ERRORS IN INSPECTION AND GRADING:
DISTRIBUTIONAL ASPECTS OF
SCREENING AND HIERARCHAL SCREENING
by
Samuel Kotz
University of Maryland
and
Norman L. Johnson
University of North Carolina at Chapel Hill
ABSTRACT
Continuing the studies of Johnson et at (1980) and Johnson
and Kotz (1981), further distributions arising from models of
errors in inspection and grading of samples from finite, possibly
stratified lots are
obta~ned.
Screening, and hierarchal screening
forms of inspection are also considered, and the effects of errors
on the advantages of these techniques assessed.
NOTATION SUMMARY
Single Sampling
Probability that a defective item is classified as defective
pI Probability that a nondefective item is classified as defective
p = N-l{Dp + N-D)pl} = Probability that an individual chosen at
random is classified as defective.
P
e
Hierarchal Sampling
Number of items tested en bloc at j-th stage
n.
J
Probability that a group containing at least one defective is
p.
J
classified as defective, at j-th stage.
Probability that a group containing no defectives is classified
p. I
J
as defective, at j-th stage
Po(n j ) = Probability that a random sample of n. items contains no defective
items (j = 1, 2).
J
* ) = Probability that a random sample of n. items contains no defective
Po(n
j
items, given that it contains at least one nondefective.
=
(p. -p!) Po(n.)
=
(p . -p!) Po* (n.)
J
J
J
J
J
J
-21.
INTRODUCTION
For convenience, we first summarize some results obtained by Johnson
et al. (1980) and Johnson
&Kotz
(1981).
A random sample of size n is chosen (without replacement) from a lot
of size N, which contains D defective (or nonconforming) members.
Suppose
that on inspection of items in the sample, the probability that a defective
item will be classified as defective is p, while the probability that a nondefective item will (erroneously) be classified as
'~efectiv~'
is pl.
If Y denotes the actual unknown number of defective items in the sample,
the distribution of Y is hypergeometric with parameters (n, D, N).
If X
denote the number (among these Y) which are (correctly) classified as
defective, and X' the number (among the (n-Y) nondefective items in the
sample) which are (incorrectly) classified as "defective" then, conditionally
~
on Y, the variables X and X' are independent binomial with parameters (Y,p)
and (n-Y, pI) respectively.
Averaging over the distribution of Y, the
distribution of
Z=X+x l
,
the total number of items described as "defective" after inspection of the
sample, can be formally expressed as
Binomial (Y,p) * Binomial en-V, pI) A Hypergeometric (n,D,N)
y
(1)
The symbol * stands for convolution, and th~ symbol
1\ indicates the "correspondY
'"
ing" operation with Y distributed as "(see, e.g. Johnson & Kotz (1969,
Chapter 8)).
distributions.
Distribution (1) is a mixture of convolutions of two binomial
The r-th descending factorial moment of Z is
(2)
-3-
where Z(r) = Z(Z-l) .•. (Z-r+l).
In particular
E(Z) = np/N
(3.1)
var(Z) = np(l-p) _ n(n-l) • Q
N-l
N
where
p=
{Op + (N_O)pl}N-
I
(l-Q)N (p_pl)2
(3.2)
is the probability that a single individual,
chosen at random from the lot, will be described as "defective" after
inspection (whether it is really defective or not).
Tables of the
distribution of Z for n=lO with p = 0.75(0.05)0.95; pI = 0(0.025)0.1;
N = 100 and 0 = 5, 10, 20; N = 200 and 0 = 10, 20, 40 are presented in
Johnson and Kotz (1981).
Johnson
If pI = 0, we have the situation described in
(19RO) and (1) becomes a hypergeometric-binomial distribution.
et~.
For fixed values of D/N = A say, the distribution (1) is quite sensitive to
changes in p and pI, but not to changes in 0 and N unless n/N (the sampling
fraction) is large.
It is easy to see that as 0, N ~
00
(with O/N = A) the
distribution of Z tends to binomial with parameters (n,Ap+l(I-A)pl).
Johnson and Kotz (1981) also, intep aZia, consider lots divided into k
strata IT , ... , IT of sizes N , ... , N (with
k
l
k
I
l.k
N.=N) and suppose that the
j =1 J
probability of an individual from the j-th stratum being classified as
defective is p ..
J
(The situation considered at the beginning of this section
corresponds to k=2, NI=O, N =N-D, PI = p,
2
Pz
= pl.)
The distribution of the
total number (Z) classified as defective is (in an obvious notation)
k
'*IBinomial (Y.,P.)
J=
J
J
A
Y
.....
multivariate Hypergeometric (n,N,N)
.....
(y=(Y , ... Y ) is the vector of numbers from IT , ... , IT in the sample
l
k
I
k
Yl
+ ••• +
Yk
= n)
(4)
The r-th factorial moment of Z is
(r)
,(
:cr) l.r rl ,r2 ,r .•. ,
(5)
-4-
where 2~ denotes sunwation subject to r
l
+ ••• +
r
k
=
~
is the multinomial coefficient rl/[
r.I].
k j=l J
In particular, with p = N- l L N.p. (again, denoting the probability that
j=l J J
an individual chosen at random is classified as "defective"
E(Z)
= np
(6.1)
k
n(n-l) \
- 2
var(Z)= np(l-p) - (N 1) L N.(p.-p)
n - j=l J J
(6.2)
(Note that var(Z) is usually less than the "binomial" value np(l-p).)
2.
GRADING
Formulae (4)-(6) can be regarded as generalizations of (1)-(3).
e
Further
useful generalization is obtained by considering judgement not to be restricted
to "defective" or "non-defective" but to inc! ude assignment to one of several
categories - as would be the case, if product were being graded in terms of
quality and/or size with regard to marketing.
(One aspect of this situation
is the multivariate topic of disimminent analysis, but here we are concerned
with the consequences rather than the methods of assignment.)
We will analyze a situation in which the aim of judgement is assign
an individual to one of s classes C , ... , Cs ' We denote the probabilith
I
that an individual, who really belongs to C. will be assigned to C. by
J
P.. , with, of course 2~ lP"
1J
1=
1J
= 1.
1
Still further generalization is possible by
introducing stratification within each class.
This leads to straightforward,
but notationally complicated elaboration, and will not be pursued here.
Let Y. denote the number of individuals belonging to C., in a random
J
J
sample of size n for a lot of size N containing N. individuals in C. (j=l,
J
J
s; 2~1= IN.=N);
and let Z..
denote the number, among these Y., assigned to
1
1J
J
... ,
-5-
C. (i=l, ... , s).
The
1
parameters
so that
(n;~;N),
PCX)=
X has a multivariate hypcrgeometric distribution with
{fT
j =1
[~JJ:]} /(~)
s
(I
j=l
Y.=n)
(7)
J
Also, given ~Y, ~J
Z. = (Z 1J
.. , ... , ZS).) has a multinomial distribution with parameters
(Y . , p .) where P. = (P .. , .•• , p .), and
J -J
-J
sJ
1J
s
~J
and
~l'
... ,
~s
J i=l
~
s
Z•.
p(z.IY) = Y.I IT (P.~J/Z
1)
(L
.. I)
1J
are mutually independent.
Z.. =Y.)
J
(8)
i=l 1J
It follows that the joint distribu-
tion of all the Z' s, Z = (Zl' ... , ,.."S
Z ) is
t"'<J
I'"tJ
(z .. ) s
{ N. 1J IT
P(~)
)
s
= i=l
L Z...
1J
where Z..
1)
i=l
z..
(P. ~J /Z .. 1)
1J
}
1J
(9)
(Note that the Y., are dete~ined by the Z.. , .)
J s
1J S
Formally we can write the distribution of
~
as
s
j=l Multinomial
(Yj'£j)~
Multivariate
Hypergeometric(n,~,N)
(10)
and it might be called a "multivariate hypergeometric-multinomial distribution.
factoria~r~~)]ents Of~ ~[
The jOint
E
r
S
S
IT
IT
j=l
i=l
1J
Z..
1J
= Ey
~
=
s
where
r .. =
J
L
i=l
5
r .. ; r .. =
J
can be ob(tain)ed]]from
()
]
S
r. .
S
r .. s r ..
E IT
IT Z.. 1J Ix = E IT Y. 1J IT P.~J
j=l i=l 1J
j=l J
i=l 1J
S
n(r .. ) s {(r .. )s r .. }
N. J IT P. 7J
IT
N(r .. ) j =1
J
i=l 1)
5
L L
i=l j=l
r ..
1)
In particular, with i # ii j f. j ,
(11)
-6E(Z •• ) = nN
-1
1)
N.P ..
J 1J
(12.1)
_ (2) (2)
(2)
E(Z .. Z.,.) - n
N. P .. P., ./N
1J 1 J
J
1J 1 J
_
E(Z .. Z. , . , ) - n
(2)
1J 1 J
N. N. ,P .. P. , . IN
J)
whence
(12.2)
(2)
(12.3)
1J 1 )
cov (Z .. ,Z.,.)
-1
} P.. P.,.N - 2
= -nN.J { (N-n)N.+N(n-l)
(N-l)
(13.1)
J
1J 1 J
cov (Z .. ,Z.,. I)
-2
-1
= -nN.N.
I (N-n)P .. P. I' IN (N-l)
J J
1J 1 J
1)
1 J
1J
1 J
Using these results we can find the covariance between Z..
1
S
and Z. "
1
= I z. ,. j =1
1
J
=
(13.2)
s
L Z..
j=l IJ
that is the total number of individuals assigned to
C., C., respectively,
J
1
cov (Z 1.. ,Z.,.)
= -n{(N-n)P.P., + (n-1)(P.P.,)}
I
1 1
1 1
where
-1 s
P. = N
1
I N.P .. ;
j=l J IJ
(14)
-1 s
(P.P. ,) = N
1
1
L N.P .. P. ,.
j=l ) 1J 1 J
(P. is the probability that an individual chosen at random is assigned to C.,
1
1
(P.P. ,) is the probability that an individual chosen at random would be assigned
1
1
to C. on one judgement and C., on another.).
1
1
The marginal distribution of
Z.. is like (4) with k replaced by sand p. by P... , so we can use the
J
1
IJ
corresponding formulae for moments.
3.
GROUP SCREENING
Further interesting distributions arise in connection with "group
screening" (Dorfman (1943)), in which groups of units can be tested for the
existence of one or more defective units among them.
This can be practicable,
for example, when testing liquids for presence of contaminants, and is then
suggested as a possible way of reducing the average total amount of testing.
-7Suppose that material from n l units is mixed and tested for presence of
"defective" material. If a negative result ("no defectives") is obtained,
no further action is taken, but if there is a positive result, each unit
is tested separately.
Let PI' pi denote the probabilities of obtaining correct or incorrect
positive results, respectively, at the first test.
As before, p, pI denote
the probabilities of correct or incorrect positive results, respectively,
when units are tested individually; D, N denote the number of defective units
and the total number of units in the population respectively, and Y denotes
the actual number of defective units among the n
tested.
l
The overall probability of obtaining a positive result on the first
test is
(15)
4It
where po(n ) = (N_D)(nl)/N(nl) is the probability that the sample contains
l
no defective items.
The distribution of Z can be represented as
(Binomial (WY,p) *Binomial(W(nl-Y),pl))y Hypergeometric (nl,D,N)
(16)
where Wis an indicator variable, defined by
w=
c
if the first test gives a positive result
otherwise
Denote that
P(W=l) ly>O) = PI ;
An
P(W=I) Iy=o) = pI
(17)
I
explicit formula for P(Z=z) is
P(Z-z) = E y[I' (W;)pxCl_P)WY-X
w,
x
[
w(nl-y)] 3w(nl_y)-z+]
pI xCI_pI)
z-x
I
where
Lx
denotes summation over (O,z-W(nl-Y))
S
x
S
min CWY,z).
-SWe .have
Noting (17) and
for i
=0
(lS.l)
(lS.2)
we find
E(Z(r))
= nCr)
1
{ (N(r))-l ~ (r)D(i)(N_D)(r-i) i 'r-i_( _ ') ,r p ( )}
PI
.L i
P P
PI PI P 0 n 1
1.=0
(19.1)
In particular
E(Z)
= n 1 {P 1P-
= n1(P1P
where
and
(19.2)
(P1-pi)p'PO(n 1 )}
- P(l)P')
= (P1-pPP (n 1 )
2
_ (2}r {-2 D(p _ p2 )+(N-D) (p,2_
var(Z) - n
N (N-I)
1 ~1 P -
-
2
-
+ n 1 (P I P - P(l)P') - n 1 (P 1P - P(l)P')
If there is no possibility of a "false positive"
E(Z)
Var(Z)
p 2 )}
_ P
p,2l
(1)
J
2
(19.3)
(so that pIp'
1
= 0)
we find
= n1P1PD/N
=
P1PD
n (n -1)pD
n P 1pD
1
1 1
N
(1--N-) +
N2 (N_I) {N(D-1)-(N-1)Dp1}
-9-
In general, it is to be expected (and hoped) that PI > pi just as
p > p', since we would expect (hope) that the probability of correct decision
would exceed that of incorrect decision.
It may well happen that PI < P
since detection of a defective may be more difficult with the mixture of
material from separate units.
More complicated distributions will be obtained
if it is supposed that PI depends on the value of Y ( the number of defective
units).
It does not seem unreasonable to suppose that PI might increase with Y.
The effectiveness of the screening procedure is measured by the three
quantities
(i)
(ii)
Probability of correct classification for defectives
= PIP
Probability of correct classification for nondefective
*
p ')
PO(nl)(I-Pi
+
*
(I-PO(nl))(I-PlP')
where
(20.1)
=
*
= 1-(Pl-P(l))P'
= (N-D-I)
(nl-l)
I(N-I)
(nl-l)
(20.2)
is the probability that the sample contains no defectives, given that one
member of the sample is nondefective.
(20.3
of course, the larger the values of (i) and (ii), and the smaller the value
of (iii), the better.
From (20.1) it is clear that the probability of correct classification of
a defective is dearea8ed by the screening process. (since PIP < p).
The value
of screening must therefore corne from increased correct classification of
nondefectives or reduction in the expected number of tests.
some relevant numerical information.
would be necessary, so
Table I contains
In the absence of screening, n
l
tests
-10-
indicates the proportionate saving from screening, and this is given (as a
percentage) in the last column of Table 1.
We have taken N = 00, so that
w = DIN is to be interpreted as proportion defective, because the values do
not depend greatly on N.
* decrease so that
As N decreases, Pel) and Pel)
both the proportionate saving in number of tests from screening and the
probability of correct identification of nondefectives decrease, (though
not sUbstantially, unless N is quite small).
It is to be noted that screening
increases the probability of correct identification of nondefectives.
4.
HIERARCHAL GROUP SCREENING
Sometimes additional saving in the expected number of tests, and improved
accuracy in classification, can be attained by using two-or more-stage
screening - that is, hierarchal screening.
For simplicity, we will consider
= hn 2 . (Generalizal
If a positive results is
two-stage procedures, with a first-stage sample of size n
tion to more than two stages follows similar lines).
obtained for the combined sample, it is split into h subsamples, each of size
n and each is then treated as in Section 3.
z
Letting PZ'
Pz
denote the
respective probabilities of correct and incorrect positive results when testing
•
each of the second stage (size n
z)
subsamples, the three quantities measuring
effectiveness are:
(i)
(ii)
Probability of correct classification for defectives
= PlPZP
Probability of correct classification for nondefectives
*
+ (1-P o(n Z»Pl(1-P2 P ')
(21.1)
-11-
= l-P I P2P
where
=
*
,
+
*
, "
PO(n l )(Pl-PI)P2 P
+
(P(l)P2
*'
+
*
P(2)P I )P
*
+
,
PO(n 2)PI (P2-P2)P
,
,
(21. 2)
,
PO(n 2) (P 2-P 2)
* (n ) - PO(n
* ) is the probability that a random sample of n ,
(Note that PO
2
l
l
known to contain at least one nondefective, also contains at least one
defective, but a randomly chosen subsample of size n, containing at least one
nondefective, jn fact contains no defectives.)
(iii)
Expected number of tests
(21. 3)
The proportional reduction in expected number of tests is
(21.4 )
Values given by (21.2) and (21.4) are shown in Table 2.
From (21.1) we see that the probability of
correct classification of a
defective is decreased more than for simple screening (cf. (20.1)).
As some
compensation, the probability of correct classification of nondefectives is
higher.
As in the case of simple screening the advantages of a screening procedure
are greater when the proportion
(~)
of defectives is smaller.
*
*
finite lot size (N) is to decrease P(l)' P(2)' Pel)'
P(2).
The effect of
From equations (21)
it can be seen that this will
(i)
not affect the probability of correct classification of defectives
(ii)
decrease the probability of correct classification of nondefectives
-12(iii)
decrease the expected number of tests.
Some analysis of the distribution of the total number (Z) of items
classified as "defectives" by this hierarchal screening procedure is given
in the Appendix.
Here we just give the expected value
(22)
ACKNOWLEDGEMENT
Samuel Kotz's work was supported by the U.S. Office of Naval Research
under Contract N00014-8l-K-030l.
Norman L. Johnson's work was supported by
the National Science Foundation under Grant MCS-8021704.
REFERENCES
Dorfman, R. (1943). The detection of defective members of large populations.
Ann. Math. Statiat. 14, 436-440.
Johnson, N.L. and Kotz, S. (1969). Distributions in Statistias--Disarete
Distributions. New York: .John Wiley and Sons.
Johnson, N.L. and Kotz, S. (1981). Faulty inspection distributions -- some
generalizations. (To be published in Proa. of ONR/AHO ReZiabiZity
Workshop, April 1981) Institute of Statistias Mimeo Series #1335,
University of North Carolina at Chapel Hill.
Johnson, N.L., Kotz, S. and Sorkin, H.L. (1980). Faulty inspection
distributions. Commun. Statist. A9(9), 917-922.
Johnson, N.L. and Leone, F.C. (1977). Statistias and E~erimental Design:
In Engineering and the Physiaal Saienaes, 2nd Ed~tion. New York: John
Wiley and Sons. (pp 914-915).
Table 1:
Simple Screening
Probability of correct classification of defective is PIP
These tables correspond to N = 00. Values of w(=D/N) are proportions of defectives in the lot.
* ) = (l-w) n1-l
po(n l ) = (l-w) nl ; po(n
. . An N decreases. Po and Po* decrease; other quantities
l
(i)
(ii)
are not changes.
Note the values shown do not depend on p.
(iii)
I
~l
PI
P
0.98
0.05
0.98
P
0.05
w
0.05
0.1
t")
-t
I
0.2
e
nl
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTlVE
EXPECTED PERCENT
REDUCTION IN TESTS
6
0.9870
53.7
8
0.9835
51.2
10
0.9803
47.7
12
0.9774
43.9
6
0.9785
34.8
8
0.9732
29.5
10
0.9690
12
0.9656
20.9
6
0.9662
9.7
8
0.9608
5.1
10
0.9572
2.0
12
0.9550
0.0
e
-
24.4
e
Table 1:
Simple Screening (cont'd)
PI
PI
0.95
0.05
P
0.95
P
0.05
w
0.05
0.1
-
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTIVE
EXPECTED PERCENT
REDUCTION IN TESTS
0.9873
54.5
8
0.9839
52.2
10
0.9804
48.9
12
0.9781
45.3
6
0.9791
36.2
8
0.9740
31.2
10
0.9699
26.4
12
0.9666
22.1
6
0.9672
11.9
8
0.9619
7.6
10
0.9589
4.7
12
0.9564
2.9
n
1
6
I
o:::t
0.2
·1
-
e
e
Table 1:
Simple Screenin[
PI
PI
P
P
0.90
0.05
0.95
0.05
(cont'd)
w
0.05
0.1
I
Lf)
~
0.2
I
e
n
1
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTIVES
EXPECTED PERCENT
REDUCTION IN TESTS
-%
6
0.9878
55.8
8
0..9847
53.9
10
0.9818
50.9
12
0.9792
47.6
6
0.9801
38.5
8
0.9753
34.1
10
0.9715
29.6
12
0.9683
25.7
6
0.9689
15.5
8
0.9639
11.8
10
0.9607
9.1
12
0.9587
7.5
e
e
Table 1:
PI
0.95
Simple Screening
PI
0.10
P
0.90
P
0.10
(cont'd)
w
0.05
0.1
I
0.2
\0
.-4
I
n
1
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTIVE
EXPECTED PERCENT
REDUCTION IN TESTS
----
6
0.9718
60.8
8
0.9644
58.9
10
0.9586
55.9
12
0.9533
52.6
6
0.9552
43.5
8
0.9457
39.1
10
0.9379
34.6
12
0.9317
30.7
6
0.9329
20.5
8
0.9228
16.8
10
0.9164
14.1
12
0.9123
12.5
Note that in the last set, the expected percent reduction in expected number of tests is always 5%
greater than for the corresponding case in the penultimate set.
,
This is because the value of PI-PI
is the same (0.85) in the two sets while the value of PI is 0.05 greater in the last set.
e
e
e
Table 2:
Two-Stage Hierarchal Screening
PI
PI
P2
0.98
0.05
0.98
P2
0.05
P
w
0.05
0.05
0.1
I
l"-
,
..-4
0.2
e
EXPECTED PERCENT
REDUCTION IN TESTS %
2
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTIVES
6
2
0.9971
65.2
6
3
0.9949
63.7
12
2
0.9966
56.5
12
3
0.9944
60.3
12
4
0.9924
60.4
12
6
0.9886
57.1
6
2
0.9943
39.3
6
3
0.9902
40.0
12
2
0.9937
34.9
12
3
0.9896
39.5
12
4
0.9859
38.8
12
6
0.9796
33.4
6
2
0.9892
10.1
6
3
0.9819
10.6
12
2
0.9886
8.5
12
3
0.9813
12.1
12
4
0.9755
10.4
12
6
0.9671
4.6
n
1
n
e
e
Table 2:
Two-Stage Hierarchal Screening
PI
PI
P2
0.98
0.05
0.90
P2
0.05
P
0.05
(cont'd)
til
0.05
0.1
0.2
I
00
I
e
n
1
6
n
2
2
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTIVES
EXPECTED PERCENT
REDUCTION IN TESTS(%)
0.9973
58.9
6
3
0.9959
60.1
12
2
0.9968
57.3
12
3
0.9948
61.5
6
2
0.9948
40,8
6
12
3
0.9910
42.1
2
0.9941
36.4
12
3
0.9904
41.6
6
2
0.9900
12.9
6
3
0.9833
14.5
12
2
0.9894
11.3
12
3
0.9828
15.9
e
e
Table 2:
Two-Stage Hierarchal Screening
PI
PI
P2
P2
P
0.95
0.05
0.95
0.05
0.05
(cont'd)
w
0.05
0.1
I
O'l
r-4
n
l
6
n
2
PROBABILITY OF CORRECT
CLASSIFICATION FOR NONDEFECTIVES
EXPECTED PERCENT
REDUCTION IN TESTS(%)
2
0.9972
59.2
6
3
0.9954
60,2
12
2
0.9968
57.8
12
3
0.9947
61.7
6
2
0.9947
40.9
6
3
0.9908
42.0
12
2
0.9941
37.2
12
3
0.9902
41.8
6
2
0.9898
13.4
6
3
0.9830
14.3
12
2
0.9893
11.9
12
3
0.9825
15.9
I
0.2
e
e
e
Table 2:
Two-Stage Hierarchal Screening
PI
PI
P2
P2
0.95
0.10
0.95
0.05
P
0.05
(cont'd)
w
0.25
0.1
I
0
n
l
n
2
PROBABILITY OF CORRECT
CLASSIFICATION OF NONDEFECTIVES
EXPECTED PERCENT
REDUCTION IN TESTS(%)
6
2
0.9971
57.2
6
3
0.9953
58.8
12
2
0.9967
56.3
12
3
0.9946
60.7
6
2
0.9946
39.4
6
3
0.9907
41.0
12
2
0.9941
36.4
12
3
0.9902
41.3
6
2
0.9898
12.7
6
12_
3
0.9830
13.8
2
0.9893
11.7
12
3
0.9825
15.8
N
I
0.2
~ote
that in the last set the probabilities of correct classification of nondefectives are very slightly
greater (no more than 0.0001) that the corresponding probabilities for the penultimate set, while the
expected reduction in number of tests is slightly (c. 1-2%) less.
,
(This comparison reflects the effect
of changing PI from 0.05 to 0.10).
e
e
e
-ZlAPPENDIX
In order to study the distribution of the total number (Z) of items
declared "defective" we introduce the auxiliary indicator variables:
= {l
Wj
if items in the j-th subsample are tested individually
otherwise
(AI)
0
Conditional on WI' ... , W and the aatuaZ numbers
h
Y , ... , Y of defectives in the corresponding subsamples, Z is distributed as
h
l
for j = 1, •.. , h.
h
Binomial (
h ,
L W.Y.,p) * Binomial (L W.(nz-Y')'p )
j=l J J
j=l J
J
(AZ)
1be conditional r-th factorial moment of Z is
r
r
i=O
3.
h
L (.) (L
W.Y.)
j=l J J
Conditional on y for any a, a
,
(i)
{.
(r-i)
h
L W.(nZ-Y') }
j=l J
> 0; and j
PlPZ
E(W~Iy) = p(Wj=lIXJ = PlP Z
, ,
PlPZ
Z
PlPZ
~
E(W~~,
IY) ~ P(W.=l
J J ,..,
J
w.
'J'
j
if Y. > 0
J
h
LY.
if Y. = 0,
1 3.
h
J
> 0
'"
Pl P2
1 3.
J
if Y. > 0, Y.
J
J
,
,2
PlPZ
if Y. = Y.
J
(A4)
LY.
if Y. = 0,
,Z
=lly)=
(A3)
J
,
> 0
,
PlPZP Z if Y.J > 0, Y.J =
,
. ,r-i
p3.p
J
,
o or
Y. = 0, Y. ' > 0
J
J
h
= 0,
if Y. = Y. , = 0 =
J
J
L Y. > 0
1 3.
h
LY.
1 3.
(AS)
-22We have
(A6.1)
(A6.2)
(A6.3)
P(Y j > 0, Yj ' > 0)
P(Y .
J
= l-2P O(n 2)
h
r Y.
= YJ.' = 0,
i=l
> 0)
1
+
PO(2n 2)
= PO(2n 2)
h
P(Y.
J
= 0 = r Y.) = P(Y. = Y.' = 0 =
i=l
1
J
J
(A6.4)
(A6.S)
- PO(n l )
h
r
i=l
Y.) =
1
PO(n l )
(A6.6)
whence
(A7.l) .
•
(A7.2)
Also, from (A4) and (AS), with 8,
a..8
E(W.Y.)
J J
=
8
PlP2E(Y.)
J
.. ~ '8 8' )
E(W.¥f:,Y.Y.
J J J J
2
8 8' )
= PlP2c(y.y.
J J
8'
> 0
(AB.l)
(AB.2)
-23-
The joint distribution function of
P(X)
Xis
(A9)
=
[ Y , ... ,
I
and the r-th joint factorial moment of
X is
(AID)
E
h
where r
=. L1r 1..
In particular for all j ; j
1=
E(Y j ) = nzU/N
(AIl.I)
Z
E(Y j )= nzO{(nz-I)D+N-nz}/N(Z)
(All. Z)
E(Y.Y.') = nZzu(Z)/N(Z)
(All. 3)
J J
E(Yj!Y j ' = 0)
= n 2D/(N-n z); E(YjIYj' > 0) = n2Dh-n2N-I-Po(nz)}{I-po(nz)}-1
x (Nn
-1
z)
(All A )
Applying formulae (A3) - (All) we get after some algebraic rearrangement the
formula (ZZ) for E(Z).
In order to calculate the variance of Z, we have to
calculate
E(Z
(2)
Z
) = E[p (
h
L W.Y.)( hL W.Y.-I)
j=l J J
j=l J J
,h
h
+ Zpp ( L W.Y.){ L w.(nZ-Y')}
j=l J J j=l J
J
h
{
h
L w.(nZ-Y')}{ L W.(nZ-Y')- I}]
j=l J
J
j=l J
J
-24-
'2
h
2
2 ,2
h
= (p-p ) E[( I W.Y.) ] - (p -p )E( I w.Y.)
j=l J J
j=l J J
+
,
,h
h
2n 2P (p-p )E[( I W.Y.)( L w.)]
j=l J J j=l J
2 ,2
+ n P
2
Z ,Z
+
nZp
2
E[(Iw j ) ]
[hE(W .)
+
J
h(h-I)E(W.W.')]
(AIZ)
J J
where E(W.) and E(W.W.') are given by (A7), and
J
E(W.Y.)
J J
= nZPIPZDN
J J
-1
(Al3.!)
(Al3 .Z)
(AI3.3)
The resulting formulas for E(Z(2)) and
var(Z) = E(Z(Z))
+
E(Z) _ {E(Z)}Z
are complicated, but numerical calculation is straightforward.
,
Pz = 0 (so that there are no false positives), then
E(Z(2)) = p2[hE(W.y. 2)
JJ
and also
+
h(h-I)E(W.W.'Y.Y. ') - hE(W.Y.)] ~
JJ
JJ
JJ
,
If P
= P1 =
-25E(Z) =
(A14)
so that
w
(A15)