Lecture #11: Continuity of Probability (continued)
We will now apply the continuity of probability theorem to prove that the function F (x) =
P cfw_(, x], x R, dened last lecture is actua
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 11, 2013
Lecture #4: There is no uniform probability on ([0, 1], 2[0,1] )
Our goal for today is to prove the rst of the claims made last lecture, namely that there
does not exist a uniform probabi
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 23, 2013
Lecture #9: Construction of a Probability (Part II)
Recall that we have been trying to construct a uniform probability on [0, 1]. As we saw in
Lecture #4, it is not possible to construct
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 20, 2013
Lecture #8: Independence and Conditional Probability
Denition. Let (, F , P) be a probability space. The events A, B F are said to be
independent if
P cfw_A B = P cfw_A P cfw_B .
A coll
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 16, 2013
Lecture #6: Construction of a Probability (Part I)
As we showed in Lecture #4, when the sample space is = [0, 1], it is not possible to
construct a probability P : 2 [0, 1] satisfying bot
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 13, 2013
Lecture #5: The Borel Sets of R
We will now begin investigating the second of the two claims made at the end of Lecture #3,
namely that there exists a -algebra B1 of subsets of [0, 1] on
Statistics 451 (Fall 2013)
Prof. Michael Kozdron
September 18, 2013
Lecture #7: Proof of the Monotone Class Theorem
Our goal for today is to prove the monotone class theorem. We will then deduce an extremely
important corollary which we will ultimately us
Stat 451: Solutions to Assignment #1
(2.1) By denition, 2 is the set of all subsets of . Therefore, to show that 2 is a -algebra we
must show that the conditions of the denition -algebra are met. In particular,
so that 2 ,
so that 2 ,
if A , then Ac =
Stat 851: Solutions to Assignment #3
(7.11) To see that A is a Borel set write A as
A = cfw_x0 = cfw_(, x0 ) (x0 , )c .
()
Since open intervals are Borel, so too are unions of open intervals, as are complements of unions of
open intervals. Using () and e
Stat 451: Solutions to Assignment #2
(3.1) If A B = , then by Theorem 2.2 we conclude P (A B ) = P () = 0. Hence, in order for A
and B to be independent, it must be the case that P (A B ) = P (A) P (B ) = 0. The product of
two real numbers is 0 if and onl
Stat 451 Fall 2013
Assignment #2
This assignment is due on Monday, September 23, 2013.
1. Complete the following exercises from pages 1920.
#3.1, 3.3, 3.6, 3.8, 3.11, 3.12, 3.13
2. Complete the following exercises from page 26.
#4.1
Stat 451 Fall 2013
Assignment #3
This assignment is due on Friday, October 25, 2013.
1. Complete the following exercises from pages 4546.
#7.11, 7.12, 7.13, 7.14, 7.15, 7.17, 7.18
Stat 451 Fall 2013
Assignment #1
This assignment is due on Monday, September 16, 2013.
1. Complete the following exercises from pages 1213.
#2.1, 2.2, 2.3, 2.6, 2.9, 2.10, 2.11, 2.12, 2.14, 2.15, 2.17