Page 205 - Fundamentals of Probability and Statistics for Engineers

188 Fundamentals of Probability and Statistics for Engineers

6.26 Each air traffic controller at an airport is given the responsibility of monitoring at
most 20 takeoffs and landings per hour. During a given period, the average rate of
takeoffs and landings is 1 every 2 minutes. Assuming Poisson arrivals and depar-
tures, determine the probability that 2 controllers will be needed in this time
period.
6.27 The number of vehicles crossing a certain point on a highway during a unit time
period has a Poisson distribution with parameter . A traffic counter is used to
record this number but, owing to limited capacity, it registers the maximum
number of 30 whenever the count equals or exceeds 30. Determine the pmf of Y
if Y is the number of vehicles recorded by the counter.
6.28 As an application of the Poisson approximation to the binomial distribution,
estimate the probability that in a class of 200 students exactly 20 will have birth-
days on any given day.
6.29 A book of 500 pages contains on average 1 misprint per page. Estimate the
probability that:
(a) A given page contains at least 1 misprint.
(b) At least 3 pages will contain at least 1 misprint.
6.30 Earthquakes are registered at an average frequency of 250 per year in a given
region. Suppose that the probability is 0.09 that any earthquake will have a
magnitude greater than 5 on the Richter scale. Assuming independent occurrences
of earthquakes, determine the pmf of X, the number of earthquakes greater than 5
on the Richter scale per year.
6.31 Let X be the number of accidents in which a driver is involved in t years. In
proposing a distribution for X, the ‘accident likelihood’ varies from driver to
driver and is considered as a random variable. Suppose that the conditional pmf
p X (xj ) is given by the Poisson distribution,
k t
t† e
p X kj †ˆ ; k ˆ 0; 1; 2; ... ;
k!
and suppose that the probability density function (pdf) of is of the form
(a, b > 0)
8 a 1
a a
> a =b
< e ; for 0,
f †ˆ b a† b

>
0; elsewhere,
:
where (a) is the gamma function, defined by
Z 1
a†ˆ x a 1 x
e dx:
0
Show that the pmf of X has a negative binomial distribution in the form
a
a ‡ k† a bt k
p k†ˆ ; k ˆ 0; 1; 2; .. . :
X
k! a† a ‡ bt a ‡ bt

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