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 romsek
Reply
 
LinkBack Thread Tools Display Modes
September 26th, 2017, 10:51 PM   #1
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 128
Thanks: 0

Exclamation Probability of Infinite Union of Events



Honestly have no idea where to go with this one. Since Pk is not assumed to be a constant (it could be any probability mass function), I find it difficult to take advantage of any algebraic properties. I suspect that there may be some pattern by using the inclusion-exclusion formula, but I'm not seeing anything when writing it out. I'm just really not sure how to start. Any recommendations?
John Travolski is offline  
 
September 26th, 2017, 11:37 PM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,324
Thanks: 1233

well... this isn't rigorous but...

$B=\bigcup_{k=1}^\infty~A_k$

$B^c = \left(\bigcup_{k=1}^\infty~A_k\right)^c = \bigcap_{k=1}^\infty~A_k^c$

$P[B]=1 \Rightarrow P[B^c]=0$

$P[B^c] = P\left[\bigcap_{k=1}^\infty~A_k^c\right] = \prod_{k=1}^\infty~(1-p_k)$

$\prod_{k=1}^\infty~(1-p_k) = 0 \Rightarrow

\sum_{k=1}^\infty~\ln(1-p_k) = -\infty$
Thanks from John Travolski

Last edited by romsek; September 27th, 2017 at 12:26 AM.
romsek is online now  
September 26th, 2017, 11:39 PM   #3
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 128
Thanks: 0

That's definitely enough. That's so clever. I wish I would be able to think of things like that. It makes so much sense considering that they're independent events. Thank you.
John Travolski is offline  
September 27th, 2017, 12:27 AM   #4
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,324
Thanks: 1233

note that you have to show the other direction as well
romsek is online now  
Reply

  My Math Forum > College Math Forum > Advanced Statistics

Tags
events, infinite, probability, union



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Probability of Union of Events John Travolski Advanced Statistics 3 September 5th, 2017 08:41 AM
Probability of Events, Simultaneously or One by One Events palongze Probability and Statistics 8 January 5th, 2017 07:39 PM
Prove the theorem on probability of union of two events. beesee Probability and Statistics 10 October 29th, 2014 04:19 AM
Is the union of infinite disjoint closed sets closed? 03sqq Real Analysis 4 November 13th, 2012 04:40 AM
Probability: non mutually exclusive union of several events Caranfin Probability and Statistics 2 January 31st, 2012 09:28 AM





Copyright © 2019 My Math Forum. All rights reserved.