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) ( , ) 0 ( , )

(2) ( , ) 1

XY

XYx y

f x y for all x y

f x y

Example:7.1 page 191

For what value of the constant k the function

given by

Is a joint probability density function of some random variables X , Y ?

1,2,3; 1,2,3( , )

0

k x y if x yf x y

otherwise

Marginal probability density function Example:

المحاضرة 10/4/1435اإلثنين :الثالثة

7.2. Bivariate Continuous Random Variables

7.2. Bivariate Continuous Random Variables

In this section, we shall extend the idea of probability density functionsof 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

2(1) ( , ) 0 ( , )

(2) ( , ) 1

XY

XY

f x y for all x y

f x y dx dy

7.2. Bivariate Continuous Random Variables

Example 7.6. Let the joint density function of X and Y be given by

2 0 1( , )

0

k x y if x yf x y

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:

( ) ( , )XYAP A f x y dx dy

Example 7.7. Let the joint density of the ontinuous random variables X and Y be

26( 0 1;0 1

( , ) 50

x x y if x yf x y

otherwise

What is the probability of the event ( )?x y

Bivariate Continuous Random Variables

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

1( ) ( , )f x f x y dy

is called the marginal probability density function of X. Similarly, the function

Marginal probability density function:

1( ) ( , )f y f x y dx

is called the marginal probability density function of Y.

Similarly, the function

Marginal probability density function:

Example 7.9. If the joint density function for X and Y is given by:

2 0( , )

0

x ye if x yf x y

otherwise

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

2

( , ) ( , ) ( , )

( , ) .

y x

XYF x y P X x Y y f u v du dv

for all x y

The joint cumulative distributionfunction F(x, y):

From the fundamental theorem of calculus, we again: obtain

2 ( , )( , ) .

F x yf x y

x y

The joint cumulative distributionfunction F(x, y):

Example 7.11. If the joint cumulative distribution function of X and Y is given by

3 2 21( 2 3 ) 0 , 1

( , ) 20

x y x y for x yF x y

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 yg 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

( , )( / ) , ( ) 0

( )

f x yg x y f y

f y

7.3. Conditional Distributions

Similarly, the conditional probability density function h of Y , given (the event)

X = x, is defined as

( , )( / ) , ( ) 0

( )

f x yh x y f x

f y

7.3. Conditional Distributions

Example 7.14. Let X and Y be discrete random variables with joint probability function

1( ) 1,2,3; 1,2

( , ) 210

x y for x yf x y

otherwise

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

( )1, 2; 1,2,3,4

( , ) 320

x yfor x y

f x yotherwise

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 0 2 1( , )

0

x for y xf x y

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

2 124 0

( ) 20

x for xf x

otherwise

and the conditional density of Y given X = x is

2, 0 2

2( / )

0

yfor y x

xh x y

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

( , ) ( ) ( ) ( , ) .x yf x y f x f y for all x y

7.4. Independence of Random Variables

Example 7.18. Let X and Y be discrete random variables with joint density

11 6

36( , )2

1 636

for x yf x y

for x y

Are X and Y stochastically independent?

7.4. Independence of Random Variables

Example 7.19. Let X and Y have the joint density

( ) 0 ,( , )

0

x ye for x yf x y

otherwise

Are X and Y stochastically independent?

7.4. Independence of Random Variables

Example 7.20. Let X and Y have the joint density

0 1 ;0 1( , )

0

x y for x yf x y

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