Probability and Statistics Basic Probability and Statistics Math Forum

 April 23rd, 2017, 12:54 PM #1 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 Calculating expected time Lets say we have 3 states 0 - Infection Dormant 1 - Infection Active 2 - Individual Dead Transitions 0 -> 1, at rate a 1 -> 2, at rate b 1 -> 0, at rate g The question asks to prove that the expected time until an individual with a dormant infection will die is: $\displaystyle E[T] = \frac{a + b + g}{ab}$ where T denotes the length of time until death. Any ideas how to solve this, I thought I would be able to solve in the following way: $\displaystyle M_{02} = 1 + aM_{12}$ $\displaystyle M_{12} = 1 + bM_{22} + gM_{02}$ $\displaystyle M_{22} = 1$ $\displaystyle M_{02} = \frac{1 + a + ab}{1 - ab}$ However as you can see, my answer is not even close to what the question states the answer should be. April 23rd, 2017, 03:04 PM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,638 Thanks: 1474 are these rates Poisson? April 23rd, 2017, 11:16 PM #3 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 The rates are instantaneous transition rates April 24th, 2017, 11:28 AM #4 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 After a bit of googling, it looks like you can calculate expected time for a state transition of a continuous markov chain with the following method: $\displaystyle E[0->2] = \frac{1}{a} + E[1->2]$(1) $\displaystyle E[1->2] = \frac{1}{g+b} + \frac{b}{g+b} .0 + \frac{g}{g+b}E[0->2]$ (2) when you sub (2) into (1), $\displaystyle E[0->2] = \frac{a + b+ g}{ab}$ as required Tags calculating, expected, time 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 lark Computer Science 1 June 25th, 2013 06:55 AM jayz657 Calculus 3 October 13th, 2012 07:38 PM mathbeauty Algebra 0 August 10th, 2010 02:10 PM Artemis Advanced Statistics 1 November 11th, 2009 08:08 PM jayz657 Complex Analysis 0 December 31st, 1969 04:00 PM

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