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# 458 Class Note for STAT 416 at PSU

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COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
6
WORDS
KARMA
25 ?

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 22 views.

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Date Created: 02/06/15
Mean Time Spent in Transient States 0 Consider a nite state Markov Chain with a set of transient states T 1 2 t o Gambler s ruin problem states 0 l N The transient states are l 2 N l T12N 1 e Let the transition probability matrix be P o A part of P formed by probabilities from transient states to transient states P11 P12 P12 21 P22 P22 PT PH 352 PttJ 0 Examples Gamblers ruin problem P P00 P01 P10 P11 PN0 PN1 PN 11 PN 12 Pow 1 Pow PLN l PLN PNN 1 PNN PLN l 1 J PN LN 1 Suppose a four state MC with states 0 1 2 3 has two transient states 0 and 3 Then P03 P13 P23 P33J PT P00 P03 P30 P33 o For transient states 239 and j 8M expected number of time periods the MC is in state j given that it starts in state 2 Special case 3M starting from 239 the number of time periods in z Transient states fi lt 1 Recall that fi is the prob ability of ever revisit state 239 starting from state 2 De ne fZj the probability that the MC ever visits state j given that it starts in 2 Special case f fl The above equation can be written as t E Visits to j after the initial state Z Bkskj k1 T0 combine Eq 1 and 2 into a uni ed form n 1 2 j ampJ0ij Hence t Sz39j 51739 Z Pikskj 3 k1 for allij 6 12 t Use matrix 811 812 Sit 821 822 822 S J 8251 822 Stt Eq 3 is equivalent to S I PTS SU P 4 To obtain fZj the probability of ever transit to j starting in 239 compute expectation by conditioning sZj Pever transit to j i start in gtlt Etime spent in j i start in 239 and transit to j Pnever transit to j i start in gtlt Etime spent in j i start in 239 never transit to j fz39jwz39j Sjjlt1 fz39j5z39j 517 fz39jsjj Hence 4 Special case Example Gambler s ruin problem Suppose p 04 N 7 Start with 3 a The expected amount of time the gambler has 5 units or 2 units b The probability that the gambler ever has a fortune of 1 Solution States 0 12 3456 7 Transient states 1 2 3 4 5 6 0040 0 0 0 h0lt3040 0 01 PT 0 0040 0 0 000J040 390 0 006JO4 0 0 0 0060 Solve S I PT1 we get 8375 8372 Hence the expected number of times the gambler has 5 units is 09228 2 units is 23677 hi ng gqmmznmmwwmw that the gambler ever has a fortune of l is 08797

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