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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