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Statistics 270 - Lecture 18
• Will begin Chapter 5 today
• Many situations where one is interested in more than one random
variable
• Have a joint distribution for such cases
Example
• Let X and Y be random variables with pmf
x
y
1
2
1
2
3
0.10 0.5 0.10
0.05 0.10 0.15
Definition
• Let X and Y be rv’s on a sample space S
• Discrete rv’s: The joint prob. mass function for each (x,y) is
defined by p(x,y)=P(X=x, Y=y)
• If A is an event then, P( X , Y )  A  
 p ( x, y )
( x , y )A
Discrete RV’s
• Usual properties of pmf’s still hold
Example
• Let X and Y be random variables with pmf
x
y
• Observations:
• P(X=2,Y=2)=
• P(X>1, Y=1)
1
2
1
2
3
0.10 0.5 0.10
0.05 0.10 0.15
Example
• Let X be the number of Canon digital cameras sold in a week at a
certain store
• The pmf for X is
x
p(x)
0
.1
1
.2
2
.3
3
.25
4
.15
• 60% of all customers who purchase camera also purchase the longterm warranty
• Determine the joint pdf of X and Y
Definition
• The marginal probability mass function for discrete random
varaibles X and Y, denote by pX(x) and pY(y), respectively, are given
by
p X ( x)   p( x, y ) and pY ( y )   p( x, y )
y
x
Example
• Let X be the number of Canon digital cameras sold in a week at a
certain store
• The pmf for X is
x
p(x)
0
.1
1
.2
2
.3
3
.25
4
.15
• 60% of all customers who purchase camera also purchase the longterm warranty
• Find the marginal distributions of X and Y
Definition
• Let X and Y be rv’s on a sample space S
• Continuous rv’s: The joint prob. Distribution function for (x,y) is
defined by f(x,y)
• If A is an event then, P( X , Y )  A    f ( x, y)dxdy
A
Continuous rv’s
• Usual properties of pdf’s still hold
Example:
• The front tire on a particular type of car is suppose to be filled to a
pressure of 26 psi
• Suppose the actual air pressure in EACH tire is a random variable (X
for the right side; Y for the left side) with joint pdf
f ( x, y)  K ( x 2  y 2 ) for 20  x  30 and 20  y  30
• Notice that they seem to vary jointly