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 September 26th, 2017, 09:51 PM #1 Senior Member   Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 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? September 26th, 2017, 10:37 PM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,424 Thanks: 1311 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 26th, 2017 at 11:26 PM. September 26th, 2017, 10:39 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. September 26th, 2017, 11:27 PM #4 Senior Member   Joined: Sep 2015 From: USA Posts: 2,424 Thanks: 1311 note that you have to show the other direction as well Tags events, infinite, probability, union Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post John Travolski Advanced Statistics 3 September 5th, 2017 07:41 AM palongze Probability and Statistics 8 January 5th, 2017 06:39 PM beesee Probability and Statistics 10 October 29th, 2014 03:19 AM 03sqq Real Analysis 4 November 13th, 2012 03:40 AM Caranfin Probability and Statistics 2 January 31st, 2012 08:28 AM

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