exercises - Mathematics 741 - Advanced Probability Theory I...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Mathematics 741 - Advanced Probability Theory I - Fall 2008 - Zirbel Exercises to accompany the lecture notes These exercises may or may not be assigned as homework for Math 741. Some of them are referred to in the lecture notes by number. Let ( , H , IP) be a probability space. 1. a) Let A 1 A 2 be sets in H . Let A = i =1 A i . Show that IP( A i ) increases to IP( A ) as i . b) Let A 1 A 2 be sets in H . Let A = i =1 A i . Show that IP( A i ) decreases to IP( A ) as i . 2. Let F be the smallest -algebra on R containing all intervals of the form ( ,b ], b R . Show that F = B R . We say that B R is generated by the open subsets of R or by intervals of the form ( ,b ]. We write B R = ( { ( ,b ] : b R } ). Hint: Show that F B R and then that B R F . Use the fact that B R contains all sets of the form ( ,b ]. Show that F contains all open intervals in R and thus all open subsets of R (each open subset is a countable union of open intervals). 3. a) Let and S be sets and let S be a -algebra on S . Let X : S . Define a collection ( X ) of subsets of by ( X ) = { X- 1 ( B ) : B S} . Show that ( X ) is a -algebra on . Show all details. b) Let H be a -algebra on . Let D = { B S : X- 1 ( B ) H} . Show that D is a -algebra on S . Think of D as all subsets of S whose inverse images under X are nice. Notes: (i) ( X ) is called the -algebra generated by X . It is the collection of all events concerning X . (ii) The function X : ( , H ) ( S, S ) is measurable if and only if ( X ) H . Think of ( X ) as the subsets of that need to be measurable for the mapping X to be measurable. 4. Represent by a square in the plane with sides of length 1. Divide in half hor- izontally and vertically. Let X : R take on the values 1, 2, 3, 4 in the four smaller squares. Represent X by showing the four values in the four smaller squares, and represent subsets of by shading in the appropriate squares. For example, the set { X = 1 } is the upperleft smaller square. Represent all elements of ( X ) by shading and in the form X- 1 ( B ) for different sets B . Check ( X ) to make sure it is a -algebra. 5. Let limsup n A n n =1 m = n A m . Show that IP(limsup n A n ) limsup n IP( A n )....
View Full Document

This note was uploaded on 01/01/2011 for the course MATHSTAT 741 taught by Professor Zirbel during the Spring '10 term at Bowling Green.

Page1 / 4

exercises - Mathematics 741 - Advanced Probability Theory I...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online