Applied Statistics and Probability for Engineers

EXAMPLE 3-21 The probability that a wafer contains a large particle of contamination is 0.01. If it is assumed

that the wafers are independent, what is the probability that exactly 125 wafers need to be

analyzed before a large particle is detected?

Let Xdenote the number of samples analyzed until a large particle is detected. Then Xis

a geometric random variable with p0.01. The requested probability is

The derivation of the mean and variance of a geometric random variable is left as an exercise.

Note that g^ k 1 k 11 p 2 k^1 pcan be shown to equal (^1) p. The results are as follows.
P 1 X 1252  1 0.99 2124 0.010.0029
012345678910
0
0.2
0.6
0.8
1.0
x
f (x)
11 1213 1415 16 1718 1920
0.4
p
0.1
0.9
Figure 3-9 Geometric
distributions for
selected values of the
parameter p.
EXAMPLE 3-22 Consider the transmission of bits in Example 3-20. Here, p0.1. The mean number of
transmissions until the first error is 10.1 10. The standard deviation of the number
of transmissions before the first error is
Lack of Memory Property
A geometric random variable has been defined as the number of trials until the first success.
However, because the trials are independent, the count of the number of trials until the next
 311 0.1 (^2) 0.1^241 ^2 9.49
If Xis a geometric random variable with parameter p,
E 1 X 2  (^1) p and 2 V 1 X 2  11 p (^2) p^2 (3-10)
3-7 GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS 79
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