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BIVARIATE DISTRIBUTIONS
Most often, we may be concerned with the study of two or more random variables
simultaneously. The joint probability distribution of the two random variables is called a
Bivariate Distribution.
Note
X Y , is a discrete bivariate random variable, if each of the random variables X and Y
is discrete.
X Y , is a continuous bivariate random variable, if each of the random variables is
continuous.
- There are cases where one variable is discrete and the other continuous.
Joint Probability Distributions
1. Discrete Case
Let X and Y be discrete random variables with possible values ; 1, 2,...,
i
x i = m and
j
y j = n respectively. The joint (or bivariate) probability distribution for X and Y is
given by
i j i j i j
P x y = P X = x Y = y x y
Or
i j
P x y = P X = x Y = y
The function
i j
P x y is referred to as the joint probability mass function (pmf) of X
and Y. This function gives the probability that X will assume a particular value x while at the
same time Y will assume a particular value y.
In tabular form,
X
Y
Row Totals
1
y
2
y
n
y
1
x
1 1
p x , y
1 2
p x , y
1
n
p x y
1
p x
2
x
2 1
p x , y
2 2
p x , y
2
n
p x y
2
p x
m
x
1
m
p x y
2
m
p x y
m n
p x y
m
p x
Column
Totals
1
p y
2
p y
n
p y
Example
Consider tossing a fair coin 3 times. Then the sample space, S is
S = HHH HHT HTH THH TTH THT HTT TTT , , , , , , ,
We define two random variables on this sample space:
- X − the number of heads in the 3 tosses;
- Y − the number of tails prior to the last head.
The distribution of the two random variables is given below.
Event HHH HHT HTH THH HTT THT TTH TTT
X 3 2 2 2 1 1 1 0
Y 0 0 1 1 0 1 2 3
From the table above,
P X = Y = = P = =
Similarly,
Example
Given the function
0, otherwise
k x y x y
p x y
(a) Find the constant k such that the
p x y , is a joint pmf.
(b) Present in a tabular form for the probabilities associated with the sample points ( x , y ).
Obtain the row and column totals.
Solution
(a) Clearly,
p x y , 0
Also,
1 2 1
0 0 0
1
0
x y x
x
k x y k x x x
k x
k
k
= = =
=
For
p x y , to be a joint pmf, we must have
21 k = 1
Therefore
k =
(b) For
p X = 0, Y = 0 = p 0, 0
Similarly,
p
p
p
p
p
These results are presented in the following table.
X
Y
Row Totals 0 1 2
Column Totals 3 21 7 21 11 21
2. Continuous Case
Let X and Y be two-dimensional random variables. The joint (or bivariate) probability
density function
f x y , of X and Y is given by
( ) ( )
2 2
1 1
1 2 1 2
y x
y x
P x X x y Y y = f x y dxdy
for two pairs
1 2
x , x and
1 2
y , y with
2 1 2 1
x x , y y.
Definition
Let ( X , Y ) be a continuous bivariate random variable assuming all values in the R. The
joint pdf ( )
f x y , is a function satisfying the following properties:
(b) The probability is calculated as
( )
( )
( )
1
2
1
4
1
2
1
4
1
2
1
4
1
2
1
4
2
1
0
1
2 2
0
2
3
x y
P x y A dxdy
x y
dy
y dy
y y
Exercise
Given the following function of a two-dimensional continuous random variable ( X , Y ):
2
0, elsewhere.
xy
x x y
f x y k
where, k is a constant.
(a) Find the value of k 0 such that
f x y , is a pdf.
(b) Find P ( 0 x 1, 1 y 2 ).
Joint Cumulative Distribution Functions
The joint cumulative distribution function (cdf) of two random variables X and Y is
called the bivariate cumulative distribution function , or simply the joint or bivariate
distribution function.
For any random variables X and Y , the joint (bivariate) cumulative distribution function
F x y , , is given by
F x y , = P { X x } { Y y = P X ( x Y , y ).
Definition 1
The joint distribution function of two discrete random variables X and Y is
( )
i j
i j
x x y y
F x y P x y
Definition 2
The cumulative distribution function of two-dimensional continuous random variables
X Y , is defined as
y x
F x y f s t dsdt
− −
where f ( s , t ) is the value of the joint pdf of X and Y at ( s , t ).
Example 2
Given the following joint pdf of
X Y , :
2
0, elsewhere.
xy
x x y
f x y
Calculate
P X 1, Y 1.
Solution
1 1
2
0 0
1
3 2
1 0 0 1 0 1
2
0
xy
P X Y x dxdy
x x y
dy
y
dy
y y
Theorem
If F is the cdf of a two-dimensional random variable with joint pdf
f x y , , then
2
F x y , f x y ,
x y
where F is differentiable.
Example
Let
( )( )
x y
F x y e e x y
− −
Find the joint pdf
f x y ,.
Solution
( )
( )
2
x y
x y
x y
F x y e e
x
F x y e e
x y
e x y
− −
− −
− +
Hence,
( )
x y
f x y e x y
− +
Marginal Distribution of Bivariate Random Variables
Definition 1
Let X and Y be discrete random variables with joint probability function ( )
i j
P x y.
Then the marginal distributions of X and Y , respectively, are given by
( ) ( ) ( )
( ) ( ) ( )
1
1
m
i i i j
j
n
j j i j
i
g x P X x p x y i n
h y P Y y p x y j m
=
=
Note
In a bivariate pmf table, the row and column totals represent the marginal probabilities of the
respective random variables.
j
y
( ) j
h y
Note
( ) ( ) ( )
1 1 1 1
n m m n
i j i j i j
i j j i
p x y g x y h x y
= = = =
Example 2
Let the joint pmf of X and Y be as follows:
0, elsewhere.
x y
x y
p x y
(a) Find the marginal distribution of X , and hence the probability that X = 3.
(b) Find the marginal distribution of Y , and hence
P y = 2.
Solution
(a) The marginal distribution of X ,
g x is given by
2
1
y y
x y
g x p x y
x x
x
=
Thus,
0, elsewhere
x
x
g x
Therefore,
P x
(b) The marginal distribution of Y ,
h y is given by
3
1
x
x y
h y
y y y
y
y
=
Thus,
0, elsewhere
y
y
h y
Therefore,
P y
Definition 2
Suppose f be the joint pdf of the continuous two-dimensional random variables
( X Y , ). We define g x ( ) and h ( y ), the marginal probability density functions of X and Y ,
respectively, by
g ( x ) f ( x y dy , )
−
and
h ( y ) f ( x y dx , ).
−
Note
Given the joint probability distribution
p x y , and marginal probability functions
g ( x )and h ( y ), respectively, the conditional discrete probability function of X given Y is
p x y
p x y h y
h y
Similarly, the conditional discrete probability function of Y given X is
p x y
p y x g x
g x
Example
Define the joint pmf of
X Y , by:
X
Y
Find
(a)
P Y = 10 | X = 0
(b)
P X = 1| Y = 0
Solution
(a)
p
P Y X
g
But
y
g x p x y
p x p x p x
Implies,
g = p + p + p
Therefore,
P Y = X = = =
(b) Exercise
Continuous Case
Let ( X Y , )be a continuous two-dimensional random variable with joint pdf f ( x y , ).
Let g and h be the marginal pdfs of X and Y , respectively. The conditional probability of X for
a given Y = y is defined by
f x y
f x y h y
h y
and the conditional probability of Y for a given X = x is given by
f x y
f y x g x
g x
Example
Let the continuous random vector
X Y , have a joint pdf given by
0, elsewhere.
y
e x y
f x y
−
F x y , = F x F y
for every pair of real numbers
x y ,.
Definition
If X and Y are discrete random variables with joint probability function
p x y , and
marginal probability function
g x and
h y , respectively, then X and Y are independent if
and only if
p x y , = g x h y ,
for all pairs of real numbers
x y ,.
Example
Consider the discrete bivariate random vector
x y , with joint pmf given by:
p p p
p p
p
Verify whether X and Y are independent.
Solution
For X and Y to be independent,
p x y , = g x h y ,∀ ( x , y )
y
g x p x y
p x p x p x
When x = 10,
g = p + p + p
Also,
x
h y p x y
p y p y
When y = 1,
h = p + p
Implies,
g h = =
Therefore,
p 10,1 = g 10 h 1
Also,
h = p + p
Implies,
g h = =
Therefore,
p 10,3 g 10 h 3
Hence, X and Y are not independent, since
p x y , g x h y , x y ,.
