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5-4 MULTIPLE CONTINUOUS RANDOM VARIABLES 169

As for two random variables, a probability involving only one random variable, say, for
example P1a X b2, can be determined from the marginal probability distribution of X i
i
or from the joint probability distribution of X , X , p , X . That is,
1
2
p
P1a X b2 P1
X
, p ,
X i 1
, a X b,
1
i
i

X i 1
, p ,
X
2
p
Furthermore, E1X 2 and V1X 2, for i 1, 2, p , p, can be determined from the marginal prob-
i
i
ability distribution of X i or from the joint probability distribution of X 1 , X , p , X p as follows.
2
Mean and
Variance from
Joint E1X 2 p x f 1x , x , p , x 2 dx dx p dx
Distribution i i X 1 X 2 p X p 1 2 p 1 2 p

and (5-24)

2
2 p 1x 2 f 1x , x , p , x 2 dx dx p dx
V1X i i X i X 1 X 2 p X p 1 2 p 1 2 p

The probability distribution of a subset of variables such as X , X , p , X , k p, can be
2
k
1
obtained from the joint probability distribution of X , X , X , p , X p as follows.
3
2
1
Distribution of
a Subset of If the joint probability density function of continuous random variables X , X , p , X p
1
2
Random is f 1x , x , p , x 2, the probability density function of X , X , p , X , k p,
1
2
p
2
1
k
Variables X 1 X 2 p X p
is
f 1x , x , p , x 2
2
k
1
X 1 X 2 p X k
p f 1x , x , p , x 2 dx k 1 dx k 2 p dx p (5-25)
p
2
1
X 1 X 2 p X p
R x 1 x 2 p x k
where R denotes the set of all points in the range of X , X , p , X k for which
1
2
x 1 x 2 p x k
x , X x , p , X x .
X 1 1 2 2 k k
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.
1x , x , x , x , x 2
f X 1 X 2 X 3 X 4 X 5 1 2 3 4 5
f 1x , x , x 2
1
2
3
X 1 X 2 X 3 | x 4 x 5
f 1x , x 2
4
5
X 4 X 5
1x , x 2
0.
for f X 4 X 5 4 5