Page 111 - Applied statistics and probability for engineers

Section 3-7/Geometric and Negative Binomial Distributions 89

Example 3-23 Lack of Memory Property In Example 3-20, the probability that a bit is transmitted in error is
equal to 0.1. Suppose that 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 Ð rst error.
.
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 Digital Channel As in Example 3-20, suppose that the probability that a bit transmitted through
a digital transmission channel is received in error is 0.1. Assume that the transmissions are inde-
pendent 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. For example,
(

P X = ) is the probability that exactly three errors occur in the irst 9 trials and then trial 10 results in the fourth error.
10

The probability that exactly three errors occur in the irst 9 trials is determined from the binomial distribution to be
⎛ 9⎞ 3 0 9) 6
0 1) ( .
⎜ ⎟ ( .
3⎠
⎝

Because the trials are independent, the probability that exactly three errors occur in the irst 9 trials and trial 10 results
in the fourth error is the product of the probabilities of these two events, namely,
⎛ 9⎞ 3 0 9) ( ⎛ 9⎞ 4 0 9) 6
6
0 1) ( .
0 1) ( .
.
⎜ ⎟ ( . 0 1) = ⎜ ⎟ ( .
3⎠
⎝
3⎠
⎝
(
In general, probabilities for X can be determined as follows. Here P X = x) implies that r −1
successes occur in the i rst x −1 trials and the rth success occurs on trial x. The probability
that r −1 successes occur in the i rst x −1 trials is obtained from the binomial distribution to be
⎛ x − ⎞ 1 r−1 x r −
⎜ ⎝ r − ⎠ 1 ⎟ p 1 ( − p)
for r ≤ x. The probability that trial x is a success is p. Because the trials are independent, these
probabilities are multiplied so that
⎛ x − ⎞ 1
−
P X = x) = ⎜ ⎝ r − ⎠ 1 ⎟ p r−1 ( − p) x r p
(
1
This leads to the following result.
Negative Binomial
Distribution In a series of Bernoulli trials (independent trials with constant probability p of a suc-
cess), the random variable X that equals the number of trials until r successes occur is a
negative binomial random variable with parameters 0 < p < 1 and r = 1 2 3, , ,…, and
⎛ x − ⎞ 1 x r
−
,
f x ( ) = ⎜ ⎝ r − ⎠ 1 ⎟ (1 − p) p r x = r r +1, r + 2,… (3-11)
Because at least r trials are required to obtain r successes, the range of X is from r to ∞. In the
special case that r = 1, a negative binomial random variable is a geometric random variable.
Selected negative binomial distributions are illustrated in Fig. 3-10.
The lack of memory property of a geometric random variable implies the following. Let
X denote the total number of trials required to obtain r successes. Let X 1 denote the num-
ber of trials required to obtain the i rst success, let X 2 denote the number of extra trials
required to obtain the second success, let X 3 denote the number of extra trials to obtain the
third success, and so forth. Then the total number of trials required to obtain r successes is
X = X + X + … + X r . Because of the lack of memory property, each of the random variables
2
1