theorem: 7.1. p 191 a real valued function f of two variables is joint probability density function...
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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
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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
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Marginal probability density function Example:
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المحاضرة 10/4/1435اإلثنين :الثالثة
7.2. Bivariate Continuous Random Variables
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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
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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 ?
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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
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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
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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
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Marginal probability density function:
1( ) ( , )f y f x y dx
is called the marginal probability density function of Y.
Similarly, the function
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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?
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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
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The joint cumulative distributionfunction F(x, y):
From the fundamental theorem of calculus, we again: obtain
2 ( , )( , ) .
F x yf x y
x y
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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 ?
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EXERCISES:
Page 208-209
1 , 2 , 3 , 4 , 7 , 8 , 10 , 11
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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).
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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
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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
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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
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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
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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 ?
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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 ?
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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 ?
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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
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7.3. Conditional Distributions
What is the conditional density of X given Y = y over the appropriate domain?
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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.
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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
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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?
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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?
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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?
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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.
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EXERCISES:
Page 210-211
14 , 16 , 21