Page 260 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 243

7.28 Let random variable X be 2 -distributed with parameter . Show that the limiting
n
distribution of
X n
2n† 1=2
as n !1 is N(0, 1).
7.29 Let X 1 , X 2 ,..., X n be independent random variables with common PDF F X (x)
and pdf f (x). Equations (7.89) and (7.91) give, respectively, the PDFs of their
X
maximum and minimum values. Let X (j) be the random variable denoting the
jth-smallest value of X 1 , X 2 ,... , X n . Show that the PDF of X (j) has the form
n
n
X
k
x†ˆ ‰F X x†Š ‰1 F X x†Š n k ; j ˆ 1; 2; ... ; n:
F X j†
k
kˆj
7.30 Ten points are distributed uniformly and independently in interval (0, 1). Find:
(a) The probability that the point lying farthest to the right is to the left of 3/4.
(b) The probability that the point lying next farthest to the right is to the right of 1/2.
7.31 Let the number of arrivals in a time interval obey the distribution given in Problem
6.32, which corresponds to a Poisson-type distribution with a time-dependent
mean rate of arrival. Show that the pdf of time between arrivals is given by
8 v
v t

t exp ; for t 0;
< v 1
f t†ˆ w w
T
:
0; elsewhere:
As we see from Equation (7.124), it is the Weibull distribution.
7.32 A multiple-member structure in a parallel arrangement, as shown in Figure 7.17,
supports a load s. It is assumed that all members share the load equally, that their
resistances are random and identically distributed with common PDF F R (r), and
that they act independently. If a member fails when the load it supports exceeds
its resistance, show that the probability that failure will occur to n k members
among n initially existing members is
s
h i n
1 F R ; k ˆ n;
n
and
n k
s
X n h i j n
F R p n j†k s†; k ˆ 0; 1; ... ; n 1;
j n
jˆ1
where
s
h i k
n
p s†ˆ 1 F R ;
kk
k
j k r

X j s s j
i
p s†ˆ F R F R p j r†k s†; n i > j > k:
jk
rˆ1 r j i

TLFeBOOK