Applied Statistics and Probability for Engineers

164 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

EXAMPLE 5-18 For the random variables that denote times in Example 5-15, determine the conditional mean

for Ygiven that x1500.

The conditional probability density function for Ywas determined in Example 5-17.

Because fY 1500 (y) is nonzero for y 1500,

Integrate by parts as follows:

With the constant 0.002e^3 reapplied

5-3.4 Independence

The definition of independence for continuous random variables is similar to the definition for

discrete random variables. If fXY(x, y)fX(x) fY(y) for all xand y, Xand Yare independent.

Independence implies that fXY(x, y)fX(x) fY(y) for all xand y. If we find one pair of xand y

in which the equality fails, Xand Yare not independent.

E 1 Y^0 x 15002  2000



1500

0.002

e^3 

e^3

1 0.002 21 0.002 2



e^3

0.002

120002



1500

0.002

e^3 a

e0.002y

1 0.002 21 0.002 2

`

1500

b

(^) 
1500
ye0.002y dyy
e0.002y
0.002
`
1500
 (^) 
1500
a
e0.002y
0.002
b dy

E 1 Y  x 15002  

1500

y 1 0.002e0.002^115002 0.002y 2 dy0.002e^3 

1500

ye0.002y dy

For continuous random variables Xand Y, if any one of the following properties is

true, the others are also true, and Xand Yare said to be independent.

(1)

(2)

(3)

(4) for any sets Aand Bin the range

of Xand Y, respectively. (5-21)

P 1 XA, YB 2 P 1 XA 2 P 1 YB 2

fX 0 y 1 x 2 fX 1 x 2 for all x and y with fY 1 y 2

0

fY 0 x 1 y 2 fY 1 y 2 for all x and y with fX 1 x 2

0

fXY 1 x, y 2 fX 1 x 2 fY 1 y 2 for all x and y

Definition

EXAMPLE 5-19 For the joint distribution of times in Example 5-15, the

Marginal distribution of Ywas determined in Example 5-16.

Conditional distribution of Ygiven Xxwas determined in Example 5-17.

Because the marginal and conditional probability densities are not the same for all values of

x, property (2) of Equation 5-20 implies that the random variables are not independent. The

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