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162 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

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

0.002y 0.001y
3
P1Y
20002 6 10 e 11 e 2 dy
2000
e 0.002y
e 0.003y
3
6 10 ca ` b a ` bd
0.002 2000 0.003 2000
e 4 e 6
3
6 10 c d 0.05
0.002 0.003
5-3.3 Conditional Probability Distributions

Analogous to discrete random variables, we can define the conditional probability distribution
of Y given X x.

Definition
Given continuous random variables X and Y with joint probability density function
f (x, y), the conditional probability density function of Y given X x is
XY
f XY 1x, y2
f Y |x 1y2 for f X 1x2
0 (5-18)
f X 1x2

The function f Y|x (y) is used to find the probabilities of the possible values for Y given
that X x. Let R x denote 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 Y in the set R x .

Because the conditional probability density function f Y | x 1y2 is a probability density
function for all y in R , the following properties are satisfied:
x
(1) f 0 x 1 y2 0
Y
f
(2) Y 0 x 1y2 dy 1

R x
(3) P 1Y B 0 X x2 f 1y2 dy for any set B in the range of Y

Y 0 x

B
(5-19)

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 Y given that X x.
First the marginal density function of x is determined. For x
0