Fundamentals of Probability and Statistics for Engineers

7.28 Let random variable X be^2 -distributed with parameter. Show that the limiting
distribution of

as is N (0, 1).

7.29 Let X 1 ,X 2 ,...,Xn be independent random variables with common PDF FX (x)
and pdf fX (x). Equations (7.89) and (7.91) give, respectively, the PDFs of their
maximum and minimum values. Let X(j) be the random variable denoting the
jth-smallest value of X 1 ,X 2 ,...,Xn. Show that the PDF of X(j) has the form

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

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 FR(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

and

where

Some Important Continuous Distributions 243

 n

Xn

… 2 n†^1 =^2

n!1

FX…j†…x†ˆ

Xn

kˆj

n

k



‰FX…x†Šk‰ 1 FX…x†Šnk; jˆ 1 ; 2 ;...;n:

fT…t†ˆ

v

w



tv^1 exp 

tv

w



; fort0;

0 ; elsewhere:

8

<

:



1 FR

s

n

hin

; kˆn;

Xnk

jˆ 1

n

j



FR

s

n

hij

pn…nj†k…s†; kˆ 0 ; 1 ;...;n 1 ;

pnkk…s†ˆ 1 FR

s

k

hik

;

pijk…s†ˆ

Xjk

rˆ 1

j

r



FR

s

j



FR

s

i

r

pj…jr†k…s†; ni>j>k: