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5-4 MULTIPLE CONTINUOUS RANDOM VARIABLES 167

(c) Why is the joint probability distribution not needed to (a) Graph f Y ƒ X 1 y2 xe xy for y
0 for several values of x.
answer the previous questions? Determine
5-54. The conditional probability distribution of Y given (b) P1Y 2 ƒ X 22 (c) E1Y ƒ X 22
X x is f Y ƒ x 1 y2 xe xy for y
0 and the marginal probabil- (d) E1Y ƒ X x2 (e) f XY 1x, y2
ity distribution of X is a continuous uniform distribution over (f) f Y 1 y2
0 to 10.

5-4 MULTIPLE CONTINUOUS RANDOM VARIABLES

As for discrete random variables, in some cases, more than two continuous random variables
are defined in a random experiment.

EXAMPLE 5-22 Many dimensions of a machined part are routinely measured during production. Let the ran-
dom variables, X , X , X , and X denote the lengths of four dimensions of a part. Then, at least
1
4
3
2
four random variables are of interest in this study.
The joint probability distribution of continuous random variables, X , X , X p , X p
1
2
3
can be specified by providing a method of calculating the probability that X , X , X , p , X p
2
1
3
assume a value in a region R of p-dimensional space. A joint probability density function
f 1x , x , p , x 2 is used to determine the probability that 1X , X , X , p , X 2 assume a
p
1
1
2
3
p
2
X 1 X 2 p X p
value in a region R by the multiple integral of f 1x , x , p , x 2 over the region R.
p
2
1
X 1 X 2 p X p
Definition
A joint probability density function for the continuous random variables X , X ,
1
2
X , p , X , denoted as f X 1 X 2 p X p 1x , x , p , x 2, satisfies the following properties:
2
p
3
1
p
(1) f X 1 X 2 p X p 1x , x , p , x 2 0
2
p
1

(2) p f X 1 X 2 p X p 1x , x , p , x 2 dx dx p dx 1
1
p
1
2
2
p

(3) For any region B of p-dimensional space
, X , p , X 2 B4 p f 1x , x , p , x 2 dx dx p dx (5-22)
P31X 1 2 p X 1 X 2 p X p 1 2 p 1 2 p
B
Typically, f 1x , x , p , x 2 is defined over all of p-dimensional space by assum-
p
1
2
X 1 X 2 p X p
ing that f 1x , x , p , x 2 0 for all points for which f 1x , x , p , x 2 is not
1
2
p
2
1
p
X 1 X 2 p X p X 1 X 2 p X p
specified.
EXAMPLE 5-23 In an electronic assembly, let the random variables X , X , X , X 4 denote the lifetimes of four
1
3
2
components in hours. Suppose that the joint probability density function of these variables is
f 1x , x , x , x 2 9 10 e
2 0.001x 1 0.002x 2 0.0015x 3 0.003x 4
3
4
1
2
X 1 X 2 X 3 X 4
for x 0, x 0, x 0, x 0
4
3
1
2
What is the probability that the device operates for more than 1000 hours without any failures?