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Unformatted text preview: = ( .02 0.45)⇡2 or .5 times the second gives
⇡2 = 16/47 Since the three probabilities add up to 1, ⇡3 = 9/47.
Using the TI83 calculator is easier. To begin we write (1.10) in matrix
form as
0
1
.2 .1 1
⇡1 ⇡2 ⇡3 @ .2
.4 1A = 0 0 1
.3
.3 1
If we let A be the 3 ⇥ 3 matrix in the middle this can be written as ⇡ A = (0, 0, 1).
Multiplying on each side by A 1 we see that
⇡ = (0, 0, 1)A 1 which is the third row of A 1 . To compute A 1 , we enter A into our calculator
(using the MATRX menu and its EDIT submenu), use the MATRIX menu to
put [A] on the computation line, press x 1 , and then ENTER. Reading the
third row we ﬁnd that the stationary distribution is
(0.468085, 0.340425, 0.191489)
Converting the answer to fractions using the ﬁrst entry in the MATH menu
gives
(22/47, 16/47, 9/47)
Example 1.19. Brand Preference (continuation of 1.5).
1
2
3 1
.8
.2
.3 2
.1
.6
.3 3
.1
.2
.4 21 1.4. STATIONARY DISTRIBUTIONS
Using the ﬁrst two equations and the fact that the sum of the ⇡ ’s is 1
.8⇡1 + .2⇡2 + .3⇡3
.1⇡1 + ....
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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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