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 defined as
p
X,Y
(
x, y
) :=
P
(
X
=
x
∩
Y
=
y
)
Example:
A box contains 5 unmarked PowerPC G4 processors of different
speeds:
2
400 mHz
1
450 mHz
2
500 mHz
Select
two
processors
out
of
the
box
(without
replacement)
and
let
X
= speed of the first selected processor
Y
= speed of the second selected processor
1
Example (Cont’d)
Summary:
For 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
1
400
2
450
500
1
500
2
400
1

x
x
x
x
400
2
x

x
x
x
2nd processor
450
x
x

x
x
500
1
x
x
x

x
500
2
x
x
x
x

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:
2
Example (Cont’d)
Probabilities:
2nd processor
mHz
400
450
500
400
0.1
0.1
0.2
1st proc.
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 Spring '11
 Zhou
 Probability theory, ρ, (Cont’d), probability mass

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