Applied Statistics and Probability for Engineers

5-2 MULTIPLE DISCRETE RANDOM VARIABLES 153

That is, is the sum of the probabilities over all points in the

range of for which X 1  x 1 , X 2  x 2 , , and Xk xk. An example is

presented in the next section. Any krandom 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

for The conditional joint probability mass function of X 1 , X 2 , X 3 given X 4 , X 5

provides the conditional probabilities at all points in the range of X 1 , X 2 , X 3 , X 4 , X 5 for which

X 4 x 4 and X 5 x 5.

The concept of independence can be extended to multiple discrete random variables.

fX 4 X 51 x 4 , x 52

 0.

fX 1 X 2 X 3 0 x 4 x 51 x 1 , x 2 , x 32 

fX 1 X 2 X 3 X 4 X 51 x 1 , x 2 , x 3 , x 4 , x 52

fX 4 X 51 x 4 , x 52

X 1 , X 2 , X 3 ,p, Xp p

P 1 X 1 x 1 , X 2 x 2 ,p, Xkxk 2

If are discrete random variables with joint probability mass function

the joint probability mass functionof X 1 , X 2 ,, Xk,

k p, is

(5-11)

where denotes the set of all points in the range of for which

X 1 x 1 , X 2 x 2 ,p, Xkxk.

Rx 1 x 2 pxk X 1 , X 2 ,p, Xp

 a

Rx 1 x 2 pxk

P 1 X 1 x 1 , X 2 x 2 ,p, Xkxk 2

fX 1 X 2 pXk 1 x 1 , x 2 ,p, xk 2 P 1 X 1 x 1 , X 2 x 2 ,p, Xkxk 2

fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 , p

X 1 , X 2 , X 3 ,p, Xp

Distribution of

a Subset of

Random

Variables

Discrete variables are independentif and only if

(5-12)

for allx 1 , x 2 ,p, xp.

fX 1 X 2 pXp 1 x 1 , x 2 ,p, xp 2 fX 1 1 x 12 fX 2 1 x 22 pfXp 1 xp 2

X 1 , X 2 ,p, Xp

Definition

Similar to the result for bivariate random variables, independence implies that Equation 5-12

holds for all If we find one point for which the equality fails,

are not independent. It can be shown that if are independent,

for anysets A 1 , A 2 ,, p Ap.

P 1 X 1 A 1 , X 2 A 2 ,p, XpAp 2 P 1 X 1 A 12 P 1 X 2 A 22 p P 1 XpAp 2

X 1 , X 2 ,p, Xp

x 1 , x 2 ,p, xp. X 1 , X 2 ,p, Xp

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