Applied Statistics and Probability for Engineers

162 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

We have obtained the marginal probability density function of Y. Now,

5-3.3 Conditional Probability Distributions

Analogous to discrete random variables, we can define the conditional probability distribution

of Ygiven Xx.

 610 ^3 c

e^4

0.002



e^6

0.003

d0.05

 610 ^3 ca

e0.002y

0.002

`

2000

ba

e0.003y

0.003

`

2000

bd

P 1 Y

20002  6 10 ^3 

2000

e0.002y 11 e0.001y 2 dy

Given continuous random variables Xand Ywith joint probability density function

fXY(x, y), the conditional probability density function of Ygiven Xxis

fY |x 1 y 2  (5-18)

fXY 1 x, y 2

fX 1 x 2

for fX 1 x 2

0

Definition

The function fY|x(y) is used to find the probabilities of the possible values for Ygiven

that X x. Let Rxdenote the set of all points in the range of (X, Y) for which X x. The

conditional probability density function provides the conditional probabilities for the values

of Yin the set Rx.

Because the conditional probability density function is a probability density

function for all yin Rx, the following properties are satisfied:

(1)

(2)

(3)

(5-19)

P 1 YB 0 Xx 2 

B

fY 0 x 1 y 2 dy for any set B in the range of Y



Rx

fY 0 x 1 y 2 dy 1

fY (^0) x 1 y 2  0
fY | x 1 y 2
It is important to state the region in which a joint, marginal, or conditional probability
density function is not zero. The following example illustrates this.
EXAMPLE 5-17 For the random variables that denote times in Example 5-15, determine the conditional prob-
ability density function for Ygiven that Xx.
First the marginal density function of xis determined. For x
0
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