Page 258 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 241
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7.15 If X 1 , X 2 , . . ., X n are independent random variables, all having distribution N(m, ),
determine the conditions that must be imposed on c 1 , c 2 , . . ., c n such that the sum
Y ˆ c 1 X 1 ‡ c 2 X 2 ‡ ‡ c n X n

is also N(m, 2 ). Can all cs be positive?
U . Then, X
7.16 Let U be the standardized normal random variable, and define X ˆj j
is called the folded standardized normal random variable. Determine f (x).
X
7.17 The Cauchy distribution has the form
1
f x†ˆ ; 1 < x < 1:
X
1 ‡ x †
2
(a) Show that it arises from the ratio X 1 /X 2 , where X 1 and X 2 are independent and
distributed as N(0, 2 ).
(b) Show that the moments of X do not exist.
7.18 Let X 1 and X 2 be independent normal random variables, both with mean 0 and
standard deviation 1. Prove that:

X 2
Y ˆ arctan
X 1
is uniformly distributed from to .
7.19 Verify Equations (7.48) for the lognormal distribution.
7.20 The lognormal distribution is found to be a good model for strains in structural
members caused by wind loads. Let the strain be represented by X, with m X ˆ 1
2
and ˆ : 0 09.
X
(a) Determine the probability P(X > 1 2).
:
(b) If stress Y in a structural member is related to the strain by Y ˆ a ‡ bX , with
b > 0, determine f (y) and m Y .
Y
7.21 Arrivals at a rural entrance booth to the New York State Thruway are considered
to be Poisson distributed with a mean arrival rate of 20 vehicles per hour. The time
to process an arrival is approximately exponentially distributed with a mean time of
one min.
(a) What percentage of the time is the tollbooth operator free to work on opera-
tional reports?
(b) How many cars are expected to be waiting to be processed, on average, per hour?
(c) What is the average time a driver waits in line before paying the toll?
(d) Whenever the average number of waiting vehicles reaches 5, a second tollbooth
will be opened. How much will the average hourly rate of arrivals have to
increase to require the addition of a second operator?
7.22 The life of a power transmission tower is exponentially distributed, with mean life
25 years. If three towers, operated independently, are being erected at the same
time, what is the probability that at least 2 will still stand after 35 years?
7.23 For a gamma-distributed random variable, show that:
(a) Its mean and variance are those given by Equation (7.57).
(b) It has a positive skewness.

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