Dr. R. Sujatha / Dr. B. Praba, Maths Dept., SSNCE.

Two Dimensional Random Variables

Two dimensional random variable

Let S be the sample space of a random experiment. Let X and Y be two random

variables defined on S. Then the pair (X,Y) is called a two dimensional random variable.

Discrete bivariate random variable

If both the random variables X and Y are discrete then (X,Y) is called a discrete random

variable.

Joint Probability mass function

Let X take values },,,{ 21 nxxx … and Y take values },,,{ 21 myyy … . Then

)(),(),( jijiji yYxXPyYxXPyxp =∩===== . {xi, yj, p(xi,yj)} is called joint

probability mass function.

Marginal Probability Mass Function of X

)}(,{ iXi xpx is called the marginal probability mass function of X where

∑=

=m

jjiiX yxpxp

1

),()( .

Marginal Probability Mass Function of Y

)}(,{ JYj ypy is called the marginal probability mass function of Y where

∑=

=n

ijijY yxpyp

1

),()( .

Conditional Probability Mass Function

• Of X given Y=yj n,1,2, i yp

yxpyxP

jY

ji

jiYX …== ,)(

),()/(/

• Of Y given X=xi m,1,2, j xp

yxpxyP

iX

ji

ijXY …== ,)(

),()/(/

Independent Random Variables

Two random variables X and Y are said to be independent if

m,1,2,j ; n,1,2, i ,ypxpyxp jYiXji …… === )()(),(

Dr. R. Sujatha / Dr. B. Praba, Maths Dept., SSNCE.

Continuous bivariate random variable

If X and Y are both continuous then (X,Y) is a continuous bivariate random variable.

Joint Probability Density Function

If (X,Y) is a two dimensional continuous random variable such that

dxdyyxfdy

yYdy

y dx

xXdx

xP XY ),(2222

=

+≤≤−∩+≤≤− then f(x,y) is called the

joint pdf of (X,Y) provided (i) XYRyxyxf ∈∀≥ ),(,0),( (ii) 1),( =∫∫ dxdyyxfXYR

.

Joint Distribution Function

∫ ∫∞− ∞−

=≤≤=x y

XY dydxyxfyYxXPyxF ),(),(),(

Note: yx

yxFyxf

∂∂

∂=

),(),(

2

Marginal Probability Density Function

• Of X ∫∞

∞−

= dyyxfxf X ),()(

• Of Y ∫∞

∞−

= dxyxfyfY ),()(

Marginal Probability Distribution Function

• Of X ∫∞−

=x

XX dxxfxF )()(

• Of Y ∫∞−

=y

YY dyyfyF )()(

Conditional Probability Density Function

• Of Y given X = x )(

),()/(/

xf

yxfxyf

X

XY =

• Of X given Y = y )(

),()/(/

yf

yxfyxf

Y

YX =

Independent random variables

X and Y are said to be independent if )()(),( yfxfyxf YX= i.e, )()(),( yFxFyxF YX= .

Dr. R. Sujatha / Dr. B. Praba, Maths Dept., SSNCE.

Moments of two dimensional random variable

The (m, n)th

moment of a two dimensional random variable (X,Y) is

∑∑==i j

ji

n

j

m

i

nm

mn yxpyxYXE ),()(µ (discrete case)

= ∫ ∫∞

∞−

∞

∞−

dxdyyxfyxnm ),( (continuous case)

Note : (1) when m=1,n=1 we have E(XY)

(2) when m = 1, n = 0 we have ∑∑ ∑==i j i

iXijii xpxyxpxXE )(),()( (discrete)

= ∫ ∫∫∞

∞−

∞

∞−

∞

∞−

= dxxxfdxdyyxxf X )(),( (continuous)

(3) when m = 0, n = 1 we have ∑∑ ∑==i j j

jYjjij ypyyxpyYE )(),()( (discrete)

= ∫ ∫∫∞

∞−

∞

∞−

∞

∞−

= dyyyfdxdyyxyf Y )(),( (continuous)

(4) ∑=i

iXi xpxXE )()(22 (discrete)

= ∫∞

∞−

dxxfx X )(2 (continuous)

(5) ∑=j

jYj ypyYE )()(22 (discrete)

= ∫∞

∞−

dyyfy Y )(2 (continuous)

Covariance

YXXYEYYXXEYXCov −=−−= )()])([(),(

Note: (1) If X and Y are independent then E(XY)=E(X)E(Y). (Multiplication theorem

for expectation). Hence Cov(X,Y)=0.

(2) Var(aX+bY)=a2Var(X)+b

2Var(Y)+2abCov(X,Y).

Correlation Coefficient

This is measure of the linear relationship between any two random variables X and Y.

The Karl Pearson’s Correlation Coefficient is

Dr. R. Sujatha / Dr. B. Praba, Maths Dept., SSNCE.

YX

YXCovYXr

σσρ

),(),( ==

Note: Correlation coefficient lies between -1 and 1.

Regression lines

The regression line of X on Y: )( YYrXXY

X −=−σ

σ

The regression line of Y on X: )( XXrYYX

Y −=−σ

σ

r- Correlation coefficient of X and Y.

Transformation of random variables

Let X, Y be random variables with joint pdf ( )yxf XY , and let u(x, y) and be v(x, y) be

two continuously differentiable functions. Then U = u(x, y) and V = v(x, y) are random

variables. In other words, the random variables (X, Y) are transformed to random

variables (U, V) by the transformation u = u(x, y) and v = v(x, y).

The joint pdf ( )vugUV , of the transformed variables U and V is given by

( ) ( ) Jyxfvug XYUV ,, =

where |J| = ( )( )

v

y

v

xu

y

u

x

vu

yx

∂

∂

∂

∂∂

∂

∂

∂

=∂

∂

,

, i.e., it is the modulus value of the Jacobian of

transformation and ( )yxf XY , is expressed in terms of U and V.