Applied Statistics and Probability for Engineers

168 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

The requested probability is P(X 1

 1000,X 2

 1000,X 3

 1000,X 4

 1000), which

equals the multiple integral of over the region x 1

 1000, x 2

 1000,

x 3

 1000,x 4

 1000. The joint probability density function can be written as a product of

exponential functions, and each integral is the simple integral of an exponential function.

Therefore,

Suppose that the joint probability density function of several continuous random vari-

ables is a constant, say cover a region R(and zero elsewhere). In this special case,

by property (2) of Equation 5-22. Therefore, (R). Furthermore, by property (3)

of Equation 5-22.

When the joint probability density function is constant, the probability that the random vari-

ables assume a value in the region Bis just the ratio of the volume of the region to the

volume of the region Rfor which the probability is positive.

EXAMPLE 5-24 Suppose the joint probability density function of the continuous random variables Xand Yis

constant over the region Determine the probability that

The region that receives positive probability is a circle of radius 2. Therefore, the area of

this region is 4. The area of the region is . Consequently, the requested prob-

ability is     14  2  1   4.

x^2 y^21

x^2 y^2 4. X^2 Y^2 1.

B ̈R



volume 1 B ̈R 2

volume 1 R 2



B

pfX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxpc volume 1 B ̈R 2

P 31 X 1 , X 2 ,p, Xp 2 B 4

c 1 volume





(^) 


p 



fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxpc 1 volume of region R 2  1

P 1 X 1

 1000, X 2

 1000, X 3

 1000, X 4

10002 e^1 ^2 1.5^3 0.00055

fX 1 X 2 X 3 X 4 1 x 1 , x 2 , x 3 , x 42

If the joint probability density function of continuous random variables

is the marginal probability density functionof is

(5-23)

where denotes the set of all points in the range of for which

Xixi.

Rxi X 1 , X 2 ,p, Xp

fXi 1 xi 2 

Rxi

p fX 1 X 2 pXp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxi 1 dxi 1 pdxp

fX 1 X 2 pXp 1 x 1 , x 2 p, xp 2 Xi

X 1 ,X 2 ,p, Xp

Definition

c 05 .qxd 5/13/02 1:50 PM Page 168 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: