Theorem: 7.1. p 191
A real valued function f of two variables is joint
probability density function of a pair of
discrete random variables X and Y if and
only if :
(1)f XY (x , y )  0 for all ( x , y )
(2) f XY (x , y )  1
x
y
Example:7.1 page 191
For what value of the constant k the function
given by
if x  1, 2,3; y 1, 2,3
k x y
f (x , y )  
otherwise
0
Is a joint probability density function of some
random variables X , Y ?
Marginal probability density function
Example:
:‫ المحاضرة الثالثة‬10/4/1435 ‫اإلثنين‬
7.2. Bivariate Continuous
Random Variables
7.2. Bivariate Continuous Random
Variables
In this section, we shall extend the idea of probability
density functions
of one random variable to that of two random variables.
Definition 7.5. The joint probability density function of
the random variables X and Y is an integrable function
f(x, y) such that
(1)f X Y (x , y )  0 for all ( x , y ) 

(2)

 
 
f X Y ( x , y ) dx dy  1
2
7.2. Bivariate Continuous Random
Variables
Example 7.6. Let the joint density function of X
and Y be given by
k x y 2
f (x , y )  
0
if 0  x  y 1
otherwise
What is the value of the constant k ?
REMARK:
If we know the joint probability density function
f of the random variables X and Y , then we
can compute the probability of the event A
from:
P ( A )  
A
f X Y (x , y ) dx dy
Bivariate Continuous Random
Variables
Example 7.7. Let the joint density of the ontinuous
random variables X and Y be
6 2
if 0  x  1;0  y  1
 (x x y
f (x , y )   5
0
otherwise
What is the probability of the event ( x  y )?
Marginal probability density function:
Definition 7.6. Let (X, Y ) be a continuous bivariate
random variable. Let f(x, y) be the joint probability
density function of X and Y . The function

f 1 (x ) 
 f (x , y ) dy

is called the marginal probability density function
of X. Similarly, the function
Marginal probability density function:
Similarly, the function

f 1( y ) 
 f (x , y ) dx

is called the marginal probability density function
of Y.
Marginal probability density
function:
Example 7.9. If the joint density function for X and
Y is given by:
2e  x  y if 0  x  y 
f (x , y )  
otherwise
0
What is the marginal density of X where nonzero?
Definition 7.7. Let X and Y be the continuous
random variables with joint probability density
function f(x, y). The joint cumulative distribution
function F(x, y) of X and Y is defined as
y
F ( x , y )  P ( X  x ,Y  y ) 
x

 
for all ( x , y )
2
.
f XY (u ,v ) du dv
The joint cumulative distribution
function F(x, y):
From the fundamental theorem of calculus, we
again: obtain
2F ( x , y )
f (x , y ) 
.
x  y
The joint cumulative distribution
function F(x, y):
Example 7.11. If the joint cumulative distribution
function of X and Y is given by
1
3
2 2
(2
x
y

3
x
y ) for 0  x , y 1

F (x , y )   2
0
otherwise
then what is the joint density of X and Y ?
EXERCISES:
Page 208-209
1 , 2 , 3 , 4 , 7 , 8 , 10 , 11
7.3. Conditional Distributions
First, we motivate the definition of conditional
distribution using discrete random variables
and then based on this motivation we give a
general definition of the conditional
distribution. Let X and Y be two discrete
random variables with joint probability
density f(x, y).
7.3. Conditional Distributions
Then by definition of the joint probability
density, we have f(x, y) = P(X = x, Y = y).
If A = {X = x}, B = {Y = y} and f (y) = P(Y = y),
then from the above equation we have
P ({X = x} / {Y = y}) = P (A/B)

P (A
B
P (B )
P { X  x }and P {Y  y } f ( x , y )


P {Y  y }
f (y )
7.3. Conditional Distributions
If we write the P ({X = x} / {Y = y}) as g(x / y),
then we have
f (x , y )
g (x / y )
f (y )
7.3. Conditional Distributions
Definition 7.8. Let X and Y be any two random
variables with joint density f(x, y) and
marginals f1(x) and f2(y). The conditional
probability density function g of X, given (the
event) Y = y, is defined as
f (x , y )
g (x / y )
, f ( y ) 0
f (y )
7.3. Conditional Distributions
Similarly, the conditional probability density
function h of Y , given (the event)
X = x, is defined as
f (x , y )
h (x / y )
, f (x ) 0
f (y )
7.3. Conditional Distributions
Example 7.14. Let X and Y be discrete random
variables with joint probability function
1
( x  y ) for x 1, 2,3; y 1, 2

f ( x , y )   21

otherwise
0
What is the conditional probability density
function of Y, given X = 2 ?
7.3. Conditional Distributions
Example 7.15. Let X and Y be discrete random
variables with joint probability function
( x  y )

f ( x , y )   32

0
for x 1, 2; y 1, 2, 3, 4
otherwise
What is the conditional probability density
function of Y, given X = x ?
7.3. Conditional Distributions
Example 7.16. Let X and Y be contiuous
random variables with joint pdf
12 x
f (x , y )
0
for 0  y  2x 1
otherwise
What is the conditional probability density
function of Y, given X = x ?
7.3. Conditional Distributions
Example 7.17. Let X and Y random variables
such that X has pdf

 24 x
f (x )

0
1
2
otherwise
for 0  x 
2
and the conditional density of Y given X = x
is
 y

h (x / y )2x
0

2
, for 0  y  2x
otherwise
7.3. Conditional Distributions
What is the conditional density of X given Y = y
over the appropriate domain?
7.4. Independence of Random
Variables
In this section, we define the concept of
stochastic independence of two random
variables X and Y . The conditional robability
density function g of X given Y = y usually
depends on y. If g is independent of y, then
the random variables X and Y are said to be
independent. This motivates the following
definition.
7.4. Independence of Random
Variables
Definition 7.8. Let X and Y be any two random
variables with joint density f(x, y) and
marginals f1(x) and f2(y). The random
variables X and Y are (stochastically)
independent if and only if
f ( x , y )  f ( x ) f ( y ) for all ( x , y )
x

y
.
7.4. Independence of Random
Variables
Example 7.18. Let X and Y be discrete random
variables with joint density
 1

 36
f (x , y )
 2

 36
for 1  x  y  6
for 1  x  y  6
Are X and Y stochastically independent?
7.4. Independence of Random
Variables
Example 7.19. Let X and Y have the joint
density
e  ( x  y )
for 0  x , y  
f (x , y )
otherwise
0
Are X and Y stochastically independent?
7.4. Independence of Random
Variables
Example 7.20. Let X and Y have the joint
density
x  y
f (x , y )
0
for 0  x  1 ; 0  y 1
otherwise
Are X and Y stochastically independent?
7.4. Independence of Random
Variables
Definition 7.9. The random variables X and Y
are said to be independent and identically
distributed (IID) if and only if they are
independent and have the same distribution.
EXERCISES:
Page 210-211
14 , 16 , 21