Download Chapter 5 - Lecture 1 Jointly Distributed Random Variables

Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Chapter 5 - Lecture 1
Jointly Distributed Random Variables
Andreas Artemiou
Novemer 16th, 2009
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Introduction
Discrete random variables
Joint Probability mass function
Marginal distribution mass functions
Continuous random variables
Joint Probability density function
Marginal Probability Density Function
Area restriction
Independence
Independent random variables
Exercises
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Introduction
I
I
There are many cases that we are interested about the joint
distribution of two random variables.
Example:
I
What is the joint distribution of X = the final letter grade in
Stat 318 and Y =the final letter grade in Stat 319 of all the
students that take both classes?
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability mass function
Marginal distribution mass functions
Joint Probability mass function
I
I
Let X and Y be two discrete random variables. The joint
probability mass function p(x, y ) is defined for each pair of
numbers (x, y ) by p(x, y ) = P(X = xandY = y ).
Example: I am interested to see the joint distribution of X=
the number of kids a family in SC has and Y = the number of
bedrooms in their house. The results are summarized as
follows:
# of kids
# of Rooms
0
1
2
3
4
5
1
0.01 0.01 0.01 0.00 0.00 0.00
2
0.05 0.05 0.15 0.05 0.03 0.01
3
0.02 0.09 0.20 0.17 0.10 0.05
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability mass function
Marginal distribution mass functions
Example
I
The table in the previous slide shows the joint probability
function since we can see exactly what is P(X = xandY = y )
I
Find P(X = 4, Y = 2).
I
Find P(X = 3).
I
Find P(Y > 1).
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability mass function
Marginal distribution mass functions
Marginal distribution mass functions
I
The marginal probability mass function of X and of Y ,
denoted by pX (x) and pY (y ) respectively, are given by:
X
pX (x) =
p(x, y )
y
pY (y ) =
X
p(x, y )
x
I
In the example with X =#of children and Y =# of bedrooms
find the marginals of X and Y .
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability density function
Marginal Probability Density Function
Area restriction
Joint Probability density function
I
Let X and Y be two continuous random variables. Then
f (x, y ) is the joint probability density function for X and Y
if for any two dimensional set A
Z Z
P((X , Y ) ∈ A) =
f (x, y )dydx
A
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability density function
Marginal Probability Density Function
Area restriction
Marginal Probability Density Function
I
The marginal probability density functions of X and of Y ,
denoted by fX (x) and fY (y ) respectiuvely, are given by:
Z ∞
fX (x) =
f (x, y )dy , −∞ < x < ∞
−∞
Z
∞
f (x, y )dx, −∞ < y < ∞
fY (y ) =
−∞
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability density function
Marginal Probability Density Function
Area restriction
Example
I
Prove that:
f (x, y ) = x + y , 0 < x < 1, 0 < y < 1,
is a joint probability function.
I
Find P(0 < X < 1/4, 1/2 < Y < 1).
I
Find the marginal distributions of X and Y .
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Joint Probability density function
Marginal Probability Density Function
Area restriction
Area issues
I
In the previous example the region of calculation was a
rectangle. This makes calculations very easy.
I
There are cases that calculations are not on a rectangle.
I
Work Example 5.5 page 234 in the book.
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Independent random variables
Independent random variables
I
There are cases that knowing something about random
variable X give us information for random variable Y and vice
versa. That means, X and Y are dependent.
I
There are cases that by knowing everything about random
variable X we gain no information for random variable Y .
That means, X and Y are independent.
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Independent random variables
Definition
I
Two random variable X and Y are said to be independent if
for every pair of x and y values,
I
Discrete variables:
p(x, y ) = pX (x)pY (y )
I
Continuous variables:
f (x, y ) = fX (x)fY (y )
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Independent random variables
Examples
I
In the example with the number of kids and number of
bedrooms of families in State College. Are X and Y
independent?
I
In the continuous example we had with
f (x, y ) = x + y , 0 < x < 1, 0 < y < 1 are X and Y
independent?
I
Example 5.8 page 236
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables
Outline
Introduction
Discrete random variables
Continuous random variables
Independence
Exercises
Exercises
I
Section 5.1 page 239
I
Exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17
Andreas Artemiou
Chapter 5 - Lecture 1 Jointly Distributed Random Variables