Unformatted text preview: omes smoggy. It stays smoggy for an
exponentially distributed number of days with mean 4, then rain comes. The
rain lasts for an exponentially distributed number of days with mean 1, then
sunshine returns. Remembering that for an exponential the rate is 1 over the
mean, the verbal description translates into the following Qmatrix
1
1/ 3
0
1 1
2
3 2
1/3
1/ 4
0 3
0
1/4
1 The relation ⇡ Q = 0 leads to three equations:
1
3 ⇡1
1
3 ⇡1 1
4 ⇡2
1
4 ⇡2 +⇡3
⇡3 =0
=0
=0 Adding the three equations gives 0=0 so we delete the third equation and add
⇡1 + ⇡2 + ⇡3 = 1 to get an equation that can be written in matrix form as
0
1
1/3 1/3 1
⇡1 ⇡2 ⇡3 A = 0 0 1
1/4 1A
where A = @ 0
1
0
1 This is similar to our recipe in discrete time. To ﬁnd the stationary distribution
of a k state chain, form A by taking the ﬁrst k 1 columns of Q, add a column
of 1’s and then
⇡1 ⇡ 2 ⇡ 3 = 0 0 1 A 1
i.e., the last row of A 1 . In this case we have ⇡ (1) = 3/8, ⇡ (2) = 4/8, ⇡ (3) = 1/8 To check our...
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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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