My Math Forum  

Go Back   My Math Forum > College Math Forum > Advanced Statistics

Advanced Statistics Advanced Probability and Statistics Math Forum

Thanks Tree1Thanks
  • 1 Post By mehoul
LinkBack Thread Tools Display Modes
June 28th, 2016, 09:29 AM   #1
Joined: Jun 2016
From: India

Posts: 3
Thanks: 0

Smile Probability - Sigma algebra

I read in a book "The elements of $\displaystyle sigma $- algebra over a sample space $\displaystyle omega$ are called events" and not all subsets of a sample space are events. Can you please explain this definition?
Niladri is offline  
June 28th, 2016, 01:47 PM   #2
Global Moderator
Joined: May 2007

Posts: 6,275
Thanks: 516

A (set) algebra is a collection of sets (events) which is closed under any finite combination of unions, intersects, and complements, which includes the entire set (sample space). A sigma-algebra extends this notion to include countable combinations as well.

Note that measure theory and probability theory are very similar, but the terminology differs: measure theory - sets, probability theory - events.
mathman is offline  
June 28th, 2016, 10:47 PM   #3
Joined: Jun 2016
From: India

Posts: 3
Thanks: 0

Actually I a newly studying the probability theory and would like to know why the definition of events is given as following "The elements of a sigma- algebra over a sample omega are called events" and why not all subsets of the sample space of a particular experiment can be events?
Niladri is offline  
June 29th, 2016, 03:00 AM   #4
Senior Member
Joined: Feb 2012

Posts: 144
Thanks: 16

There are two reasons why one considers only set belonging to a sigma-algebra instead of all subsets of Omega.

1) a suitable sigma-algebra is all what is needed. Suppose Omega is the set of real numbers R and X is a real random variable. All the interesting questions you may ask about X is whether X<a for some real number a, or if X belongs to the interval [a,b] or (a,b), (a,b]... so that you only need to consider the sigma-algebra generated by theses sets (which is much smaller than the power set of R)

2) you might nevertheless consider all subsets of Omega. However, if you do that, you will find that for an Omega big enough there are some probability measures which behave pathologically. This is related to the so-called Banach-Tarski paradox: in the 3-dimensional euclidean space there is no rigid-motion-invariant measure on all subsets (but there is one on the borelian sigma-algebra).
Thanks from Niladri
mehoul is offline  

  My Math Forum > College Math Forum > Advanced Statistics

algebra, probability, probaility, sigma, sigma algebra

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
$\sigma$-algebra parrsecc Advanced Statistics 0 May 6th, 2014 06:53 AM
Sigma algebra limes5 Real Analysis 16 July 27th, 2013 04:43 AM
Sigma algebra Lucida Algebra 9 September 21st, 2012 10:52 AM
Sigma-algebra SunIzz Real Analysis 0 September 5th, 2009 05:50 PM
sigma algebra sangfroid Advanced Statistics 1 February 18th, 2008 12:45 PM

Copyright © 2017 My Math Forum. All rights reserved.