Unformatted text preview: NS
To check this we note that
1/3 2/3 ✓ .6
.2 .4
.8 ◆ = ✓ .6 .4
+
3
3 .4 1.6
+
3
3 ◆ General two state transition probability.
1
2 1 1
a 1 b 2
a
b We have written the chain in this way so the stationary distribution has a simple
formula
b
a
⇡1 =
⇡2 =
(1.8)
a+b
a+b
As a ﬁrst check on this formula we note that in the weather chain a = 0.4 and
b = 0.2 which gives (1/3, 2/3) as we found before. We can prove this works in
general by drawing a picture:
b1
•
a+b a2
!•
b a
a+b In words, the amount of sand that ﬂows from 1 to 2 is the same as the amount
that ﬂows from 2 to 1 so the amount of sand at each site stays constant. To
check algebraically that ⇡ p = ⇡ :
b
a
b ba + ab
b
(1 a) +
b=
=
a+b
a+b
a+b
a+b
b
a
ba + a ab
a
a+
(1 b) =
=
a+b
a+b
a+b
a+b (1.9) Formula (1.8) gives the stationary distribution for any two state chain, so
we progress now to the three state case and consider the
Example 1.18. Social Mobility (continuation of 1.4).
1
2
3 1
.7
.3
.2 2
.2
.5
.4 3
.1
.2
.4 The equation ...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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