Compound p.m.f.
Motivation:
Real problems very seldom concern a single random variable.
As soon as more than 1 variable is involved it is not suﬃcient to think of
modeling them only individually  their
joint
behavior is important.
2 variables case:
Consider the two variables:
X,Y
are two discrete variables.
The
joint probability mass function
is deFned as
p
(
x, y
):=
P
X
=
x
∩
Y
y
)
Example:
A box contains 5 unmarked PowerPC G4 processors of di±erent
speeds:
2
400 mHz
1
450 mHz
500 mHz
Select two processors out of the box (without replacement) and let
= speed of the Frst selected processor
= speed of the second selected processor
Example (Cont’d)
Summary:
²or a sample space we can draw a table of all the possible
combinations of processors. We will distinguish between processors of the
same speed by using the subscripts 1 and 2
1st processor
Ω
400
450
500

xxx
xx
xx
2nd processor
xxxx
xxxx
In total we have
5
·
4=20
possible combinations.
Since we draw at random, we assume that each of the above combinations
is equally likely. This yields the following probability mass function:
Probabilities:
mHz
0.1
0.2
1st proc.
0.0
Question 1:
What is the probability for
?
This might be important if we wanted to match the chips to assemble a
dual processor machine
Solution:
)=
(400
,
400) +
(450
450) +
(500
500)
=0
.
1+0+0
1=0
3
Marginal p.m.f.
Question 2:
X>Y
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 Spring '11
 Zhou
 Probability theory, ρ, (Cont’d), probability mass

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