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.



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



A

B

C

Each year10% of the residents of A

move to B

.10

and20% of the residents

of A move to C

.20



A

B

C

.10

.20

Each year20% of the residents of B

move to A

.20


of B move to C

.30



A

B

C

.10

.20

.20

.30

Each year20% of the residents of C

move to A

.20


of C move to B

.20



A

B

C

.10

.20

.20

.30

.20

.20



A

B

C

.10

.20

.20

.30

.20

.20



A

B

C

.10

.20

.20

.30

.20

.20



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





A

B

C

.10

.20

.20

.30

.20

.20



AB

C

.10

.20

.20

.30.20

.20

An&1 ' .An



A

C

.10

.20

.20

.30.20

.20

An&1 ' .An & .2Bn & .2Cn

B



A

C

.10

.20

.20

.30.20

.20

An&1 ' .An & .2Bn & .2Cn

Bn&1 ' .1An & .#Bn & .2Cn

B



A

C

.10

.20

.20

.30.20

.20

An&1 ' .An & .2Bn & .2Cn

Bn&1 ' .1An & .#Bn & .2Cn

Cn&1 ' .2An & .3Bn & .(Cn

B



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

!"#"$"

$"%"1"

$"$"&"

B



An&1 ' .An & .2Bn & .2Cn

Bn&1 ' .1An & .#Bn & .2Cn

Cn&1 ' .2An & .3Bn & .(Cn

=

n

n

n

C

B

A

!"#"$"

$"%"1"$"$"&"

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



=

+

+

+

n

n

n

n

n

n

C

B

A

C

B

A

!"#"$"$"%"1"

$"$"&"

1

1

1

nn v M v =+1

0

/

1

#

$

$

#/

0

#

1

$

$#

0$

1$

01

v M v M v M v M v

v M v M v M v

v M v M v

v M v

====

=====

=



=

+

+

+

n

n

n

n

n

n

C

B

A

C

B

A

!"#"$"$"%"1"

$"$"&"

1

1

1

nn v M v =+1

0v M v nn =



.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,

!"#"$"

$"%"1"$"$"&"

100

010001

5

−−−−−−

/"#"$"

$"%"1"$"$"#"

)



−−−−

−−

/"#"$"

$"%"1"

$"$"#"

The n!ll sace o, )

1#

6

1/

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

=

1#0

60

1/0

!#$

$"%"1"

$"$"&"

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