Unformatted text preview: A)
= Probability of moving from A to B
= Probability of moving from B to A
= Probability of staying in B (moving from B to B)
35 Markov Transition Matrices
Let At be the number of workers in industry A at time t, while Bt is the
Let
while
number of workers in industry B at time t.
t.
Call vector xt′ = [ At Bt ] the distribution of workers across industries at
Call
time t.
t. PAA PAB Consider the following Markov transition probabilities matrix: M = PBA PBB Then the distribution of workers across industries in period t+1 will be
Then
t+1
given by:
given PAA
xt′+1 = [ At +1 Bt +1 ] = [ ( At PAA + Bt PBA ) ( At PAB + Bt PBB ) ] = [ At Bt ] × PBA PAB PBB The number
The number
(number of workers f workers in
of workers in
o in A at t)x
(Number of workers in B at t)x
(probability of staying time t+1 (probability t+1moving from B to A)
B at time of
A at in A)
=
=
Number of workers who
Number of workers who moved
stayed in A at t+1
from B to A at t+1 36 Markov Transition Matrices
By multiplying the initial distribution of workers by the Markov transition
By
probability matrix, we can find workers’ distribution by industry many
periods ahead in the future.
periods
2 Two periods ahead: PAA
[ At +2 Bt +2 ] = [ At Bt ] PBA PAB PBB N periods ahead: PAA
[ At +N Bt +N ] = [ At Bt ] PBA PAB PBB N It may happen that after a certain number of periods N 0 the vector
It
of workers’ distribution does not change any more, i.e.
of [A t + N 0 +1
[ Bt + N 0 +1 = At + N 0 Bt + N 0 We call this unchanging vector a steady state distribution. It
We
corresponds to a transition matrix with identical rows (matrix
corresponds M N0 )
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This note was uploaded on 09/14/2013 for the course STAT 1001 taught by Professor Kim during the Fall '11 term at Yonsei University.
 Fall '11
 Kim

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