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5-1 TWO DISCRETE RANDOM VARIABLES 147

The function f Y x (y) is used to find the probabilities of the possible values for Y given that X x.
That is, it is the probability mass function for the possible values of Y given that X x. More pre-
cisely, let R x denote the set of all points in the range of (X, Y) for which X x. The conditional
probability mass function provides the conditional probabilities for the values of Y in the set R x .

Because a conditional probability mass function f Y ƒ x 1y2 is a probability mass func-
tion for all y in R x , the following properties are satisfied:

(1) f Y ƒx 1 y2 0
(2) a f Y ƒ x 1y2 1
R x
(3) P1Y y 0 X x2 f Y ƒ x 1 y2 (5-5)

EXAMPLE 5-6 For the joint probability distribution in Fig. 5-1, f Y ƒ x 1 y2 is found by dividing each f XY (x, y) by
f (x). Here, f (x) is simply the sum of the probabilities in each column of Fig. 5-1. The func-
X
X
tion f Y ƒx 1 y2 is shown in Fig. 5-3. In Fig. 5-3, each column sums to one because it is a proba-
bility distribution.
Properties of random variables can be extended to a conditional probability distribution
of Y given X x. The usual formulas for mean and variance can be applied to a conditional
probability mass function.

Definition
Let R x denote the set of all points in the range of (X, Y) for which X x. The
conditional mean of Y given X x, denoted as E1Y 0 x2 or Y ƒ x , is

E1Y 0 x2 a y f Y ƒ x 1y2 (5-6)
R x
2
and the conditional variance of Y given X x, denoted as V1Y 0 x2 or Y ƒ x , is
2
2
2
V1Y 0 x2 a 1 y Y ƒx 2 f Y ƒ x 1 y2 a y f Y ƒx 1 y2 Y ƒ x
R x R x
y

0.410
4
0.410 0.511
3
0.154 0.383 0.640
2
Figure 5-3
0.0256 0.096 0.320 0.800
Conditional probability 1
distributions of Y given
X x, f Y ƒx 1 y2 in 0.0016 0.008 0.040 0.200 1.0
0
Example 5-6. 0 1 2 3 4 x