Page 98 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 76

76 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

18
Now a b 18! 32! 16!4 181172 2 153. Therefore,
2
16
2
P1X 22 15310.12 10.92 0.284
Determine the probability that at least four samples contain the pollutant. The requested
probability is
18 18
x
P1X 42 a a b 10.12 10.92 18 x
x 4 x
However, it is easier to use the complementary event,

3 18
x
P1X 42 1 P1X 42 1 a a b 10.12 10.92 18 x
x 0 x
1 30.150 0.300 0.284 0.1684 0.098

Determine the probability that 3 X 7. Now

6 18
x
P13 X 72 a a b 10.12 10.92 18 x
x 3 x
0.168 0.070 0.022 0.005
0.265

The mean and variance of a binomial random variable depend only on the parameters p
and n. Formulas can be developed from moment generating functions, and details are pro-
vided in Section 5-8, part of the CD material for Chapter 5. The results are simply stated here.

Definition
If X is a binomial random variable with parameters p and n,

2
E1X2 np and V1X2 np11 p2 (3-8)

EXAMPLE 3-19 For the number of transmitted bits received in error in Example 3-16, n 4 and p 0.1, so

E1X2 410.12 0.4 and V1X2 410.1210.92 0.36

and these results match those obtained from a direct calculation in Example 3-9.

EXERCISES FOR SECTION 3-6

3-55. For each scenario described below, state whether or transducers in a sample of size 30 selected at random from
not the binomial distribution is a reasonable model for the ran- the process.
dom variable and why. State any assumptions you make. (b) From a batch of 50 temperature transducers, a sample of
(a) A production process produces thousands of temperature size 30 is selected without replacement. Let X denote the
transducers. Let X denote the number of nonconforming number of nonconforming transducers in the sample.