Download 5.D.3. Joint Probability Distributions

5.D.3.
Joint Probability Distributions
Let X and Y be stochastic variables on S with values
X  S   x1, x2 ,

Y  S    y1, y2 ,

The product set
X S  Y S  
 x , y  ,  x , y  , ,  x , y  , 
1
1
1
2
i
is a probability space if every ordered pair
j
x , y 
i
j
is assigned a probability
P  X  xi , Y  y j   f  xi , y j 
where f  xi , y j  is called the joint probability distribution of X and Y.
The switch to continuous stochastic variables is easily accomplished by making use of
the joint probability distribution density f  x, y  defined by
P  x  X  x  x, y  Y  y  y   f  x, y  xy
and replacing all sums with integrals.
By definition, f obeys the usual properties of a probability density, namely,
1.
Positive-definiteness:
f  x, y   0 .
2.
Normalization:
  dxdy f  x, y   1 .
Making use of the addition principle (see §5.B), we have
f X  x    dyf  x, y 
fY  y    dxf  x, y 
(5.22)
The covariance of X and Y is defined by
cov  X , Y     dxdy  x  X
 y 
Y
 f  x, y 
   dxdy  xy  x Y  y X  X Y
 XY  X Y  Y
X  X Y
 XY  X Y
(5.24)
The correlation of X and Y is defined by
cov  X , Y 
corr  X , Y  
(5.25)
 XY

X
XY  X Y
 X
2
 f  x, y 
2
 Y
2
 Y
2

It is dimensionless with the following easily proved properties:
corr  X ,Y   corr Y , X  .
1.
Symmetric:
2.
corr  X , X   1 ,
3.
1  corr  X ,Y   1 .
4.
corr  aX  b, cY  d   corr  X ,Y 
corr  X ,  X   1 .
if
a, c  0 .
Note that pairs of stochastic variables with identical distributions can have different
correlations if the joint distributions are different.
If X and Y are statistically independent, we have
1.
f  x, y   f X  x  fY  y  .
2.
XY  X Y .
3.
 X Y 
4.
2
 X Y
2
 X2  X
2
 Y2  Y
2
.
cov  X , Y   0 .
Note that cov  X , Y   0 does not imply X and Y are statistically independent.
Consider the variable
z  G  x, y 
where G is a known function of x and y.
The probability density f Z  z  for the stochastic variable Z is given by
f Z  z     dxdy  z  G  x, y   f  x, y 
(5.26)
Using
  x 
1
dke  ikx
2 
we have
fZ  z 
1
2
   dkdxdye
 ik  z G  x , y 
f  x, y 
which, on comparison with
fZ  z  
1
dke  ikzZ  k 
2 
gives
Z  k     dxdyeikG x , y  f  x, y 
(5.28a)
If X and Y are statistically independent, i.e.,
f  x, y   f X  x  fY  y 
we have
Z  k     dxdyeikG x , y  f X  x  fY  y 
(5.28)
For example, if
G  x, y   x  y
then
Z  k     dxdyeik  x  y  f X  x  fY  y 
 X  k  Y  k 
(5.29)
i.e., the characteristic function of the sum of 2 independent stochastic variables is the
product of their characteristic functions.