Page 187 - Applied statistics and probability for engineers

Section 5-1/Two or More Random Variables 165

5-1.4 INDEPENDENCE
In some random experiments, knowledge of the values of X does not change any of the prob-
abilities associated with the values for Y. In this case, marginal probability distributions can be
used to calculate probabilities more easily.

Example 5-10 Independent Random Variables An orthopedic physician’s practice considers the number
of errors in a bill and the number of X-rays listed on the bill. There may or may not be a relation-
ship between these random variables. Let the random variables X and Y denote the number of errors and the number
of X-rays on a bill, respectively.
Assume that the joint probability distribution of X and Y is dei ned by f XY ( x, y) in Fig. 5-10(a). The marginal probability
distributions of X and Y are also shown in Fig. 5-10(a). Note that
(
(
)
f XY ( x, y) = f x f y).
X
Y
The conditional probability mass function f Y x| ( y) is shown in Fig. 5-10(b). Notice that for any x, f Y x ⏐ y ( ) = f y).
(
Y
That is, knowledge of whether or not the part meets color specii cations does not change the probability that it meets
length specii cations.
By analogy with independent events, we dei ne two random variables to be independent
)
(
(
whenever f XY ( x, y) = f x f y) for all x and y. Notice that independence implies that
X
Y
)

(
f XY ( x, y) = f x f y) for all x and y. If we ind one pair of x and y in which the equality fails,
(
X
Y
X and Y are not independent.
If two random variables are independent, then for f x) > 0,
(
X
f Y x | ( ) f XY ( x, y ) 5 f X ( ) ( y) 5 f Y ( y)
x f Y
y 5
f X ( x) f X ( x)
With similar calculations, the following equivalent statements can be shown.
Independence
For random variables X and Y, if any one of the following properties is true, the
others are also true, and X and Y are independent.
(1) f XY ( x, y) 5 f X ( ) ( y) for all and y

x
x f Y
(2) f Y x| ( ) f Y y) for all and y with f X ( )
y 5 (
x . 0
x
(3) f X yu ( x)5 f X ( x) for all and y with f Y ( y) . 0
x
(
(4) ( A, Y ∈ B)5 P X ∈ ) ( B) for any sets
A P Y ∈
P X ∈
A and B in the range of X and Y, respectively. (5-7)
f (y) 4 f Y (y) 4
Y
FIGURE 5-10 0.17 3 0.1275 0.034 0.0085 0.17 3 0.17 0.17 0.17
(a) Joint and mar-
ginal probability 0.25 2 0.1875 0.050 0.0125 0.25 2 0.25 0.25 0.25
distributions of X
and Y for Example 0.28 1 0.210 0.056 0.014 0.28 1 0.28 0.28 0.28
5-10. (b) Conditional
probability 0.3 0 0.225 0.060 0.015 0.3 0 0.30 0.30 0.30
distribution of Y 0 1 2 0 1 2
given X = for f (x) 0.7 0.2 0.05 f (x) 0.7 0.2 0.05
x
X
X
Example 5-10. (a) (b)