Page 189 - Applied statistics and probability for engineers

Section 5-1/Two or More Random Variables 167

5-1.5 MORE THAN TWO RANDOM VARIABLES
More than two random variables can be dei ned in a random experiment. Results for multiple
random variables are straightforward extensions of those for two random variables. A summary is
provided here.

Example 5-13 Machined Dimensions Many dimensions of a machined part are routinely measured during
production. Let the random variables, X , X , X 3 , and X 4 denote the lengths of four dimensions of
1
2
a part. Then at least four random variables are of interest in this study.
The joint probability distribution of random variables X , X , X ,… , X p can be specii ed with
3
1
2
a method to calculate the probability that X , X , X ,… , X p assume a value in any region Rof
1
2
3
p-dimensional space. For continuous random variables, a joint probability density function
f X X 2 … ( x , x ,… , x p) is used to determine the probability that X , X , X ,… , X p) ∈ R by the
1 (
1
2
multiple integral of f X X 2 … ( x , x ,… , x p) over the region R. 2 3
1
X p
1 X p 1 2
Joint Probability
Density Function A joint probability density function for the continuous random variables X , X , X ,
…, X , denoted as f X X 2 … ( x , x ,… , x , satisies the following properties:
p)
1
2
3

p
2
1
X p
1
(1) f X X 2 … ( x , x ,… , x p) ≥ 0
1 X p 1 2
∞ ∞ ∞
(2) ∫ ∫ ??? ∫ f X X 2 … ( x , x ,… , x p) dx dx 2 … dx p = 1
1
1
2
X p
1
2 ∞ 2 ∞ 2 ∞
(3) For any region B of p-dimensional space,
1 (
P X , X ,… , X p) ∈ ⎤ 5 B ∫ ∫ f X X 2 ??? X p ( x , x ,… , x p) dx dx 2 … dx p (5-8)
⎡
B
⎦
⎣
1
2
1
2
1
( x , x ,…

Typically, f X X 2 ??? X p 1 2 , x p) is deined over all of p-dimensional space by assuming that
1
( x , x ,… , x p) = ( x , x ,… , x p) is not specii ed.
f X X 2 ??? 1 2 0 for all points for which f X X 2 ??? 1 2
1 X p 1 X p
Example 5-14
1
2
3
Component Lifetimes In an electronic assembly, let the random variables X , X , X , X 4 denote
the lifetime of four components, respectively, in hours. Suppose that the joint probability density
function of these variables is
( 2 12 e 2 ? x 1 0 002 x 2 0 0015 3 0 0032 ? 4 x
2 ?
0 001 2 ?
x
3
9
3
2
1
f X X X X 4 x , x , x , x 4) 5 3 10
1 2 3
for 1 x ≥ 0, 2 x ≥ 0, 3 x ≥ 0, 4 x ≥ 0
What is the probability that the device operates for more than 1000 hours without any failures? The requested prob-
1 (
ability is P X >1000 , X 2 >1000 , X 3 >1000 , X 4 >1000), which equals the multiple integral of f X X X X 4 ( x , x , x , x 4)
3
1
2
1 2 3
over the region x >1000 , x 2 >1000 , x 3 >1000 , x 4 >1000. The joint probability density function can be written as a
1
product of exponential functions, and each integral is the simple integral of an exponential function. Therefore,
(
P X >1000 , X >1000 , X >1000 , X >1000)
2
3
1
4
−
−
−
.
= e − 1 2 1 5 3 = .
0 00055
Suppose that the joint probability density function of several continuous random variables
is a constant c over a region R (and zero elsewhere). In this special case,
∫ ∫ … f X X 2 …X p (x , x , … ) dx dx 2 … dx p = × (volume of region R ) = 11
∫
c
1
1
2
, x p
1
R