Fundamentals of Probability and Statistics for Engineers

7.15 If X 1 ,X 2 ,...,Xn are independent random variables, all having distribution N(m,^2 ),
determine the conditions that must be imposed on c 1 ,c 2 ,...,cn such that the sum
Yc 1 X 1 c 2 X 2 cnXn

is also N(m,^2 ). Can all cs be positive?

7.16 Let U be the standardized normal random variable, and define X U. Then, X
is called the fo ld ed standardized normal random variable. Determine fX (x).

7.17 The Cauchy distribution has the form

(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:

Yarctan

X 2

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 mX 1
and^2 X 09.
(a) Determine the probability P(X > 1 2).
(b) If stress Y in a structural member is related to the strain by Y a bX, with
b > 0, determine fY (y) and mY.

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.

Some Important Continuous Distributions 241

ˆ‡ ‡‡





ˆjj

fX…x†ˆ

1

… 1 ‡x^2 †

; 1<x< 1 :



ˆ



ˆ

 ˆ 0 :

:

ˆ‡