maths

Unit – III

Joint Probability and Morkov Chain
Joint Probability distribution: Joint Probability distribution for two discrete random
variables (both discrete and continuous cases), expectation, covariance, correlation coefficient.
Stochastic processes- Stochastic processes, probability vector, stochastic matrices, fixed
points, regular stochastic matrices, Markov chains, higher transition probability-simple
problems.


Joint Probability Distribution
If X and Y are two discrete random variables, we define the joint probability function of X and
Y by P(X = x, Y = y) = f(x, y) where f(x, y) satisfies the conditions

(i) f(x, y)  0, (ii)  f ( x, y)  1
x y


Also, if X and Y are two continuous random variables, we define the joint probability function
for the random variables X and Y which is also called the joint density function by f(x, y)
where
 
(i) f(x, y)  0 (ii) 
 
f ( x, y ) dxdy  1 .


Suppose X and Y takes any one of the values from the set of values {x1, x2, x3, …….xm} and
{y1, y2, y3, …….yn} respectively then P(X = xi, Y = yj) = f(xi, yj) is represented in the
following two way table

Y y1 y2 -- yn Total
X
x1 f ( x1 , y1 ) f ( x1 , y2 ) -- f ( x1 , yn ) f1 ( x1 )

x2 f ( x2 , y1 ) f ( x2 , y2 ) -- f ( x2 , yn ) f1 ( x 2 )

--

--

--

xm f ( xm , y1 ) f ( xm , y2 ) -- f ( xm , yn ) f1 ( xm )

Total f 2 ( y1 ) f 2 ( y2 ) -- f 2 ( yn ) 1




1

,f1 ( xi ) or simply f1 ( x ) and f 2 ( y j ) or simply f 2 ( y ) are known as the marginal probability
functions of X and Y respectively as it can be observed that
m n m n

 f ( x )  1 ,  f ( y )  1 which can jointly written as  f ( x , y )  1.
i 1
1 i
j 1
2 j
i 1 j 1
i j



Expectation: (One Variable): The mean value of the distribution of a variate X is commonly
known as its expectation and is denoted by E(X). If f(x) is the probability density function of
the variate X then

E( X )   xi f (xi ) (Discrete distribution)


E( X )   xf ( x)dx

(Continuous distribution)


In general expectation of any function (x) is given by

E ( x)  (xi ) f (xi ) (Discrete distribution)


E  ( x )    ( x) f ( x)dx (Continuous distribution)



Expectation: (Two Variables): If X and Y are two discrete random variables having the joint
probability function f(x, y) then the expectations of X and Y are defined as

Mean of X  x  E( X )   xf ( x, y)   x f ( x )
x y i
i 1 i




Mean of Y   y  E(Y )   yf ( x, y)   y
x y j
j f2 ( y j )


and E( XY )  x y J
ij
i j ij




The Covariance of X and Y = Cov( X ,Y )  E( XY )  E( X )E(Y )

If X and Y are two continuous random variables having the joint probability function f(x, y)
then the expectations of X and Y are defined as
 
x  E( X )    xf ( x, y)dxdy
 


 
 y  E (Y )    yf ( x, y)dxdy
 



2

,  
Variance V ( X )   x2    (x  
 
x ) 2 f ( x, y )dxdy

 
V (Y )      (y  
2
y y )2 f ( x, y )dxdy
 


 
Cov ( X , Y )    ( x   )( y  
 
x y ) f ( x, y ) dxdy  E ( XY )  E ( X ) E (Y )


Cov( X , Y )
Correlation Co efficient  ( X , Y ) 
 x y

Note 1: The probability that x  [a, b] and y [c, d ] is defined as

b d
P ( a  x  b, c  y  d )    f ( x, y )dxdy
a c



Note 2: The variables x and y are said to be independent if f1 ( x)  f 2 ( y )  f ( x, y ) .


Note 3: Marginal density function of X is f1 ( x)  

f ( x, y )dy


Marginal density function of Y is f 2 ( y )  

f ( x, y ) dx


Properties of Expectation:

(i) E(kX )  kE( X )

(ii) E( X  k )  E( X )  k

(iii) E( X  Y )  E( X )  E(Y )

(iv) E( XY )  E( X )E(Y )

Properties of Variance:

(i) V ( X )  E( X )   E( X )
2 2




(ii) V (kX )  k 2V ( X )

(iii) V ( X  k )  V ( X )

(iv) V ( X  Y )  V ( X )  V (Y )


3

, Problem: Find E(X), E(X2) and 2 for the probability function P(X) defined by the following
data

xi 1 2 3 --------- n

p(xi) k 2k 3k --------- nk

Here k is an appropriate constant.

Solution: The condition  p ( x )  1 gives
i



k  2 k  3 k  4 k  .........  nk  1

k 1  2  3  4  .........  n  1

n( n  1)
k 1
2

2
k
n(n  1)

E(X )   x p(x )
i i



E( X )  (1)(k )  (2)(2k)  3(3k)  ............  (n)(nk)

2 n(n  1)(2n  1) (2n  1)
E( X )  k (12  22  32  ..........  n2 )  
n(n  1) 6 3

E(X 2)  x i
2
p ( xi )

E ( X 2 )  (12 )(k )  (22 )(2k )  (32 )(3k )  ............  ( n 2 )( nk )

2 n2 (n  1)2 n(n  1)
E ( X 2 )  k (13  23  33  ..........  n3 )  
n(n  1) 4 2

 2  E( X 2 )   2
2
n(n  1)  2n  1 
  2
 
2  3 




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