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Lahari

The document explains stochastic processes, probability vectors, and stochastic matrices, detailing their definitions and properties. It introduces regular stochastic matrices and provides examples, including how to find unique fixed probability vectors for given matrices. Additionally, it includes problems with solutions demonstrating the application of these concepts.

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Rocky Charan
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18 views5 pages

Lahari

The document explains stochastic processes, probability vectors, and stochastic matrices, detailing their definitions and properties. It introduces regular stochastic matrices and provides examples, including how to find unique fixed probability vectors for given matrices. Additionally, it includes problems with solutions demonstrating the application of these concepts.

Uploaded by

Rocky Charan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Stochastic Process:

State: the value assumed by the random variable are


called state.
Stochastic process consists of sequence of
experiments in which each experiment has a finite
number of outcomes with the given probabilities.

Probability Vector:
A vector V = (𝑉1, 𝑉2,……………………… 𝑉𝑛 ) is called a
probability vector if each one of its components are
non-negative and their sum is equal to unity or 1.
1 1 1 2 3
Eg: V= (1,0) , V= ( , ) , V = ( , , ) ,
2 2 6 6 6
V = ( 0.1 ,0.6 ,0.3 )

Stochastic Matrix:
A square matrix A=aij having every row in the form
of a probability vector is called a Stochastic Matrix.
1 0 0 1 1
1 0
Eg: A = [ ] B = [0 1 0] C = [ 2 2 ]
0 1 0 1
0 0 1
Regular Stochastic Matrix:
A Stochastic Matrix A is said to Regular Stochastic
Matrix, if all the enteries of some power An are positive.
0 1
Eg: A=[ 1 1 ]
2 2
0 1 0 1
2
Consider, A = A . A = [ 1 1 ] [ 1 1 ]
2 2 2 2
1 1
0+ 0+
2 2
=[ 1 1 1]
0+ +
4 2 4
1 1

A2 = [21 2
3]
4 4

∴ 𝐴 𝑖𝑠 𝑎 𝑅𝑒𝑔𝑢𝑙𝑎𝑟 𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 𝑚𝑎𝑡𝑟𝑖𝑥.


Properties of a regular stochastic matrix:
If A is the regular stochastic matrix of order n
i) A as a unique fixed point.
x = ( x1 , x2 , ……………,xn ) such that xA=x
ii) A as a unique probability vector.
V = ( V1 , V2 ,…………..., Vn ) such that VA=A
Problems:
1) Find the unique fixed probability vector for the
0 1 0
1 1 1
regular stochastic matrix A =[ 6 2 3]
2 1
0
3 3

Soln: we have to find V=(x , y ,z) where x+y+z=1→*


VA=V
0 1 0
1 1 1
[ x y z ] [6 2 3] =[x y z]
2 1
0
3 3
𝑦 𝑦 2𝑧 𝑦 𝑧
[0+ +0 , x+ + , 0+ + ]=[x y z]
6 2 3 3 3
𝑦 6𝑥+3𝑦+4𝑧 𝑦+𝑧
[ ]=[x y z]
6 6 3
𝑦 6𝑥+3𝑦+4𝑧 𝑦+𝑧
=x = y =z
6 6 3
y = 6x 6𝑥 + 3𝑦 + 4𝑧 = 6y y + z =3z
y = 2z
but y = 6x
6x = 2z
z = 3x
*→ x + y + z =1 y = 6x z = 3x
6 3
x + 6x + 3x =1 y= z=
10 10
10 x = 1
1
x=
10
∴ Unique fixed probability vector is
1 6 3
V=[x y z]=[ , , ]
10 10 10
2)Find the Unique fixed probability vector for the
1 1 1
2 4 4
regular stochastic matrix A= 0 [1 1]
2 2
0 1 0
Soln: we have V= [ x y z ] then x+y+z=1
VA=V
1 1 1
2 4 4
[x y z] [1 0 =[x y 1] z]
2 2
0 1 0
𝑥 𝑦 𝑥 𝑥 𝑦
[ + , +z , + ]=[x y z]
2 2 4 4 2
𝑥+𝑦 𝑥+4𝑧 𝑥+2𝑦
[ , , ]=[x y z]
2 4 4
𝑥+𝑦 𝑥+4𝑧 𝑥+2𝑦
=x , =y , =z
2 4 4
x + y = 2x , x+4z =4y , x+2y = 4z
x + y = 2x x + 2y = 4z
y=x x + 2x = 4z
3x = 4z
3𝑥
z=
4
x+y+z=1
3𝑥 4 3 4
x+x+ =1 y= z= ( )
4 11 4 11
3𝑥 3
4x + =1 z=
4 11
8𝑥+3𝑥
=1
4

11x = 4
4
x = 11
∴ Unique fixed probability vector is
4 4 3
V=[x,y,z]=[ , , ]
11 11 11

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