Applied Statistics and Probability for Engineers

5-1 TWO DISCRETE RANDOM VARIABLES 147

The function fYx(y) is used to find the probabilities of the possible values for Ygiven that Xx.

That is, it is the probability mass function for the possible values of Ygiven that Xx. More pre-

cisely, let Rxdenote the set of all points in the range of (X, Y) for which Xx. The conditional

probability mass function provides the conditional probabilities for the values of Yin the set Rx.

EXAMPLE 5-6 For the joint probability distribution in Fig. 5-1, is found by dividing each fXY(x,y) by

fX(x). Here, fX(x) is simply the sum of the probabilities in each column of Fig. 5-1. The func-

tion 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 Ygiven X x. The usual formulas for mean and variance can be applied to a conditional

probability mass function.

fYƒx 1 y 2

fYƒx 1 y 2

Because a conditional probability mass function is a probability mass func-

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

(1)

(2)

(3) P 1 Yy 0 Xx 2 fYƒx 1 y (^2) (5-5)
a
Rx
fYƒx 1 y 2  1
fYƒx 1 y 2  0
fYƒx 1 y 2
x
y
0.008 0.040 0.200 1.0
0.096 0.320 0.800
0.383 0.640
0.511
0 1 2 3 4
0
1
2
3
4
0.0016
0.0256
0.154
0.410
0.410
Figure 5-3
Conditional probability
distributions of Ygiven
Xx,in
Example 5-6.
fY ƒx 1 y 2
Let Rxdenote the set of all points in the range of (X, Y) for which Xx. The
conditional meanof Ygiven Xx, denoted as or , is
(5-6)
and the conditional variance of Ygiven Xx, denoted as or , is
V 1 Y 0 x 2  a
Rx
1 yYƒx 22 fYƒx 1 y 2  a
Rx
y^2 fYƒx 1 y 2 ^2 Yƒx
V 1 Y 0 x 2 ^2 Yƒx
E 1 Y 0 x 2  a
Rx
y fYƒx 1 y 2
E 1 Y 0 x 2 Yƒx
Definition
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