Applied Statistics and Probability for Engineers

5-4 MULTIPLE CONTINUOUS RANDOM VARIABLES 169

As for two random variables, a probability involving only one random variable, say, for

example can be determined from the marginal probability distribution of

or from the joint probability distribution of That is,

Furthermore, and for can be determined from the marginal prob-

ability distribution of Xior from the joint probability distribution of X 1 , X 2 ,p, Xpas follows.

E 1 Xi 2 V 1 Xi 2 , i1, 2,p, p,



Xi 1 

,p, 

Xp

 2

P 1 aXib 2 P 1 

X 1 

,p, 

Xi 1 

, aXib,

X 1 , X 2 ,p, Xp.

P 1 aXi b 2 , Xi

and (5-24)

V 1 Xi 2  



(^) 


p 



1 xiXi 22 fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxp

E 1 Xi 2 



(^) 


p 



xi fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxp

Mean and

Variance from

Joint

Distribution

If the joint probability density function of continuous random variables X 1 ,X 2 ,,Xp

is the probability density functionof X 1 , X 2 ,, Xk, k p,

is

(5-25)

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, Xk



Rx 1 x 2 pxk

pfX 1 X 2 pXp 1 x 1 , x 2 ,p, xp 2 dxk 1 dxk 2 pdxp

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

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

p

Distribution of

a Subset of

Random

Variables

The probability distribution of a subset of variables such as can be

obtained from the joint probability distribution of X 1 , X 2 , X 3 ,p, Xpas follows.

X 1 , X 2 ,p, Xk, kp,

Conditional Probability Distribution

Conditional probability distributions can be developed for multiple continuous random vari-

ables by an extension of the ideas used for two continuous random variables.

for fX 4 X 5 1 x 4 , x 52

 0.

fX 1 X 2 X 3 | 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

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