Unformatted text preview: gives
✓ ◆x
✓ ◆x
q
q
Px (V0 = 1) = 1
and Px (V0 < 1) =
.
(1.23)
p
p
To see that the form of the last answer makes sense, note that to get from x to
0 we must go x ! x 1 ! x2 . . . ! 1 ! 0, so
Px (V0 < 1) = P1 (V0 < 1)x . 49 1.9. EXIT TIMES 1.9 Exit Times To motivate developments we begin with an example.
Example 1.45. Two year college. In Example 1.39 we introduced a Markov
chain with state space 1 = freshman, 2 = sophomore, G = graduate, D =
dropout, and transition probability
1
2
G
D 1
0.25
0
0
0 2
0.6
0.2
0
0 G
0
0.7
1
0 D
0.15
0.1
0
1 On the average how many years does a student take to graduate or drop out?
Let g (x) be the expected time for a student starting in state x. g (G) =
g (D) = 0. By considering what happens on one step
g (1) = 1 + 0.25g (1) + 0.6g (2)
g (2) = 1 + 0.2g (2)
where the 1+ is due to the fact that after the jump has been made one year
has elapsed. To solve for g , we note that the second equation implies g (2) =
1/0.8 = 1.25 and then the ﬁrst that...
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 Spring '10
 DURRETT
 The Land

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