4 markovchain isox upon a time fake pdfsox upon a time fake pdfsox upon a time fake pdf
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7/24/2019 4 MarkovChain ISox upon a time fake pdfSox upon a time fake pdfSox upon a time fake pdf
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MARKOV CHAIN
A, B and C are three towns. Each year:
10% of the residents of A move to B
20% of the residents of A move to C
20% of the residents of B move to A
30% of the residents of B move to C
20% of the residents of C move to A20% of the residents of C move to B
Everybody else stays !t. "o one dies. "o one is born.
7/24/2019 4 MarkovChain ISox upon a time fake pdfSox upon a time fake pdfSox upon a time fake pdf
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A
B
C
A, B, and C are
three towns.
10 eole live in A
1# eole live in C
20 eole live in B
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A
B
C
Each year10% of the residents of A
move to B
.10
and20% of the residents
of A move to C
.20
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A
B
C
.10
.20
Each year20% of the residents of B
move to A
.20
and30% of the residents
of B move to C
.30
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A
B
C
.10
.20
.20
.30
Each year20% of the residents of C
move to A
.20
and20% of the residents
of C move to B
.20
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A
B
C
.10
.20
.20
.30
.20
.20
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A
B
C
.10
.20
.20
.30
.20
.20
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A
B
C
.10
.20
.20
.30
.20
.20
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B
A
C
.10
.20
.20
.30
.20
.20
1$1$
1
AB
C
1020
1#
This year
Next year
Where will everyone be in 10 years
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A
B
C
.10
.20
.20
.30
.20
.20
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AB
C
.10
.20
.20
.30.20
.20
An&1 ' .An
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A
C
.10
.20
.20
.30.20
.20
An&1 ' .An & .2Bn & .2Cn
B
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A
C
.10
.20
.20
.30.20
.20
An&1 ' .An & .2Bn & .2Cn
Bn&1 ' .1An & .#Bn & .2Cn
B
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A
C
.10
.20
.20
.30.20
.20
An&1 ' .An & .2Bn & .2Cn
Bn&1 ' .1An & .#Bn & .2Cn
Cn&1 ' .2An & .3Bn & .(Cn
B
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A
C
.10
.20
.20
.30.20
.20
An&1 ' .An & .2Bn & .2Cn
Bn&1 ' .1An & .#Bn & .2Cn
Cn&1 ' .2An & .3Bn & .(Cn
=
n
n
n
C
B
A
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B
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An&1 ' .An & .2Bn & .2Cn
Bn&1 ' .1An & .#Bn & .2Cn
Cn&1 ' .2An & .3Bn & .(Cn
=
n
n
n
C
B
A
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Note' "&("1("$ ) 1" This a**o+nts ,or 100- o, those in town A in year n
Cn&1 ' .2An & .3Bn & .(CnCn&1 ' .2An & .3Bn & .(Cn
Note' "$("%("# ) 1" This a**o+nts ,or 100- o, those in town . in year n
Note' "$("$("! ) 1" This a**o+nts ,or 100- o, those in town C in year n
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n
n
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C
B
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A
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.e*a+se the s+ o, the n+bers in ea*h *ol+n o, M is 1
1 is an ei)enval!e ,or M and
An ei2enve*tor belon2in2 to 1 will des*ribe stable 3ro3ortions"
The n+ll s3a*e o, 1I 4 M ) the n+ll s3a*e o,
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7+33ose there are this year 1/0 3eo3le in A 60 3eo3le in . and
1#0 3eo3le in C" How any will be in ea*h town next year
=
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