Applied Statistics and Probability for Engineers

5-4 MULTIPLE CONTINUOUS RANDOM VARIABLES 167

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 1 , X 2 , X 3 , and X 4 denote the lengths of four dimensions of a part. Then, at least

four random variables are of interest in this study.

The joint probability distribution of continuous random variables,

can be specified by providing a method of calculating the probability that

assume a value in a region Rof p-dimensional space. A joint probability density function

is used to determine the probability that assume a

value in a region Rby the multiple integral of fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 over the region R.

fX 1 X 2 pXp 1 x 1 , x 2 ,p, xp 2 1 X 1 , X 2 , X 3 ,p, Xp 2

X 1 , X 2 , X 3 ,p, Xp

X 1 , X 2 , X 3 p, Xp

(c) Why is the joint probability distribution not needed to

answer the previous questions?

5-54. The conditional probability distribution of Ygiven

Xxis for y 0 and the marginal probabil-

ity distribution of Xis a continuous uniform distribution over

0 to 10.

(a) Graph for y 0 for several values of x.

Determine

(b) (c)

(d) (e)

(f) fY 1 y 2

E 1 YƒXx 2 fXY 1 x, y 2

P 1 Y 2 ƒX 22 E 1 YƒX 22

fY ƒ X 1 y 2 xexy

fY ƒ x 1 y 2 xexy

A joint probability density functionfor the continuous random variables

denoted as satisfies the following properties:

(1)

(2)

(3) For any region Bof p-dimensional space

P 31 X 1 , X 2 ,p, Xp 2 B 4  (5-22)

B

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









p 



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

fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2  0

X 3 ,p, Xp, fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 ,

X 1 , X 2 ,

Definition

Typically, is defined over all of p-dimensional space by assum-

ing that for all points for which is not

specified.

EXAMPLE 5-23 In an electronic assembly, let the random variables denote the lifetimes of four

components in hours. Suppose that the joint probability density function of these variables is

What is the probability that the device operates for more than 1000 hours without any failures?

for x 1 0, x 2 0, x 3 0, x 4  0

fX 1 X 2 X 3 X 4 1 x 1 , x 2 , x 3 , x 42  910 ^2 e0.001x^1 0.002x^2 0.0015x^3 0.003x^4

X 1 , X 2 , X 3 , X 4

fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2  0 fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2

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

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