Page 102 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 80

80 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

success can be started at any trial without changing the probability distribution of the random
variable. For example, in the transmission of bits, if 100 bits are transmitted, the probability
that the first error, after bit 100, occurs on bit 106 is the probability that the next six outcomes
5
are OOOOOE. This probability is 10.92 10.12 0.059 , which is identical to the probability
that the initial error occurs on bit 6.
The implication of using a geometric model is that the system presumably will not wear
out. The probability of an error remains constant for all transmissions. In this sense, the geo-
metric distribution is said to lack any memory. The lack of memory property will be dis-
cussed again in the context of an exponential random variable in Chapter 4.

EXAMPLE 3-23 In Example 3-20, the probability that a bit is transmitted in error is equal to 0.1. Suppose
50 bits have been transmitted. The mean number of bits until the next error is 1 0.1 10—
the same result as the mean number of bits until the first error.

3-7.2 Negative Binomial Distribution

A generalization of a geometric distribution in which the random variable is the number of
Bernoulli trials required to obtain r successes results in the negative binomial distribution.

EXAMPLE 3-24 As in Example 3-20, suppose the probability that a bit transmitted through a digital transmis-
sion channel is received in error is 0.1. Assume the transmissions are independent events, and
let the random variable X denote the number of bits transmitted until the fourth error.
Then, X has a negative binomial distribution with r 4. Probabilities involving X can be
found as follows. The P(X 10) is the probability that exactly three errors occur in the first
nine trials and then trial 10 results in the fourth error. The probability that exactly three errors
occur in the first nine trials is determined from the binomial distribution to be

9
3
a b 10.12 10.92 6
3
Because the trials are independent, the probability that exactly three errors occur in the first
9 trials and trial 10 results in the fourth error is the product of the probabilities of these two
events, namely,
9 9
3
6
4
a b 10.12 10.92 10.12 a b 10.12 10.92 6
3 3
The previous result can be generalized as follows.

Definition
In a series of Bernoulli trials (independent trials with constant probability p of a suc-
cess), let the random variable X denote the number of trials until r successes occur.
Then X is a negative binomial random variable with parameters 0 p 1 and
r 1, 2 3, p , and

x 1
p x r, r 1, r 2, p .
f 1x2 a b 11 p2 x r r (3-11)
r 1