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St a t i s t i c s 2 7 0 - L e c t u r e 1 8
•
Will begin Chapter 5 today
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Many situations where one is interested in more than one random
variable
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Have a joint distribution for such cases
Ex a m p l e
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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
De f i n i t i o n
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Let X and Y be rv’s on a sample space S
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Discrete rv’s: The joint prob. mass function for each (x,y) is
defined by p(x,y)= P(X= x, Y= y)
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If A is an event then, P ( X , Y )
A
p ( x, y )
( x, y ) A
Di s c r e t e RV ’s
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Usual properties of pmf’s still hold
Ex a m p l e
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Let X and Y be random variables with pmf
x
y
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Observations:
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P(X= 2,Y= 2)=
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P(X> 1, Y= 1)
1
2
1
2
3
0.10 0.5 0.10
0.05 0.10 0.15
Ex a m p l e
•
Let X be the number of Canon digital cameras sold in a week at a
certain store
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The pmf for X is
x
p(x)
0
.1
1
.2
2
.3
3
.25
4
.15
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60% of all customers who purchase camera also purchase the longterm warranty
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Determine the joint pdf of X and Y
De f i n i t i o n
•
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 )
y
p ( x, y )
x
Ex a m p l e
•
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
De f i n i t i o n
•
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)
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If A is an event then, P ( X , Y )
A
f ( x, y )dxdy
A
Co n t i n u o u s r v ’s
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Usual properties of pdf’s still hold
Ex a m p l e :
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The front tire on a particular type of car is suppose to be filled to a
pressure of 26 psi
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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 )
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K (x 2
y 2 ) for 20
x
Notice that they seem to vary jointly
30 and 20
y
30
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