Applied Statistics and Probability for Engineers

62 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

For example, in Example 3-4,

and Check that the sum of the probabilities in Example 3-4 is 1.

EXAMPLE 3-5 Let the random variable Xdenote the number of semiconductor wafers that need to be ana-

lyzed in order to detect a large particle of contamination. Assume that the probability that a

wafer contains a large particle is 0.01 and that the wafers are independent. Determine the

probability distribution of X.

Let pdenote a wafer in which a large particle is present, and let adenote a wafer in which

it is absent. The sample space of the experiment is infinite, and it can be represented as all pos-

sible sequences that start with a string of a’s and end with p. That is,

Consider a few special cases. We have Also, using the inde-

pendence assumption

A general formula is

Describing the probabilities associated with Xin terms of this formula is the simplest method

of describing the distribution of Xin this example. Clearly. The fact that the sum of

the probabilities is 1 is left as an exercise. This is an example of a geometric random variable,

and details are provided later in this chapter.

f 1 x 2  0

1 x 12 a’s

P 1 Xx 2 P 1 aa (^) p ap 2 0.99x^1 1 0.01 2 , for x1, 2, 3, (^) p
P 1 X 22 P 1 ap 2 0.99 1 0.01 2 0.0099
P 1 X 12 P 1 p 2 0.01.
s 5 p, ap, aap, aaap, aaaap, aaaaap, and so forth 6
f 142 0.0001.
f 102 0.6561, f 112 0.2916, f 122 0.0486, f 132 0.0036,
For a discrete random variable Xwith possible values , a probability
mass function is a function such that
(1)
(2)
(3) f 1 xi 2 P 1 Xxi 2 (3-1)
a
n
i 1
f 1 xi 2  1
f 1 xi 2  0
x 1 , x 2 ,p, xn
Definition
0 1 234 x
0.2916 0.0036
0.0001
0.0486
0.6561
f (x)
Figure 3-1 Probability distribution
for bits in error.
Figure 3-2 Loadings at discrete points on a
long, thin beam.
Loading
x
μ
PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 62