Page 115 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 93

3-9 POISSON DISTRIBUTION 93

Because this sum is tedious to compute, many computer programs calculate cumulative
Poisson probabilities. From one such program, P1X 122 0.791 .

The derivation of the mean and variance of a Poisson random variable is left as an exer-
cise. The results are as follows.

If X is a Poisson random variable with parameter , then

2
E1X2 and V1X2 (3-16)

The mean and variance of a Poisson random variable are equal. For example, if particle counts
follow a Poisson distribution with a mean of 25 particles per square centimeter, the variance
is also 25 and the standard deviation of the counts is 5 per square centimeter. Consequently,
information on the variability is very easily obtained. Conversely, if the variance of count data
is much greater than the mean of the same data, the Poisson distribution is not a good model
for the distribution of the random variable.

EXERCISES FOR SECTION 3-9
3-97. Suppose X has a Poisson distribution with a mean of (c) What is the probability that there are no flaws in 20 square
4. Determine the following probabilities: meters of cloth?
(a) P1X 02 (b) P1X 22 (d) What is the probability that there are at least two flaws in
(c) P1X 42 (d) P1X 82 10 square meters of cloth?
3-98. Suppose X has a Poisson distribution with a mean of 3-102. When a computer disk manufacturer tests a disk, it
0.4. Determine the following probabilities: writes to the disk and then tests it using a certifier. The certi-
(a) P1X 02 (b) P1X 22 fier counts the number of missing pulses or errors. The num-
(c) P1X 42 (d) P1X 82 ber of errors on a test area on a disk has a Poisson distribution
3-99. Suppose that the number of customers that enter with 0.2.
a bank in an hour is a Poisson random variable, and sup- (a) What is the expected number of errors per test area?
pose that P1X 02 0.05. Determine the mean and (b) What percentage of test areas have two or fewer errors?
variance of X. 3-103. The number of cracks in a section of interstate high-
3-100. The number of telephone calls that arrive at a phone way that are significant enough to require repair is assumed
exchange is often modeled as a Poisson random variable. to follow a Poisson distribution with a mean of two cracks
Assume that on the average there are 10 calls per hour. per mile.
(a) What is the probability that there are exactly 5 calls in one (a) What is the probability that there are no cracks that require
hour? repair in 5 miles of highway?
(b) What is the probability that there are 3 or less calls in one (b) What is the probability that at least one crack requires
hour? repair in 1 2 mile of highway?
(c) What is the probability that there are exactly 15 calls in (c) If the number of cracks is related to the vehicle load on
two hours? the highway and some sections of the highway have a
(d) What is the probability that there are exactly 5 calls in heavy load of vehicles whereas other sections carry
30 minutes? a light load, how do you feel about the assumption of a
3-101. The number of flaws in bolts of cloth in textile man- Poisson distribution for the number of cracks that
ufacturing is assumed to be Poisson distributed with a mean of require repair?
0.1 flaw per square meter. 3-104. The number of failures for a cytogenics machine
(a) What is the probability that there are two flaws in 1 square from contamination in biological samples is a Poisson random
meter of cloth? variable with a mean of 0.01 per 100 samples.
(b) What is the probability that there is one flaw in 10 square (a) If the lab usually processes 500 samples per day, what is
meters of cloth? the expected number of failures per day?