Inspection Paradox 检验悖论
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THE INSPECTION PARADOX
S
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DO
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M. R
O
O
OS
S
SS
S
S
Department of Industrial Engineering and Operations Research
University of California
Berkeley, CA 94720
E-mail: smross
@
newton.berkeley.edu
We relate two variants of the inspection paradox by showing that both represent the
length of a renewal interval that contains a chosen point, the difference being in the
way that point is chosen+ We show, in both cases, that the length of the interval is
likelihood ratio larger than that of an ordinary renewal interval+
1. AN INSPECTION PARADOX
A total of n families have children who attend a certain school+ Arbitrarily number
these families, and for i 5 1,+++,n, let X
i
denote the number of children of family i
who attend the school+ Assume that the X
i
are independent and identically distrib-
uted with mass function p
j
5 P$X
i
5 j%, j $ 1, and finite mean µ 5
(
j
jp
j
+
Suppose that one of the school children is randomly chosen+ We are inter-
ested in the distribution of the number of school children in the selected child’s
family; that is, if I is the family to which the selected child belongs, we are inter-
ested in X
I
+
Theorem 1.1:
(a) X
I
is likelihood ratio ordered larger than X
1
.
(b) P$X
I
$ r% $
(
j$r
p
j
+
(c) P$X
I
5 j% $
jp
j
µ
nµ
j 1 ~n 2 1!µ
(d) lim
nr`
P$X
I
5 j% 5
jp
j
µ
+
Probability in the Engineering and Informational Sciences, 17, 2003, 47–51+ Printed in the U+S+A+
© 2003 Cambridge University Press 0269-9648003 $12+50
47
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