Applied Statistics and Probability for Engineers

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

are OOOOOE. This probability is , 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 10.1 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 rsuccesses 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 Xdenote the number of bits transmitted until the fourtherror.

Then, Xhas a negative binomial distribution with r4. Probabilities involving Xcan be

found as follows. The P(X10) 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

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,

The previous result can be generalized as follows.

a

9

3

b 1 0.1 231 0.9 261 0.1 2 a

9

3

b 1 0.1 241 0.9 26

a

9

3

b 1 0.1 231 0.9 26

1 0.9 251 0.1 2 0.059

In a series of Bernoulli trials (independent trials with constant probability pof a suc-

cess), let the random variable Xdenote the number of trials until rsuccesses occur.

Then Xis a negative binomial random variablewith parameters and

r1, 2 3, p, and

f 1 x 2 a (3-11)

x  1

r  1

b 11 p 2 xrpr xr, r 1, r 2,p.

0 p 1

Definition

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