Chapter 3 discusses multiple random variables, focusing on joint cumulative distribution functions (CDF), joint probability density and mass functions, marginal statistics, independence, conditional distributions, and correlation. The document includes detailed definitions, properties, and examples related to these concepts, providing a foundation for understanding probabilistic analysis of random variables. Key topics also include methods for calculating marginal statistics and the implications of variable independence on joint distributions.

1. Chapter 3:- Multiple Random Variables
Prepared by Getahun Sh.(MSc)
Ambo University
Hachalu Hundessa Campus
School of Electrical Engineering and Computing
Department of Electrical & Computer Engineering
Probability and Random Process (ECEg-2114)

2. Multiple Random Variables
Outline
The Joint Cumulative Distribution Function
The Joint Probability Density and Mass Functions
Marginal Statistics
Independence
Conditional Distributions
Correlation and Covariance
2
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3. The Joint Cumulative Distribution Function
 The joint cdf of two random variables X and Y denoted by
FXY(x, y) is a function defined by:
Properties of the Joint cdf, FXY(x, y):
numbers.
real
arbitrary
are
and
where
)
,
(
)
,
(
]
)
(
and
)
(
[
)
,
(
y
x
y
Y
x
X
P
y
x
F
y
Y
x
X
P
y
x
F
XY
XY






 

1
)
,
(
0
. 
 y
x
F
i XY
1
)
,
(
)
,
(
lim
. 







XY
XY
y
x
F
y
x
F
ii
0
)
,
(
)
,
(
lim
. 







XY
XY
y
x
F
y
x
F
iii
3
11/04/24

4. The Joint Probability Density Function
 The joint probability function (pdf) of two continuous random
variables X and Y is defined as:
 Thus, the joint cumulative distribution function (cdf) is given
by:
y
x
y
x
F
y
x
f XY
XY




)
,
(
)
,
(
2

 

y x
XY
XY dxdy
y
x
f
y
x
F
- -
)
,
(
)
,
(
4
11/04/24

5. The Joint Probability Density Function Cont’d…..
Properties of the Joint pdf, fXY(x, y):
 
 











2
1
2
1
)
,
(
)
,
(
.
3
1
)
,
(
.
2
0
)
,
(
.
1
2
1
2
1
- -
y
y
x
x
XY
XY
XY
dxdy
y
x
f
y
Y
y
x
X
x
P
dxdy
y
x
f
y
x
f
5
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6. The Joint Probability Mass Function
 The joint probability mass function (pmf) of two discrete
random variables X and Y is defined as:
 The joint cdf can be written as:
Properties of the Joint pmf, PXY (xi , yj ):
)
,
(
)
,
( j
i
j
i
XY y
Y
x
X
P
y
x
P 


 
 

x
x y
y
j
i
XY
XY
i j
y
x
P
y
x
F )
,
(
)
,
(
 


i j
x y
j
i
XY
j
i
XY
y
x
P
y
x
P
1
)
,
(
.
2
0
)
,
(
0
.
1
6
11/04/24

7. Marginal Statistics
 In the case of two or more random variables, the statistics of
each individual variable are called marginal statistics.
i. Marginal cdf of X and Y
ii. Marginal pdf of X and Y
)
,
(
)
,
(
lim
)
( 




x
F
y
x
F
x
F XY
XY
y
X
)
,
(
)
,
(
lim
)
( y
F
y
x
F
y
F XY
XY
x
Y 








-
)
,
(
)
( dy
y
x
f
x
f XY
X




-
)
,
(
)
( dx
y
x
f
y
f XY
Y
7
11/04/24

8. Marginal Statistics Cont’d…..
iii. Marginal pmf of X and Y




j
y
)
,
(
)
(
)
( i
i
XY
i
X
i y
x
P
x
P
x
X
P




i
x
)
,
(
)
(
)
( i
i
XY
j
Y
j y
x
P
y
P
y
Y
P
8
11/04/24

9. Independence
 If two random variables X and Y are independent, then
i. from the joint cdf
ii. from the joint pdf
iii.from the joint pmf
)
(
)
(
)
,
( y
F
x
F
y
x
F Y
X
XY 
)
(
)
(
)
,
( y
f
x
f
y
x
f Y
X
XY 
)
(
)
(
)
,
( j
Y
i
X
j
i
XY y
P
x
P
y
x
P 
9
11/04/24

10. Conditional Distributions
i. Conditional Probability Density Functions
ii. Conditional Probability Mass Functions
0
)
(
,
)
(
)
,
(
)
/
(
/ 
 y
f
y
f
y
x
f
y
x
f Y
Y
XY
Y
X
0
)
(
,
)
(
)
,
(
)
/
(
/ 
 x
f
x
f
y
x
f
x
y
f X
X
XY
X
Y
0
)
(
,
)
(
)
,
(
)
/
(
/ 
 j
Y
j
Y
j
i
XY
j
i
Y
X y
P
y
P
y
x
P
y
x
P
0
)
(
,
)
(
)
,
(
)
/
(
/ 
 i
X
i
X
j
i
XY
i
j
X
Y x
P
x
P
y
x
P
x
y
P
10
11/04/24

11. Correlation and Covariance
i. Correlation
ii. Covariance
iii. Correlation Coefficient
)
(
)
,
( XY
E
Y
X
Cor
RXY 

)
(
)
(
)
(
)
,
(
)]
)(
[(
)
,
(
Y
E
X
E
XY
E
Y
X
Cov
Y
X
E
Y
X
Cov
XY
Y
X
XY












Y
X
XY
Y
X
XY
Y
X
Cov





 

)
,
(
11
11/04/24

12. Examples on Two Random Variables
Example-1:
The joint pdf of two continuous random variables X and Y is
given by:
.
and
of
pdf
l
conditiona
the
Find
.
)
1
(
Find
.
t?
independen
and
Are
.
.
and
of
pdf
marginal
the
Find
.
.
of
value
the
Find
.
constant.
a
is
re
whe
otherwise
,
0
1
0
,
1
0
,
)
,
(
Y
X
e
Y
X
P
d
Y
X
c
Y
X
b
k
a
k
y
x
kxy
y
x
fXY




 




12
11/04/24

13. Examples on Two Random Variables Cont’d……
Solution:
4
1
4
0
1
4
2
1
0
1
2
1
1
)
,
(
.
2
1
0
1
0
2
1
0
1
0
-






























 





k
k
y
k
ydy
k
x
y
k
kxydxdy
dxdy
y
x
f
a XY
13
11/04/24

14. Examples on Two Random Variables Cont’d……
Solution:


 















 



otherwise
,
0
1
0
,
2
)
(
2
0
1
2
4
)
(
4
)
,
(
)
(
of
pdf
Marginal
.
and
of
pdf
Marginal
.
2
1
0
x
x
x
f
x
y
x
x
f
xydy
dy
y
x
f
x
f
X
i
Y
X
b
X
X
XY
X
14
11/04/24

15. Examples on Two Random Variables Cont’d……
Solution:


 















 



otherwise
,
0
1
0
,
2
)
(
2
0
1
2
4
)
(
4
)
,
(
)
(
of
pdf
Marginal
.
and
of
pdf
Marginal
.
2
1
0
y
y
y
f
y
x
y
y
f
xydx
dx
y
x
f
y
f
Y
ii
Y
X
b
Y
Y
XY
Y
15
11/04/24

16. Examples on Two Random Variables Cont’d……
Solution:
t
independen
are
and
)
(
)
(
)
,
(
.
Y
X
y
f
x
f
y
x
f
c Y
X
XY


6
/
1
)
1
(
6
/
1
)
4
/
3
/
2
2
/
(
2
)
2
(
2
]
)
1
(
2
/
1
[
4
0
1
2
4
4
)
1
(
.
4
3
2
1
0
3
2
1
0
2
1
0
1
0
1
0
2



























 

Y
X
P
y
y
y
dy
y
y
y
dy
y
y
dy
x
y
xydxdy
Y
X
P
d
y
16
11/04/24

17. Examples on Two Random Variables Cont’d……
Solution:




 








otherwise
,
0
1
0
,
1
0
,
2
)
/
(
2
2
4
)
(
)
,
(
)
/
(
of
pdf
l
Conditiona
.
and
of
pdf
l
Conditiona
.
/
/
y
x
x
y
x
f
x
y
xy
y
f
y
x
f
y
x
f
X
i
Y
X
e
Y
X
Y
XY
Y
X
17
11/04/24

18. Examples on Two Random Variables Cont’d……
Solution:




 








otherwise
,
0
1
0
,
1
0
,
2
)
/
(
2
2
4
)
(
)
,
(
)
/
(
of
pdf
l
Conditiona
.
and
of
pdf
l
Conditiona
.
/
/
y
x
y
x
y
f
y
x
xy
x
f
y
x
f
x
y
f
Y
ii
Y
X
e
X
Y
X
XY
X
Y
18
11/04/24

19. Examples on Two Random Variables Cont’d……
Example-2:
The joint pdf of two continuous random variables X and Y is
given by:
.
and
of
pdf
l
conditiona
the
Find
.
)
2
/
1
0
(
Find
.
t?
independen
and
Are
.
.
and
of
pdf
marginal
the
Find
.
.
of
value
the
Determine
.
constant.
a
is
re
whe
otherwise
,
0
1
0
,
)
,
(
Y
X
e
X
P
d
Y
X
c
Y
X
b
k
a
k
x
y
k
y
x
fXY






 



19
11/04/24

20. Examples on Two Random Variables Cont’d……
Solution:
 
2
1
2
0
1
2
)
1
(
1
1
1
1
)
,
(
.
2
1
0
1
0
1
0
1
-
























 





k
k
y
y
k
dy
y
k
y
x
k
kdxdy
dxdy
y
x
f
a
y
XY
20
11/04/24

21. Examples on Two Random Variables Cont’d……
Solution:
 


 







 



otherwise
,
0
1
0
,
2
)
(
2
0
2
)
(
2
)
,
(
)
(
of
pdf
Marginal
.
and
of
pdf
Marginal
.
0
x
x
x
f
x
x
y
x
f
dy
dy
y
x
f
x
f
X
i
Y
X
b
X
X
x
XY
X
21
11/04/24

22. Examples on Two Random Variables Cont’d……
Solution:
 


 









 



otherwise
,
0
1
0
),
1
(
2
)
(
)
1
(
2
1
2
)
(
2
)
,
(
)
(
of
pdf
Marginal
.
and
of
pdf
Marginal
.
1
y
y
y
f
y
y
x
y
f
dx
dx
y
x
f
y
f
Y
ii
Y
X
b
Y
Y
y
XY
Y
22
11/04/24

23. Examples on Two Random Variables Cont’d……
Solution:
t
independen
not
are
and
)
(
)
(
)
,
(
.
Y
X
y
f
x
f
y
x
f
c Y
X
XY


4
/
1
)
2
/
1
0
(
4
/
1
0
2
/
1
2
0
)
2
(
2
)
,
(
)
2
/
1
0
(
.
2
/
1
0
2
2
/
1
0 0
2
/
1
0
2
/
1
0 0













  
 
X
P
x
xdx
dx
x
y
dydx
dydx
y
x
f
X
P
d
x
x
XY
23
11/04/24

24. Examples on Two Random Variables Cont’d……
Solution:
















otherwise
,
0
1
0
,
1
1
)
/
(
)
1
(
1
)
1
(
2
2
)
(
)
,
(
)
/
(
of
pdf
l
Conditiona
.
and
of
pdf
l
Conditiona
.
/
/
x
y
y
y
x
f
y
y
y
f
y
x
f
y
x
f
X
i
Y
X
e
Y
X
Y
XY
Y
X
24
11/04/24

25. Examples on Two Random Variables Cont’d……
Solution:













otherwise
,
0
1
0
,
1
)
/
(
1
2
2
)
(
)
,
(
)
/
(
of
pdf
l
Conditiona
.
and
of
pdf
l
Conditiona
.
/
/
x
y
x
x
y
f
x
x
x
f
y
x
f
x
y
f
Y
ii
Y
X
e
X
Y
X
XY
X
Y
25
11/04/24

26. Examples on Two Random Variables Cont’d……
Example-3:
The joint pmf of two discrete random variables X and Y is given
by:
t?
independen
and
Are
.
.
and
of
pmf
marginal
the
Find
.
.
of
value
the
Find
.
constant.
a
is
re
whe
otherwise
,
0
2
,
1
2;
,
1
,
)
2
(
)
,
(
Y
X
c
Y
X
b
k
a
k
y
x
y
x
k
y
x
P
i
j
i
j
i
XY


 



26
11/04/24

27. Examples on Two Random Variables Cont’d……
Solution:
18
/
1
1
18
1
)]
2
4
(
)
1
4
(
)
2
2
(
)
1
2
[(
1
)
2
(
1
)
,
(
.
2
1
2
1

















 

 
k
k
k
y
x
k
y
x
P
a
i j
i j
x y
j
i
x y
j
i
XY
27
11/04/24

28. Examples on Two Random Variables Cont’d……
Solution:
















 
 
otherwise
,
0
2
,
1
),
3
4
(
18
1
)
(
)
2
2
(
18
1
)
1
2
(
18
1
)
(
)
2
(
18
1
)
,
(
)
(
of
pmf
Marginal
.
and
of
pmf
Marginal
.
2
1
i
i
i
X
i
i
i
X
j
i
y
y
j
i
XY
i
X
x
x
x
P
x
x
x
P
y
x
y
x
P
x
P
X
i
Y
X
b
j
j
28
11/04/24

29. Examples on Two Random Variables Cont’d……
Solution:
















 
 
otherwise
,
0
2
,
1
),
6
2
(
18
1
)
(
)
4
(
18
1
)
2
(
18
1
)
(
)
2
(
18
1
)
,
(
)
(
of
pmf
Marginal
.
and
of
pmf
Marginal
.
2
1
j
j
j
Y
j
j
j
Y
j
i
x
x
j
i
XY
j
Y
y
y
y
P
y
y
y
P
y
x
y
x
P
y
P
Y
ii
Y
X
b
i
i
29
11/04/24

30. Examples on Two Random Variables Cont’d……
Solution:
t.
independen
not
are
and
)
(
)
(
)
,
(
.
Y
X
y
P
x
P
y
x
P
c j
Y
i
X
j
i
XY


30
11/04/24

31. 11/04/24 31