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152 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

Definition
If X , X , X , p , X p are discrete random variables with joint probability mass func-
1
3
2
tion f 1x , x , p , x 2, the marginal probability mass function of any X is
2
p
i
1
X 1 X 2 p X p
f 1x 2 P1X x 2 a f 1x , x , p , x 2 (5-9)

X i i i i X 1 X 2 p X p 1 2 p
R x i
denotes the set of points in the range of (X , X , p , X ) for which X x .
where R x i 1 2 p i i
EXAMPLE 5-11 Points that have positive probability in the joint probability distribution of three random variables
X , X , X are shown in Fig. 5-5. The range is the nonnegative integers with x x x 3.
3
3
2
2
1
1
The marginal probability distribution of X is found as follows.
2
12, 0, 12
P 1X 2 02 f X 1 X 2 X 3 13, 0, 02 f X 1 X 2 X 3 10, 0, 32 f X 1 X 2 X 3 11, 0, 22 f X 1 X 2 X 3
P 1X 12 f X 1 X 2 X 3 12, 1, 02 f X 1 X 2 X 3 10, 1, 22 f X 1 X 2 X 3 11, 1, 12
2
P 1X 22 f X 1 X 2 X 3 11, 2, 02 f X 1 X 2 X 3 10, 2, 12
2
P 1X 32 f X 1 X 2 X 3 10, 3, 02
2
Furthermore, E(X ) and V(X ) for i 1, 2, p , p can be determined from the marginal
i
i
, X , p , X as
probability distribution of X or from the joint probability distribution of X 1 2 p
i
follows.
Mean and
Variance from E1X 2 a x f 1x , x , p , x 2

Joint i R i X 1 X 2 p X p 1 2 p
Distribution
and
2
V1X 2 a 1x 2 f X 1 X 2 p X p 1x , x , p , x 2 (5-10)
p
i

1
2

i
X i
R
where R is the set of all points in the range of X , X , p , X .
1
2
p
With several random variables, we might be interested in the probability distribution of some
subset of the collection of variables. The probability distribution of X , X , p , X , k p can
1
2
k
be obtained from the joint probability distribution of X , X , p , X p as follows.
1
2
x 3
3 x 2
3
2
2
Figure 5-5 Joint 1 1
probability distribution
0
of X 1 , X 2 , and X 3 . 0 1 2 3 x 1