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5-2 MULTIPLE DISCRETE RANDOM VARIABLES 153

Distribution of
a Subset of If X , X , X , p , X p are discrete random variables with joint probability mass function
1
3
2
Random
f 1x , x , p , x 2, the joint probability mass function of X , X , p , X ,
k
2
1
p
1
2
Variables X 1 X 2 p X p
k p, is
f 1x , x , p , x 2 P1X x , X x , p , X x 2
k
1
2
1
1
k
2
k
2
X 1 X 2 p X k
a P1X x , X x , p , X x 2 (5-11)
2
k
2
1
1
k
R x 1 x 2 p x k
where R denotes the set of all points in the range of X , X , p , X p for which
1
2
x 1 x 2 p x k
x , X x , p , X x .
X 1 1 2 2 k k
That is, P 1X x , X x , p , X x 2 is the sum of the probabilities over all points in the
2
1
2
1
k
k
range of X 1 , X 2 , X 3 , p , X p for which X 1 x 1 , X 2 x 2 , p , and X k x k . An example is
presented in the next section. Any k random variables can be used in the definition. The first k
simplifies the notation.
Conditional Probability Distributions
Conditional probability distributions can be developed for multiple discrete random variables
by an extension of the ideas used for two discrete random variables. For example, the condi-
tional joint probability mass function of X 1 , X 2 , X 3 given X 4 , X 5 is
f X 1 X 2 X 3 X 4 X 5 1x , x , x , x , x 2
4
5
2
3
1
f 1x , x , x 2
1
2
3
X 1 X 2 X 3 0 x 4 x 5
f 1x , x 2
5
4
X 4 X 5
1x , x 2
0. The conditional joint probability mass function of X , X , X given X , X
for f X 4 X 5 4 5 1 2 3 4 5
provides the conditional probabilities at all points in the range of X , X , X , X , X for which
3
2
1
5
4
X x and X x .
4
5
4
5
The concept of independence can be extended to multiple discrete random variables.
Definition
Discrete variables X , X , p , X p are independent if and only if
2
1
f 1x , x , p , x 2 f 1x 2 f 1x 2 p f 1x 2 (5-12)
1
2
2
1
p
p
X 1 X 2 p X p X 1 X 2 X p
, x , p , x .
for all x 1 2 p
Similar to the result for bivariate random variables, independence implies that Equation 5-12
holds for all x , x , p , x . If we find one point for which the equality fails, X , X , p , X p
1
2
1
p
2
are not independent. It can be shown that if X , X , p , X p are independent,
1
2
P1X A , X A , p , X A 2 P1X A 2 P 1X A 2 p P1X A 2
1
p
1
p
2
p
p
2
1
2
2
1

for any sets A , A , p , A .
2
1
p