FAULTY INSPECTION DISTRIBlJTIONS -- SCl4E GENERALIZATIONS
by
Nonnan L. Johnson
University of North Carolina at Chapel Hill
and
Samuel Kotz
University of Maryland, College Park
Abstract
In Johnson, Kotz and Sorkin (1980), the authors derived the
distribution of the number of items observed to be defective in samples
from a finite population, when detection is erroneous with a nonzero
probability.
We extend here the above results by taking into account incorrect
identification of nondefectives as well as defectives.
waiting time distributions are also derived.
Correspondin~
Furthennore, the case of a
stratified finite population corresponding, for example, to defective
features of differing severity is considered.
Numerical values
illustrating the dependence of the corresponding probabilities on the
two
"misidentification" parameters are presented.
Key Words and Phrases:
binomial distribution; compound distributions;
faulty identification; hypergeometric distribution;
sampling inspection; waiting time; incomplete
identification.
2
1.
Introduction
Johnson et aZ. (1980) have discussed some problems arising when
attributes inspection is "less than perfect".
They considered, in
effect, sampling without replacement from a lot of size N containing X
defective (or nonconforming) items, when inspection detects such items
with probability p.
Here this is extended (i) by allowing for a
probability, p', of erroneously deciding that an item is defective when
really it is not and (ii) by stratifying the population so that
inspection error probabilities vary from stratum to stratum.
2.
Two Kinds of Inspection Error
The mnnber of defective items, Y, in a random sample (without
replacement) of size n has a hypergeometric distribution with parameters
n,X,N.
Conditionally on Y, the number of items
ao~~eatZy
called
"defective" is distributed as binomial (Y,p) and the number
called "defective" is distributed as binomial (n - Y, p').
inao~~eatZy
Thus, the
overall distribution of the total number of items called "defective", Z
say, is
Bin(Y,p) + Bin(n - Y, p')
1\
Hypg(n,X,N)
Y
(the two binomial variables being mutually independent), where
the compounding operator (Johnson and Kotz (1969, p. 184)).
Conditional on Y, the r th factorial moment of Z is
=
Ij=O (:) yO)pj (n- Y) (r-j) p,r- j
J
1\
denotes
3
The unconditional r th factorial moment of Z is
=
j E[y(j)(n-y)(r- j )]
Ij=O (:)pjp,rJ
= nCr)
N(r)
Ij=O (:)pjp,r-jx(j)
(N-X) (r-j)
J
(1)
In particular,
E[Z]
=
{Xp + (N -X)p' }n/N
Var(Z)
=
X (1- -N
X)(P_p,)2 ,
np(l~p) - n(n-l) N
where p = N-l{Xp + (N -X)p'}.
=
np
(2.1)
(N -1)
(2.2)
The conditional probability mass function
(pmf) of Z given Y is
Pr[Z = zlY]
(3)
(z
where (~)
=
= O,l, ... ,Y) ,
0 if a < b.
Hence the unconditional pmf of Z is
Pr[Z
= z]
when the first
I
is over max(O, n-N+X)
~
y
~
min (n,X) .
Numerical values can be obtained expeditiously if adequate tables of
hypergeometric probabilities
4
hey; n,X,N)
and binomial probabilities
bey; n,p)
are available from the fornnl1a
Pr[Z = z] =
L hey;
y
z
n,X,N)
L
b(j; y,p)b(z-j; n-y, p') .
(4)
j=O
Table 1 gives some examples of distributions, each with sample size
n = 10.
It is limited by space considerations, but fuller tables have
been calculated (for other sample sizes as well as other values of X and
N) •
I
e
'e'
e
p "' 0.75
N :: 100; X :: 5
Z\P'
0
1
2
3
4
5
6
7
8
9
10
0
0.025
0.05
0.075
0.1
.6731
.2818
.0422
.0028
.0001
.5244
.3575
.1010
.0156
.0015
.0001
.4060
.3891
.1608
.0379
.0056
.0006
.3122
.3901
.2139
.0679
.0138
.0019
.0002
.2384
.3711
.2557
.1028
.0267
.0047
.0006
-
-
-
-
II
N :: 200; X "' 10
0
0.025
0.05
0.075
0.1
.6778
.2736
.0446
.0038
.0002
.5280
.3520
.1015
.0167
.0017
.0001
.4087
.3855
.1603
.0388
.0060
.0006
.3142
.3879
.2129
.0684
.0143
.0020
.0002
-
.2399
.3698
.2545
.1030
.0272
.0049
.0006
.0001
.2122
.3591
.2695
.1182
.0336
.0064
.0008
.0001
.1627
.3265
.2917
.1529
.0521
.0120
.0019
.0002
-
-
-
N :: 100; X '"' 10
0
1
2
3
4
5
6
7
8
9
10
.4459
.3892
.1369
.0252
.0027
.0002
-
-
.3488
.3985
.1920
.0513
.0084
.0009
.0001
-
.2711
.3867
.2378
.0831
.0183
.0027
.0003
-
N .. 200; X :: 20
.2094
.3611
.2720
.1180
.0327
.0060
.0008
.0001
.1606
.3274
.2940
.1532
.0513
.0116
.0018
.0002
.4524
.3803
.1363
.0274
.0034
.0003
-
-
.3537
.3928
.1902
.0529
.0093
.0011
.0001
-
N :: 100; X. :: 20
0
1
2
3
4
5
6
7
8
9
10
.1856
.3540
.2870
.1299
.0362
.0065
.0008
.0001
-
.1465
.3209
.3028
.1619
.0542
.0119
.0017
.0002
-
.1149
.2860
.3093
.1915
.0751
.0195
.0034
.0004
-
-
.2749
.3831
.2354
.0840
.0193
.0030
.0003
-
N '"' 200; X :: 40
.0896
.2510
.3077
.2174
.0980
.0295
.0060
.0008
.0001
.0694
.2173
.2992
.2386
.1221
.0419
.0098
.0015
.0002
.1913
.3507
.2813
.1299
.0382
.0075
.0010
.0001
-
.1509
.3193
.2977
.1609
.0559
.0130
.0021
.0002
-
.1183
.2856
.3051
.1899
.0762
.0206
.0038
.0005
-
.0922
.2514
.3043
.2154
.0987
.0306
.0005
.0009
.0001
.0714
.2181
.2966
.2364
.1223
.0429
.0103
.0017
.0002
II
VI
e
'e'
e
p
= 0.8
N = 100; X = 5
z\p'
0
1
2
3
4
5
6
7
8
9
10
0
0.025
0.05
0.075
0.1
.6545
.2946
.0474
.0034
.0001
.5096
.3642
.1073
.0172
.0017
.0001
.3942
.3915
.1670
.0405
.0062
.0006
.3029
.3896
.2192
.0712
.0148
.0021
.0002
.2311
.3687
.2597
.1065
.0282
.0050
.0006
.0001
-
-
-
-
-
II
N = 200; X = 10
0
0.025
0.05
0.075
0.1
.6598
.2855
.0499
.0046
.0002
.5136
.3581
.1077
.0184
.0020
.0001
.3973
.3875
.1664
.0414
.0066
.0007
.0001
.3052
.3871
.2180
.0718
.0153
.0022
.0002
-
.2328
.3673
.2584
.1068
.0287
.0053
.0007
.0001
.1997
.3527
.2759
.1260
.0372
.0074
.0010
.0001
.1529
.3186
.2954
.1605
.0566
.0135
.0022
.0002
.0807
.2344
.3018
.2269
.1103
.0362
.0082
.0012
.0001
.0623
.2020
.2910
.2455
.1343
.0498
.0127
.0022
.0002
-
-
N = 100; X = 10
0
1
2
3
4
5
6
7
8
9
10
.4206
.3960
.1500
.0297
.0034
.0002
-
-
.3285
.3991
.2038
.0575
.0099
.0011
.0001
-
.2549
.3830
.2473
.0905
.0208
.0031
.0003
-
N z 200; X = 20
.1966
.3547
.2788
.1258
.0362
.0069
.0009
.0001
.1505
.3196
.2980
.1609
.0557
.0130
.0020
.0002
.4276
.3866
.1490
.0321
.0043
.0004
-
-
.3339
.3930
.2014
.0592
.ono
.0014
.0001
-
N = 100; X = 20
0
1
2
3
4
5
6
7
8
9
10
.1632
.3385
.2980
.1463
.0442
.0086
.0011
.0001
-
.1284
.3038
.3089
.1775
.0638
.0150
.0023
.0002
-
.1004
.2683
.3109
.2056
.0860
.0237
.0044
.0005
-
-
.2591
.3793
.2444
.0913
.0219
.0035
.0004
-
N = 200; X = 40
.0781
.2337
.3054
.2295
.1098
.0349
.0075
.0011
.0001
.0603
.2009
.2937
.2482
.1343
.0486
.0119
.0020
.0002
.1691
.3359
.2916
.1457
.0464
.0098
.0014
.0001
-
,,
.1330
.3028
.3033
.1760
.0655
.0163
.0028
.0003
-
.1039
.2684
.3063
.2035
.0871
.0251
.0049
.0007
.0001
0\
e
'e'
e
p .. 0.85
N • 100; X • 5
z\p'
0
1
2
3
4
5
6
7
8
9
10
N • 200; X • 10
0
0.025
0.05
0.075
0.1
0
0.025
0.05
0.075
0.1
.6363
.3068
.0528
.0040
.0001
.4950
.3705
.1136
.0189
.0019
.0001
.3827
.3936
.1732
.0431
.0067
.0007
.2938
.3889
.2244
.0746
.0158
.0023
.0002
.2240
.3661
.2637
.1103
.0298
.0054
.0007
.0001
.6421
.2968
.0553
.0054
.0003
.4995
.3637
.1140
.0203
.0023
.0002
.3861
.3892
.1724
.0442
.0073
.0008
.0(X)1
.2964
.3862
.2230
.0753
.0164
.0024
.0002
.2259
.3646
.2621
.1106
.0303
.0057
.0007
-
-
--
--
-
-
-
--
N • 100; X .. 10
0
1
2
3
4
5
6
7
8
9
10
.3964
.4012
.1632
.0347
.0042
.0003
-
.3091
.3985
.2153
.0641
.0116
.0013
.0001
-
.2396
.3785
.2563
.0980
.0235
.0037
.0004
-
.,
3
4
5
6
7
8
9
10
.1432
.3219
.3066
.1626
.0530
.0llO
.0015
.0001
-
.lI23
.2861
.3127
.1927
.0741
.0186
.0031
.0003
-
.0876
.2507
.3105
.2190
.0974
.0285
.0056
.0007
.0001
-
-
.OOll!
N • 200; X • 20
.1844
.3479
.2851
.1337
.0398
.0079
.0010
.0001
.1410
.3115
.3015
.1686
.0603
.0145
.0024
.0003
.4040
.3914
.1615
.0373
.0053
.0005
-
.3150
.3922
.2123
.0658
.0129
.0017
.0001
-
N .. 100; X • 20
0
1
-
.2441
.3747
.2529
.0988
.0248
.0042
.0005
-
.1879
.3459
.2817
.1338
.0410
.0085
.0012
.0001
.1436
.3106
.2986
.1680
.0613
.0151
.0026
.0003
.0706
.2178
.2977
.2373
.1222
.0425
.0101
.0016
.0002
.0543
.1863
.2841
.2533
.1463
.0572
.0153
.0028
.0003
N • 200; X .. 40
.0678
.2166
.3014
.2405
.1219
.0410
.0093
.0014
.0001
-
.0522
.1849
.2868
.2565
.1466
.0559
.0144
.0025
.0003
-
.1492
.3199
.2996
.1613
.0553
.0126
.0019
.0002
-
-
.1170
.2857
.3067
.1905
.0758
.0202
.0036
.0004
--
.0912
.2512
.3056
.2162
.0984
.0301
.0063
.0009
.0001
-
-
-
"-.I
e
'e'
e
p = 0.9
N .. 100; X = 5
z\p'
0
1
2
3
4
5
6
7
8
9
10
0
0.025
0.05
0.075
0.1
.6184
.3183
.0584
.0047
.0002
.4808
.3762
.1201
.0206
.0021
.0001
.3714
.3953
.1793
.0458
.0073
.0008
.0001
.2849
.3879
.2296
.0780
.0169
.0024
.0002
.2170
.3634
.2675
.1141
.0313
.0058
.0007
.0001
-
-
-
-
II
N = 200; X = 10
0
0.025
0.05
0.075
0.1
.6249
.3074
.0610
.0063
.0004
.4858
.3689
.1203
.0222
.0026
.0002
.3752
.3905
.1783
.0470
.0080
.0009
.0001
.2878
.3849
.2279
.0787
.0176
.0027
.0003
-
.2192
.3618
.2657
.1144
.0320
.0061
.0008
.0001
.1766
.3287
.2870
.1415
.0450
.0096
.0014
.0001
.1347
.3024
.3012
.1754
.0661
.0169
.0030
.0003
.0615
.2015
.2920
.2465
.1343
.0493
.0124
.0021
.0002
.0471
.1713
.2760
.2598
.1582
.0652
.0184
.0035
.0004
-
--
N .. 200; X .. 20
N '" 100; X" 10
0
1
2
3
4
5
6
7
8
9
10
.3733
.4049
.1762
.0400
.0052
.0004
-
.2907
.3968
.2263
.0709
.0135
.0016
.0001
-
.2250
.3734
.2648
.1057
.0263
.0043
.0005
-
.1729
.3407
.2908
.1417
.0437
.0089
.0012
.0001
.1319
.3032
.3045
.1762
.0651
.0161
.0027
.0003
.3815
.3947
.1738
.0428
.0065
.0006
-
.2971
.3904
.2227
.0726
.0150
.0020
.0002
-
N = 100; X = 20
0
1
2
3
4
5
6
7
8
9
10
.1253
.3044
.3128
.1787
.0626
.0140
.0020
.0002
-
.0980
.2682
.3142
.2073
.0851
.0227
.0040
.0005
-
.0761
.2331
.3081
.2314
.1093
.0339
.0070
.0010
.0001
-
.2298
.3696
.2609
.1064
.0278
.0049
.0006
-
N = 200; X = 40
.0587
.1999
.2957
.2503
.1342
.0477
.0114
.0018
.0002
.0450
.1695
.2786
.2635
.1588
.0638
.0173
.0031
.0004
.1314
.3031
.3052
.1766
.0649
.0159
.0026
.0003
-
.1027
.2684
.3079
.2043
.0867
.0246
.0047
.0006
-
.0798
.2341
.3031
.2279
.1101
.0358
.0079
.0012
.0001
00
e
'e'
e
p
= 0.95
N = 100; X = 5
z\p'
0
1
2
3
4
5
6
7
8
9
10
N = 200; X = 10
0
0.025
0.05
0.075
0.1
0
0.025
0.05
0.075
0.1
.6009
.3291
.0642
.0055
.0002
.4668
.3816
.1266
.0225
.0024
.0002
.3603
.3967
.1854
.0486
.0080
.0009
.0001
.2762
.3867
.2346
.0815
.0180
.0026
.0003
.2102
.3606
.2712
.1180
.0330
.0062
.0008
.0001
.6080
.3174
.0668
.0074
.0005
.4723
.3736
.1267
.0242
.0029
.0002
.3645
.3916
.1841
.0500
.0087
.0010
.0001
.2794
.3835
.2327
.0823
.0188
.0029
.0003
-
.2126
.3588
.2691
.1182
.0337
.0065
.0009
.0001
.1659
.3312
.2916
.1493
.0492
.0109
.0017
.0002
.1264
.2941
.3033
.1826
.0711
.0187
.0034
.0004
.0535
.1858
.2850
.2545
.1464
.0567
.0150
.0027
.0003
.0409
.1569
.2669
.2649
.1700
.0736
.0218
.0044
.0006
-
-
-
-
-
-
N = 100; X = 10
0
1
2
3
4
5
6
7
8
9
10
.3514
.4071
.1890
.0457
.0063
.0005
-
.2732
.3942
.2369
.0780
.0155
.0020
.0002
-
.2111
.3676
.2728
.1136
.0294
.0050
.0006
-
N = 200; X = 20
.1619
.3331
.2959
.1497
.0478
.0101
.0014
.0001
.1234
.2947
.3069
.1837
.0701
.0178
.0031
.0004
.3601
.3967
.1858
.0486
.0078
.0008
.0001
-
.1093
.2863
.3166
.1943
.0730
.0175
.0027
.0003
-
.0852
.2502
.3137
.2210
.0967
.0274
.0051
.0006
-
.0659
.2157
.3039
.2427
.1216
.0400
.0087
.0012
.0001
.2800
.3876
.2327
.0797
.0173
.0025
.0002
-
.2163
.3638
.2682
.1142
.0311
.0057
.0007
.0001
N = 200; X = 40
N = 100; X = 20
0
1
2
3
4
5
6
7
8
9
10
-
-
.0507
.lIn8
.2887
.2588
.1467
.0549
.0138
.0023
.0002
.0387
.1547
.2694
.2692
.1710
.0722
.0205
.0039
.0005
.1155
.2858
.3086
.1912
.0753
.0197
.0034
.0004
-
.0900
.2510
.3072
.2171
.0981
.0296
.0060
.0008
.0001
.0696
.2174
.2989
.2384
.1221
.0420
.0098
.0015
.0002
ID
10
Note that for pI = 0, the same distribution is obtained if the
values of n and X are interchanged.
As N and X are increased proportionately to each other with X/N =
w,
say, the other parameters (n,p,pl) remaining constant, the distribution
of Z tends to a binomial with parameters n, XN- l p + (1 - XN-l)pl.
Another simple special case is p = pI, leading to a binomial
distribution with parameters n,p (whatever the values of X and N).
However, this is a most unlikely situation -- it would correspond to
completely useless inspection, unable to differentiate between
satisfactory and nonsatisfactory items.
The distributions shown are quite sensitive to the value of pI
(false condenmation) because the ratio X/N is relatively small.
When
the proportion of nondefectives (1 - X!N) is lower, p I has less effect.
3.
Stratified Populations
MOre generally, we can suppose the lot divided into k strata
k of sizes N1 , NZ" .. ,Nk cI~=l Nj = N) such that for any
chosen individual in 1T., the probability of "detection as defective"
7T1' 1T
Z''''
,1T
J
(whether this is really so or not) is p..
J
The different strata may, for
example, correspond to actual defects of differing degrees of visibility.
The case considered in Section Z corresponds to k = Z, PI = p, Pz = pI,
N = X, N = N - X.
l
Z
The m.unber observed as "defective" in a random sample of size n is
then distributed as
k
z,......
L
j=l
Bin(Y.,p.)
J
J
/I
X
Mult Hypgk(n; Nl'''''~; N) .
(5)
11
X;
The binomials are mutually independent, conditional on
N -1 uk
~:
j=l
multivariate hypergeometric, Pr[Y.- = v]
= ()
~
n
Then
_
]
r. (r.)
(p.JY.]) ,
j=l ] ]
II
k
II r.!
j=l ]
,
lr
k
N].
(y.) (\lj=l Yj - n) .
k
r!
where
for the
(6)
denotes slUlIIIation over nonnegative integers r l' ... , r k such
k
that lj=l
r j = r.
Taking expectations with respect to
r!
r
II
k
II
r.!
j=l ]
r. nCr)
p.]
j=l]
k
II
j=l
X,
(r.)
N. ]
N(r)
]
(r.) r.
N. ] p.J
]
]
r· !
(7)
J
In particular,
k
L
E[Z] = n
N.p.
N j=l J J
= np
(8.1)
and
E[Z(Z-l)] = n(n-l)
k
N. (N. -l)p~
[l
N(N-l) j=l J ]
= n (n-l ) [(
N(N-l)
whence
k
L
N.p.)2 -
j=l J J
+
]
k
L
2
k
rl
p.p. ,]
j<j']]] ]
N.N.,
2
N.p.]
j=l] J
(8.2)
12
Var(Z) = np(l-p) -
-
n(n-l) k
_ 2
I N.(p.-p)
N(N-l) j=l J J
(8.3)
-1 \'k
where p = N
L.j=l NjPj.
Alternative and instructive formulae for the variance are
k
l I N.P~)
Var(Z) = n(N -n) p(l-p) + n(n-l) (p _ NN-1
N(N -1)
j=l J J
Var(Z) = n(N -n) pel-i))
N-l
(8.3) ,
k
+
n(n -1)
I N.p. (l-p.)
2
j=l
J J
J
N (N -1)
(8.3)"
The first tern in (8.3)' and (8.3)" corresponds to the variance of the
(actual) number of defectives in a random sample (without replacement) of
size n from a population of size N containing
NP
defectives.
It
follows
that the variance of Z is not less than this, while from (8.3) it cannot
exceed np(l-p)-- the value it would have in sampling with replacement
from the same population (when the distribution of Z would be binomial
with parameters n,p).
This will, of course, also be a good approximation
when N is large.
As a limiting case, we might have N. = 1 and k = N- - that is, each
J
item in the lot would have its own probability (Pj) of being declared
defective.
Note that this differs from a model in which there is supposed to be
a prior distribution of the probability of being declared defective, and
the p.'s are regarded as realized values from this distribution.
J
For
this latter model, we reach, in effect, a "with replacement" situation,
as distinguished from the
'~ithout
replacement" model we have considered,
with the distribution of Z depending on the specific values of the Pj's.
13
The "with replacement" model can be regarded as a mixture of
''without replacement" models -- the latter being conditional on the specific
sets of values Pl'PZ"",PN'
The average variance of the number bf items
declared defective for the ''without replacement" will, in general, be
smaller than that for the ''with replacement" -- because: the latter is
increased by variation among the pIS.
Formally,
Var(Z) = E [Var(Z)p)] + Var (E[Zlp]) .
}2
4.
'"
}2
(9)
'"
A Waiting Time Distribution
Suppose now that, under the conditions of Section 3, items are
inspected one at a time (without replacement) until a predetermined
number a of items have been assessed as "defective".
Denoting by M the
number of items needed to attain this goal, we have (cf. Section 4 of
Johnson et aZ. (1980))
Pr(M>m) = Pr(Z <a)
m = 1,2,. .. , N-l
(10)
(a s N) ,
where Z has the distribution (5) with n replaced by m.
Note that in this
case a can exceed the actual number of defectives in the lot because an
item can be assessed as defective even though it is not.
The distribution of M is not proper because there is a positive
probability that even when all N items in the lot have been inspected,
fewer than a items will be declared "defective".
In order to derive the distribution of M, we first consider the
possible sets of decisions for the N items.
The distribution of the
number (D) of those which will be "defective" is that of the Stnn of
independent binomial variables B , B2 , ... , 11< with parameters
1
14
(Nl ,Pl),(N2 ,P2), ... ,(Nk ,Pk) respectively.
Given D, each of the (~)
possible orderings of the D "defective" and (N-D) "not defective"
decisions is equally likely, and the number of items up to and including
the a th defective (M) has the negative hypergeometric distribution
Pr(M)
(N) -1 (m-l) (N-m)
=m = D
a-I D-a
(m = a, a+l, ... , N-D+a) ,
(11)
provided D ~ a.
The conditional expected value of Mgiven D
(~
a) is
E[MID] = a(N+1)/(D+l) .
(12)
If we neglect the possibility that D is less than a, the overall expected
value of Mis approximately
e.
a(N+l)E[(D+l)-l] ,; [{E[D] -+ U- l + {E[D] + U- 3 Var(D)] (N+l)a
•
= a(N+l)(
k
I
N.p. + 1)-1 [1 +
j=l J J
(Ik
N.p. + 1)
-2 k
j=l J J
k
1
11
=:• ap--1 [1 + -{I
+
+ I w.p.(l-p.)}] ,
N
P
p2 j=l J J
J
where w. = N./N (the proportion in the j th strattun).
J
J
We evaluate the overall variance of Mas
Var(M) = Var(E[M/D]) + E[Var(MID)]
= a2 (N+l)2 Var((D+l)-2) +
(cf. (9))
a(N+l)E~N-D)(D+l-a)J
~D+ 1) 2 (D+2)
J
After straightforward though quite tedious calculations, we find
I N.p.(l-p.)]
j=l J J
J
(13)
15
k
l\
w. p. (1- p.)
• Nj54 j =1 J J
J
Var(M) -• a
2
+
a (1- p) [1 + 1 + t.Tn
1 { 3- P
p-(l-p-) ~l w.p. ( 1
-po) -.(3 +a)}].
p2
N.,p
j=l J J
J
• a(l-p) [1 + 1
•
_2
N
P
3+a
- --
NP
+
3+a-(a+l)p·~l w.p. (l-p. )]
2
NP
(1-P)
j=l
J J
(14)
J
--1
As N -+ 00, the expected value of M tends to ap
and the variance
2
tends to ap- (1- p) . These are the mean and variance, respectively, of
the (negative binomial) waiting time distribution for occurrence of
~
"successes" in independent trials with probability of success equal to p
at each trial.
(14) can be written
Var(M) :: a(l-p)
•
Since
-2
P
~
+ 1
N
I~J= 1 w.p.(l-p.)
= P - IkJ·=l WJ.PJ~ s P - p2 = p(l-p) and a+3
J J
J
P
> a + 1,
it follows that
a(l-p) [1 + 1
-2
P
N
3t.::J s Var(M) s a(l- p) (1 +
l'p
-2
P
1:.)
N
Acknowledgement
Samuel Kotz's work was supported by the u.S. Office of Naval Research
under Contract N00014-81-K-030l.
16
References
Jo1mson, N.L. and Kotz, S. (1969). Distributions in statistics -- Discrete
Distributions. New York: Jo1m Wiley and Sons.
Jolmson, N.L., Kotz, S. and Sorkin, H.L. (1980).
distributions. Comm. statist. A9, 917-922.
Faulty inspection
Sorkin, H.L. (1977). An Empirical Study of Three Confirmation Tec1miques:
Desirability of Expanding the Respondent's Decision Field. Ph.D.
Thesis, University of Minnesota.
-.,-----_._--------:---:-1'"".,..
SECURITY CLASSIFICATION OF THIS PAr,F (IYIIM' nltl"
l. REPORT
4.
,1)
REPORT DOCUMENTAJION PAGE
NUMBER
12. GOVT
UNCLASSIFIED
READ INSTRUCTIONS
BEFORE COMPLE.TING FORM
ACCESSION NO.
TITLE (8lId Subtll/e)
Faulty Inspection Distributions - - Some
Generalizations
l
RECIPIENT'S CATALOG NUMBER
5
TYPE OF REPORT
a
PERIOD COVERED
TECHNICAL
6
PERFORMING O'G. REPORT NUMBE.R
Mimeo Series No. 1335
7.
!I
AUTHOR(s)
Norman L. Johnson and Samuel Kotz
9.
11.
CONTRACT OR GRANT NUMBER(.)
ONR Contract NOOO14-8l-K-0301
PERFORMING ORGANIZATION NAME AND ADDIlE55
---
CONTROLLING OFFICE NAME' AND ADDRESS
U.S. Office of Naval Research
Statistics and Probability Program (Code 436)
Arlington, VA 22217
10.
PROGRAM ELEMENT. PROJECT. TASK
AREA a WORK UNIT NUMBERS
12,
REPORT DATE
13
NUMBER OF PAGES
15,
SECURITY CLASS. (o/Ihls reporl)
April 1981
16
14. MONITORING AGENCY NAME 6 ADDRESS(i1 d;/(",,,,,' lrom COII/,olllnil Olfic.·)
UNCLASSIFIED
.
,
1511,
..
16.
--- -, .. __._---,,---,
._--_._-
DECLASSIFICATION 'DOWNGRADING
SCHEDULE
..
DISTRIBUTION STATEMENT (01 rll/. U"l'orlj
Approved for
e
Publi~
Release - - Distribution Unlimited
•
17.
DISTRIBUTION STATEMENT (M the /tb.trllcl .. ol",,,d In IJl0ck 20, II dll/e,enll,om Report)
18.
SUPPLEMENTARY NOTES
19.
KEY WORDS (ContinuC! on retversft .0;;'/(' Jt
flf·I·",';,,,,.f)"
---
(It,d Ic/l'Jltlly by bloc'le nun,t,,.r)
binomial distribution; compound distributions; faulty identification;
hypergeometric distribution; sampling inspection; waiting time; incomplete
identification.
20. ABSTRACT (Conrlnue on ,,,verB" side if neces.a,y and Identlly by block nllmb..,)
In Johnson, Kotz and Sorkin (1980) , the authors derived the distribution
of the number of items obsepved to be defective in samples from a finite
population, when detection is erroneous with a nonzero probability.
We extend here the above results by t~i~g into ~ccq~t i~corr~ct
identification of nondefectives as well as defectives. Corresponding
waiting time distributions are also derived. Furthermore, the case of a
. stratified finite population corresponding, for example; to defective
•
,e
DO
FORM
I JAN 73
1473
EDITION OF , NOV 65 IS OBSOLETE
•
UNCLASSIFIED
20.
features of differing severity is considered. Numerical values
illustrating the dependence of the corresponding probabilities on the
two "misidentification" parameters are presented.
,
e•
•
UNCLASSIFIED
SECURITY CLASSIFICATIO.. OF
,
.
T ...
·r
PAGE(Wh .. n DM. Ent .. ,ed)
•