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4. MULTIPLE RANDOM VARIABLES
Multiple random variables can be associated with every element
of the sample space S.
Let X and Y be two random variables. Then the pair (X, Y) is
called a bivariate random variable, or two dimensional random
vector.
The bivariate random variable is a function that maps the sample
space S into real plane – assigns to every point  in S a point
(x,y) in the real pane.
The range space of (X,
Y) is defined by:


Rxy  x, y ;   S and X    x, Y    y
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y
y( )
S
(x,y)

x( )
(X, Y) as a function from S to the real plane
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x
Joint Distribution Function
The joint cumulative distribution function of X and Y is the
function defined by:
FXY x, y   P X  x, Y  y 
S

y
y( )
(x,y)
Region included in
the definition of the
joint distribution function
x( )
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x
If event A of S is defined as
A    S; X    x;
P A  FX x
and event B as
B    S;Y    y;
PB  FY  y 
then:
FXY x, y   P A  B
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If, for particular values of x and y, A and B were independent
events, then
FXY x, y   P A  B  P APB  FX xFY  y 
If for all values of x and y
FXY x, y   FX xFY  y 
then X and Y are independent random variables.
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Properties of the distribution function FXY (x,y):
1.
0  FXY ( x, y)  1
2.
FXY ( x1 , y)  FXY ( x2 , y)
if x1  x2
FXY ( x, y1 )  FXY ( x, y2 )
if y1  y2
3.
lim FXY ( x, y)  FXY (, )  1
4.
lim FXY ( x, y )  FXY (, y )  0
x 
y 
x  
lim FXY ( x, y)  FXY ( x,)  0
y 
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lim FXY ( x, y)  FXY (a  , y)  FXY (a, y)
5.
xa
lim FXY ( x, y)  FXY ( x, b )  FXY ( x, b)
xb
6
Px1  X  x2 , Y  y   FXY ( x2 , y)  FXY ( x1 , y)
Px  X , y1  Y  y2   FXY ( x, y2 )  FXY ( x, y1 )
7.
If
x1  x2
y1  y2
and
, then:
FXY ( x2 , y2 )  FXY ( x1, y2 )  FXY ( x2 , y1 )  FXY ( x1, y1 )  0
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Marginal distribution function
lim  X  x, Y  y    X  x, Y     X  x 
y 
because the condition y
  is always satisfied. Then:
lim FXY ( x, y )  FXY (, y )  FY ( y )
x 
lim FXY ( x, y)  FXY ( x, )  FX ( x)
y 
When obtained in that way functions FX (x) and FY (y) are
referred to as the marginal cumulative distributions of X and Y.
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Joint Probability Density Function
The derivative
 FXY ( x, y)
f XY ( x, y) 
xy
2
is called the joint probability density function of the continuous
random variables X and Y.
FXY ( x, y)  
x

y
 
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f XY ( , )dd
Properties of the joint probability density function fXY (x,y):
1.
f XY ( x, y)  0

2.
f
XY
( x, y )dxdy  1

3.
fXY (x,y) is discontinuous at finite set of values of x
and y.
4.
P( X , Y )  A   f XY ( x, y )dxdy
RA
d b
Pa  X  b, c  Y  d     f XY ( x, y)dxdy
c a
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Correlation and Covariance:
The (k,n)th moment of a bivariate random variable (X,Y) is
defined by:
  x k y n p ( x , y ) or
i
j XY
i
j

mkn  E ( X kY n )   y j xi
 
k n

x
y f XY ( x, y )dxdy


  
The moment
m11  E( XY )
is called the correlation of X and Y
If m11
= 0 we say that X and Y are orthogonal
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The Covariance of random variables X and Y is defined by:
Cov( X ,Y )   XY  E X   X Y  Y 
Cov( X , Y )   XY  E XY   E X EY 
If Cov
(XY ) = 0 we say that X and Y are uncorrelated.
If two random variables are independent they are uncorrelated,
but opposite is not always true.
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30
N – Variate random variables
n – variate random variable or n – dimensional random vector



T
X  X1 X 2 X 3     X n
Covariance matrix
 11  12     1n 







12
22
2n 

K
 

 


 n1  n 2     nn 
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