on probability and statistics

1. OPERATIONS ON MULTIPLE
RANDOM VARIABLES
Prepared by
g.krishna

2. 5.1 Expected Value of a function of R.Vs
• If g(x,y) is function of two r.v.s X and Y , then the expected value
of g is:
• Note that the expected value of a jam of functions is equal to the
sum of the expected values of the functions:

3. Expected Value of a function of R.Vs

4. 5.1.1 Joint Moments about the origin:
4
Notes:
1. mn0= E[ Xn
] are the moments of X, m0k are the moments of Y.
2. The sum n+k is called the order of the moments.
Thus, m02 ,m20 m11 are called second order moments of X and Y.
3. m10 = E[X] and m01 = E[Y] are the expected values of X and Y,
respectively, and are the coordinators of the "center of gravity" of
the function fXY(x,y).

5. 5.1.1 Joint Moments about the origin:
Correlation: The second-order moment m11 = E[XY] is called the
correlation of X and Y. In fact, it is a very important statistic and
denoted by RXY.
- If RXY = E[X] E[Y] ,then X and Y are said to be uncorrelated.
- If X and Y are independent ,then fXY(x,y) = fX(x) fY(y) and

6. 5.1.1 Joint Moments about the origin:
• Therefore, if X and Y are independent they are⇒
uncorrelated.
• However, if X and Y are uncorrelated, it is not necessary
that they are independent.
• If RXY = 0 then X and Y are called orthogonal.
6

8. 5.1.2 Joint Central Moments:
• Covariance: The second order joint moment u11 is called the
covariance of X and Y and denoted by CXY
8

9. 5.1.2 Joint Central Moments:
9

10. 5.1.2 Joint Central Moments:
10
• The normalized second-order moment ρ
is known as the correlation coefficient of X and Y.

12. 5.2 Joint characteristic Functions:
12
Where w1and w2 are real numbers.
- By setting w1= 0 or w2 = 0, we obtain the marginal
characteristic function.
The joint characteristic function of two r.v.s X and Y is defined
by

13. Joint moments are obtained as :
13
Joint moments mnk can be found from the joint characteristic
function as follows:

15. 5.3 Jointly Gaussian Random Variables
• Two random variables are jointly Gaussian if their joint
density function is of the form (sometimes called bivariate
Gaussian)
15

16. Jointly Gaussian Random Variables
16
• Its maximum is located at the point

17. 17
Jointly Gaussian Random Variables
• The locus of constant values of fXY(x,y) will be an ellipse.
• This is equivalent to saying that the line of intersection formed by
slicing the function fXY(x,y) with a plane parallel to the xy plane is an
ellipse.

18. 18
Jointly Gaussian Random Variables
Where fX(x) and fY(y) are the marginal density functions of X and Y.
,0 ( , ) ( ) ( )X Y X Yf x y f x f yρ = ⇒ =
jointly gaussian & uncorr. indep.⇒

19. • Consider r.v.s Y1 and Y2 related to arbitrary r.v.s X and Y by the
coordinate relation
19
Jointly Gaussian Random Variables
1 cos sinY X Yθ θ= +
2 sin cosY X Yθ θ= − +
1 2, 1 1 2 2[( )( )]Y YC E Y Y Y Y= − −
[{( )cos ( )sin }{ ( )sin ( )cos }]E X X Y Y X X Y Yθ θ θ θ= − + − − − + −

20. 20
Jointly Gaussian Random Variables
2 2 2 2
( )sin cos [cos sin ]Y X XYCσ σ θ θ θ θ= − + −
2 2 2 21 1
( )sin 2 cos2 ( )sin 2 cos2
2 2
Y X XY Y X X YCσ σ θ θ σ σ θ ρσ σ θ= − + = − +
1 2
1
, 2 2
21
0 tan
2
X Y
Y Y
X Y
C
ρσ σ
θ
σ σ
−  
= ⇒ =  − 
• From correlation coefficient.,
• If we require Y1 and Y2 to be uncorrelated, we must have CY1Y2=0.
by equating the above equation to zero we get.,

21. N Random variables
21
• N random variables are jointly Gaussian if their joint density
function is of the form (sometimes called multivariate
Gaussian)

22. N Random variables
22
Cij is the covariance matrix , When N = 2,

23. 23
Jointly Gaussian Random Variables
Uncorrelated Gaussian random variables are also statistically
independent.
Other properties of Gaussian r.v.s include:
• Gaussian r.v.s are completely defined through their 1st- and 2nd-
order moments, i.e., their means, variances, and covariance's.
• Random variables produced by a linear transformation of jointly
Gaussian r.v.s are also Gaussian.
• The conditional density functions defined over jointly Gaussian r.v.s
is also Gaussian.
There fore we conclude that any uncorrelated Gaussian
random variables are also statistically independent.
It results that a coordinate rotation through an angle

24. Transformations of Multiple Random Variables
24
One function
1 2( , )Y g X X=
1 2, 1 2( , )X Xf x x
1 2
1 2
1 2 , 1 2 1 2
( , )
( ) [ ( , ) ] ( , )Y X X
g x x y
F y P g X X y f x x dx dx
≤
= ≤ = ∫∫
( )
( ) Y
Y
dF y
f y
dy
=

25. 25
1 2positive r.v.'s &X X
2
1 2
1
, 1 2 1 20 0
2
( ) [ ] ( , )
yx
Y X X
X
F y P y f x x dx dx
X
∞
= ≤ = ∫ ∫ 0y >
Transformations of Multiple Random Variables
1 22 , 2 2 20
( )
( ) ( , )Y
Y X X
dF y
f y x f yx x dx
dy
∞
= = ∫ 0y >
Example 1: find the density function for 1
2
X
Y
X
=

26. Transformations of Multiple Random Variables
• Multiple functions
Yi= Ti (X1, X2, X3…………XN)., i =1,2,3……….N
26
1 1 1 2
1 2
2 2 1 2
( , )
( , )
( , )
Y T X X
T X X
Y T X X
   
= =   
   
1
1 1 1 21
1 2 1
2 2 1 2
( , )
( , )
( , )
x T y y
T y y
x T y y
−
−
−
  
= =   
    
1 2 1 2, 1 2 , 1 2( , ) ( , )Y Y X Xf y y f x x J=
1 1
1 1
1 2
1 1
2 2
1 2
T T
y y
J
T T
y y
− −
− −
∂ ∂
∂ ∂
=
∂ ∂
∂ ∂
jacobian

27. Transformations of Multiple Random Variables
27
1 2
1 2
1 2 1 2
,
, 1 2
( , )
( , )
X X
Y Y
dy by cy ay
f
ad bc ad bcf y y
ad bc
− − +
− −=
−
1
1 1 1 2 11
1 2 1
2 22 1 2
( , ) 1
( , )
( , )
x T y y yd b
T y y
x c a yad bcT y y
−
−
−
  −    
= = =     −−       
1 1
1 1
1 2
1 1
2 2
1 2
1
T T
y y
J
ad bcT T
y y
− −
− −
∂ ∂
∂ ∂
= =
−∂ ∂
∂ ∂
1 1 1 2 1 2 1
1 2
2 2 1 2 1 2 2
( , )
( , )
( , )
Y T X X aX bX Xa b
T X X
Y T X X cX dX c d X
+        
= = = =        +         
Example :

28. 28
Linear Transformations of Gaussian Random Variables
1 1 11 12
21 222 2
Y X a a
Y A AX X
a aY X
     
= = = =     
    
[ ] [ ] [ ]E Y E AX AE X= =
[( )( ) ] [ ( )( ) ]T T T
YC E Y Y Y Y E A X X X X A= − − = − −
[( )( ) ]T T T
XAE X X X X A AC A= − − = 1 1 1T
Y XC A C A− − − −
⇒ =
1
1 2
1
( ) ( )
2
, 1 2 1/2/2
1
( , )
(2 )
T
Xx X C x X
X X N
X
f x x e
Cπ
−
− − −
=
1 2, 1 2( , ) ?Y Yf y y =
2
det( ) det( ) det( )Y XC A C=

29. 29
5.5 Linear Transformations of
Gaussian Random Variables
1 1 11
( ) ( )
2
1/2/2
1
(2 ) det( )
T
XA y X C A y X
N
X
e
C Aπ
− − −
− − −
=
1 2 1 2
1
, 1 2 ,
1
( , ) ( )
det( )
Y Y X Xf y y f A y
A
−
=
1 1 1 1 1
( ) ( ) ( ) ( )T T T
X XA y X C A y X y Y A C A y Y− − − − − −
− − = − −
1
( ) ( )T
Yy Y C y Y−
= − −
1
1 2
1
( ) ( )
2
, 1 2 1/2/2
1
( , )
(2 )
T
Yy Y C y Y
Y Y N
Y
f y y e
Cπ
−
− − −
⇒ =
1/2 1/2
det( )X YC A C=
(gaussian)

30. 30
5.5 Linear Transformations of
Gaussian Random Variables
1
1 2
1
( ) ( )
2
, 1 2 1/2
1
( , )
(2 )
T
Yy Y C y Y
Y Y
Y
f y y e
Cπ
−
− − −
⇒ =
Ex 5.5-1: 1 1 1
2 2 2
1 2
3 4
Y X X
A
Y X X
−      
= =      
      
1
2
[ ] 0
[ ] 0
E X
E X
   
=   
  
4 3
3 9
XC
 
=  
 
1 1
2 2
[ ] [ ] 0
[ ] [ ] 0
E Y E X
A
E Y E X
     
= =     
    
1 2 4 3 1 3 28 66
3 4 3 9 2 4 66 252
T
Y XC AC A
− −       
= = =       − −       
1 2
1 2
66
0.786
28 252
YY
Y Y
C
ρ
σ σ
−
= = = −