Appendix A 409
Appendix A. Error Function and Its Derivative [4.12]
_<" 2 -x" x -eovn2 erf x x -evn erf x D. DO. 1.1283791671 0..0.0.0.0.0. 0.0.0.0.0. 0..50. 0..87878 25789 D. 520.49 98778 0..0.1 1. 12826 63348 0..0.1128 34156 0..51 0..86995 15467 D. 52924 36198 0..0.2 1. 12792 790.57 0..0.2256 45747 0..52 0..8610.3 70.343 0..53789 8630.5 0..0.3 1. 12736 40.827 o.. 0.3384 12223 0..53 o.. 8520.4 34444 o.. 54646 40.969 0..0.4 1. 12657 520.40. 0..0.4511 110.61 0..54 o.. 84297 51813 o.. 55493 9250.5 0..0.5 1. 12556 17424 0..0.5637 19778 0..55 o.. 83383 66473 0..56332 33663 0..0.6 1.12432 430.52 0..0.6762 15944 0..56 0..82463 22395 o.. 57161 57638 0..0.7 1.12286 36333 o.. 0.7885 77198 0..57 0..81536 63461 0..57981 580.62 0..0.8 1.12118 0.60.0.4 0..0.90.0.7 81258 0..58 0..80.60.4 33431 o.. 58792 290.0.4 0..0.9 1.11927 62126 0..10.128 0.5939 0..59 o.. 79666 75911 0..59593 64972 0..10. 1. 11715 160.68 o.. 11246 29160. 0..60. D. 78724 34317 D. 60.385 60.90.8 0..11 1.11480. 80.50.0. 0..12362 28962 0..61 D. 77777 51846 D. 61168 12189 0..12 1. 11224 69379 0..13475 83518 0..62 0..76826 71442 0..61941 14619 0..13 1.10.946 97934 0..14586 71148 0..63 D. 75872 35764 D. 6270.4 64433 0..14 1.10.647 82654 D. 15694 70.331 0..64 0..74914 87161 0..63458 58291 0..15 1.10.327 41267 0..16799 59714 0..65 0..73954 67634 D. 6420.2 93274 0..16 1. 0.9985 92726 D. 1790.1 18132 0..66 D. 72992 18814 0..64937 66880. 0..17 1. 0.9623 57192 0..18999 24612 0..67 D. 720.27 81930. D. 65662 770.23 0..18 1. 0.9240. 560.0.8 0..20.0.93 58390. 0..68 0..710.61 97784 0..66378 220.27 0..19 1. 0.8837 11683 D. 21183 98922 0..69 0..70.0.95 0.6721 0..670.84 0.0.622 0..20. 1. 0.8413 47871 0..22270. 25892 0..70. 0..69127 4860.4 D. 67780. 11938 0..21 1. 0.7969 89342 0..23352 19230. D. 71 D. 68159 62792 0..68466 5550.2 0..22 1.0.750.6 61963 0..24429 59116 D. 72 0..67191 88112 0..69143 31231 0..23 1. 0.70.23 92672 0..2550.2 25996 D. 73 D. 66224 62838 D. 69810. 39429 0.24 1. 0.6522 0.9449 0..26570. 0.0.590. 0..74 0..6525824665 0..70.467 80.779 0..25 1. 0.60.0.1 41294 0..27632 6390.2 0..75 0..64293 10.692 D. 71115 56337 0..26 1. 0.5462 18194 0..28689 97232 0..76 D. 63329 57399 0..7175367528 0..27 1. 0.490.4 710.98 0..29741 82185 0..77 0..62368 0.0.626 D. 72382 16140. 0..28 1. 0.4329 31885 o.. 30.788 0.0.680. 0..78 D. 6140.8 75556 D. 730.0.1 0.4313 0..29 1. 0.3736 33334 o.. 31828 34959 0..79 0..60.452 16696 o.. 73610. 34538 0..30. 1.0.3126 0.90.96 0..32862 67595 0..80. 0..59498 57863 0..74210. 0.9647 0..31 1.0.2498 93657 0..33890. 8150.3 0..81 D. 58548 32161 0..7480.0. 3280.6 0..32 1.0.1855 22310. o.. 34912 59948 0..82 0..5760.1 71973 0..75381 0.750.9 0..33 1.0.1195 31119 o.. 35927 86550. 0..83 D. 56659 0.8944 D. 75952 37569 0..34 1.0.0.519 56887 0..36936 45293 0..84 0..55720. 73967 D. 76514 27115 0..35 0..99828 37121 o.. 37938 20.536 0..85 0..54786 97173 D. 770.66 80.576 0..36 0..99122 10.0.0.1 o.. 38932 970.11 0..86 D. 53858 0.7918 0..77610. 0.2683 0..37 0..9840.1 14337 0..39920. 59840. 0..87 0..52934 34773 0..78143 98455 0..38 0..97665 89542 0..40.90.0. 94534 0..88 0..520.16 0.5514 D. 78668 73192 0..39 0..9691675592 D. 41873 870.0.1 0..89 D. 5110.3 47116 D. 79184 32468 0..40. 0..96154 12988 0..42839 23550. 0..90. 0..50.196 85742 D. 79690. 82124 0..41 0..95378 42727 0..43796 90.90.2 0..91 0..49296 46742 D. 80.188 28258 0..42 0..94590. 0.6256 0..44746 76184 0..92 D. 4840.2 54639 0..80.676 77215 0..43 0..93789 45443 0..45688 66945 0..93 D. 47515 33132 D. 81156 35586 0..44 0..92977 0.2537 D. 46622 51153 0..94 0..46635 0.50.90. 0..81627 10.190.
0..45 0..92153 20.130. 0..47548 17198 0..95 0..45761 92546 0..820.89 0.80.73 0..46 D. 91318 41122 0..48465 5390.0. 0..96 0..44896 1670.0. 0..82542 36496 0..47 0..90.473 0.8685 0..49374 50.50.9 0..97 0..440.37 97913 0..82987 0.2930. 0..48 D. 89617 66223 0..50.274 9670.7 0..98 0..43187 55710. 0..83423 150.43 0..49 0..88752 57337 D. 51166 82612 0..99 0..42345 0.8779 D. 83850. 80.696 0..50. D. 87878 25789 0..520.49 98778 1. DO. 0..41510. 74974 0..84270. 0.7929 410 Appendix A
Appendix A (cont.) x ~e-x' erf x x -e2 -x' erfx lin lin 1.00 0.41510 74974 0.84270 07929 1.50 O. 11893 02892 0.96610 51465 1. 01 0.40684 71315 0.84681 04962 1.51 0.11540 38270 0.96727 67481 1.02 0.39867 13992 O. 85083 80177 1.52 0.11195 95356 O. 96841 34969 1. 03 0.39058 18368 0.85478 42115 1.53 0.10859 63195 0.96951 62091 1.04 0.38257 98986 O. 85864 99465 1.54 0.10531 30683 O. 97058 56899 1.05 0.37466 69570 0.86243 61061 1.55 0.10210 86576 0.97162 27333 1.06 0.36684 43034 O. 86614 35866 1.56 0.0989819506 0.97262 81220 1. 07 0.35911 31488 0.86977 32972 1.57 0.09593 17995 0.97360 26275 1.08 0.35147 46245 O. 87332 61584 1.58 O. 09295 70461 0.97454 70093 1.09 0.34392 97827 0.87680 31019 1.59 0.09005 65239 0.97546 20158 1.10 0.33647 95978 0.88020 50696 1.60 O. 08722 90586 0.97634 83833 1.11 0.32912 49667 O. 88353 30124 1.61 O. 08447 34697 0.97720 68366 1.12 0.32186 67103 0.88678 78902 1.62 0.08178 85711 0.97803 80884 1.13 0.31470 55742 0.88997 06704 1. 63 0.07917 31730 O. 97884 28397 1.14 0.30764 22299 0.89308 23276 1.64 O. 07662 60821 O. 97962 17795 1.15 0.30067 72759 O. 89612 38429 1.65 0.07414 61034 0.98037 55850 1.16 0.29381 12389 0.89909 62029 1.66 O. 07173 20405 0.98110 49213 1.17 0.28704 45748 O. 90200 03990 1.67 O. 06938 26972 0.98181 04416 1.18 0.28037 76702 0.90483 74269 1.68 0.06709 68781 0.98249 27870 1.19 0.27381 08437 0.90760 82860 1.69 O. 06487 33895 0.98315 25869 1.20 0.26734 43470 0.91031 39782 1.70 O. 06271 10405 O. 98379 04586 1.21 O. 26097 83664 0.91295 55080 1.71 0.06060 86436 O. 98440 70075 1.22 0.25471 30243 0.91553 38810 1.72 O. 05856 50157 0.98500 28274 1.23 O. 24854 83805 0.91805 01041 1.73 0.05657 89788 0.98557 84998 1.24 O. 24248 44335 0.92050 51843 1.74 0.0546493607 0.98613 45950 1.25 O. 23652 11224 0.92290 01283 1.75 0.05277 49959 0.98667 16712 1. 26 O. 23065 83281 0.92523 59418 1.76 0.0509547262 0.98719 02752 1.27 0.22489 58748 O. 92751 36293 1. 77 0.0491874012 0.98769 09422 1.28 0.21923 35317 0.92973 41930 1.78 0.04747 18791 0.98817 41959 1.29 0.21367 10145 0.93189 86327 1. 79 0.04580 70274 0.98864 05487 1. 30 0.20820 79868 0.93400 79449 1.80 0.0441917233 0.98909 05016 1.31 0.20284 40621 0.93606 31228 1.81 0.04262 48543 O. 98952 45446 1.32 O. 19757 88048 0.93806 51551 1.82 0.04110 53185 0.98994 31565 1. 33 0.19241 17326 0.94001 50262 1.83 0.03963 20255 0.99034 68051 1.34 0.18734 23172 O. 94191 37153 1.84 0.0382038966 0.99073 59476 1.35 0.18236 99865 0.94376 21961 1. 85 O. 03681 98653 0.99111 10301 1.36 0.1774941262 0.94556 14366 1.86 0.03547 88774 0.99147 24883 1.37 O. 17271 40811 0.94731 23980 1. 87 0.03417 98920 0.99182 07476 1.38 O. 16802 91568 0.94901 60353 1.88 0.03292 18811 O. 99215 62228 1.39 O. 16343 86216 0.95067 32958 1.89 0.03170 38307 0.99247 93184 1.40 0.15894 17077 0.95228 51198 1.90 0.03052 47404 0.99279 04292 1.41 0.15453 76130 0.95385 24394 1. 91 O. 02938 36241 0.99308 99398 1. 42 0.15022 55027 0.95537 61786 1.92 O. 02827 95101 0.99337 82251 1.43 0.14600 45107 0.95685 72531 1. 93 0.02721 14412 O. 99365 56502 1.44 0.14187 37413 0.95829 65696 1.94 0.02617 84752 O. 99392 25709 1.45 O. 13783 22708 O. 95969 50256 1.95 O. 02517 96849 O. 99417 93336 1.46 O. 13387 91486 0.96105 35095 1.96 O. 02421 41583 O. 99442 62755 1.47 0.13001 33993 O. 96237 28999 1.97 0.02328 09986 0.99466 37246 1.48 0.12623 40239 0.96365 40654 1. 98 0.02237 93244 0.99489 20004 1.49 O. 12254 00011 O. 96489 78648 1.99 0.02150 82701 0.99511 14132 1.50 0.11893 02892 0.96610 51465 2.00 0.02066 69854 0.99532 22650 Appendix B 411
Appendix B. Percentage Points of the xl-Distribution; Values of Xl in Terms of IX and v [4.12]
V\C( 0.995 0.99 0.975 0.95 0.9 0.75 0.5 0.25
1 -4~ 1.57088 -4~ 9.82069 (-3) 3.93214 0.0157908 0.101531 0.454937 1.32330 2 f-5r·92704-2 1.00251 !-2 2.01007 !-2 5.06356 0.102587 0.210720 0.575364 1.38629 2.77259 3 -2 7.17212 0.114832 0.215795 0.351846 0.584375 1.212534 2.36597 4.10835 4 0.206990 0.297110 0.484419 0.710721 1.063623 1.92255 3.35670 5.38527 5 0.411740 0.554300 0.831:;!11 1.145476 1.61031 2.67460 4.35146 6.62568 6 0.675727 0.872085 1.237347 1.63539 2.20413 3.45460 5.34812 7.84080 7 0.989265 1.239043 1.68987 2.16735 2.83311 4.25485 6.34581 9.03715 8 1.344419 1.646482 2.17973 2.73264 3.48954 5.07064 7.34412 10.2188 9 1.734926 2.087912 2.70039 3.32511 4.16816 5.89883 8.34283 11.3887 10 2.15585 2.55821 3.2469i 3.94030 4.86518 6.73720 9.34182 12.5489 11 2.60321 3.05347 3.81575 4.57481 5.57779 7.58412 10.3410 13.7007 12 3.07382 3.57056 4.40379 5.22603 6.30380 8.43842 11.3403 14.8454 13 3.56503 4.10691 5.00874 5.89186 7.04150 9.29906 12.3398 15.9839 14 4.07468 4.66043 5.62872 6.57063 7.78953 10.1653 13.3393 17.1170 15 4.60094 5.22935 6.26214 7.26094 8.54675 11.0365 14.3389 18.2451 16 5.14224 5.81221 6.90766 7.96164 9.31223 11.9122 15.3385 19.3688 17 5.69724 6.40776 7.56418 8.67176 10.0852 12.7919 16.3381 20.4887 18 6.26481 7.01491 8.23075 9.39046 10.8649 13.6753 17.3379 21.6049 19 6.84398 7.63273 8.90655 10.1170 11.6509 14.5620 18.3376 22.7178 20 7.43386 8.26040 9.59083 10.8508 12.4426 15.4518 19.3374 23.8277 21 8.03366 8.89720 10.28293 11.5913 13.2396 16.3444 20.3372 24.9348 22 8.64272 9.54249 10.9823 12.3380 14.0415 17.2396 21.3370 26.0393 23 9.26042 10.19567 11.6885 13.0905 14.8479 18.1373 22.3369 27.1413 24 9.88623 10.8564 12.4011 13.8484 15.6587 19.0372 23.3367 28.2412 25 10.5197 11.5240 13.1197 14.6114 16.4734 19.9393 24.3366 29.3389 26 11.1603 12.1981 13.8439 15.3791 17.2919 20.8434 25.3364 30.4345 27 11.8076 12.8786 14.5733 16.1513 18.1138 21.7494 26.3363 31.5284 28 12.4613 13.5648 15.3079 16.9279 18.9392 22.6572 27.3363 32.6205 29 13.1211 14.2565 16.0471 17.7083 19.7677 23.5666 28.3362 33.7109 30 13.7867 14.9535 16.7908 18.4926 20.5992 24.4776 29.3360 34.7998 40 20.7065 22.1643 24.4331 26.5093 29.0505 33.6603 39.3354 45.6160 50 27.9907 29.7067 32.3574 34.7642 37.6886 42.9421 49.3349 56.3336 60 35.5346 37.4848 40.4817 43.1879 46.4589 52.2938 59.3347 66.9814 70 43.2752 45.4418 48.7576 51.7393 55.3290 61.6983 69.3344 77.5766 80 51.1720 53.5400 57.1532 60.3915 64.2778 71.1445 79.3343 88.1303 90 59.1963 61.7541 65.6466 69.1260 73.2912 80.6247 89.3342 98.6499 100 67.3276 70.0648 74.2219 77.9295 82.3581 90.1332 99.3341 109.141
00 IX = [2 v/2 r(vl2)]-1 J e-1/2 t,oI2-1 dt X2 From E. S. Pearson, H. O. Hartley (eds.): Biometrika tables for statisticians, vol. I (Cambridge Univ. Press, Cambridge, England, 1954) (with permission). 412 Appendix B
Appendix B (cont.)
v\O( 0.1 0.05 0.025 O.ot 0.005 0.001 0.0005 0.0001
1 2.70554 3.84146 5.02389 6.63490 7.87944 10.828 12.116 15.137 2 4.60517 5.99147 7.37776 9.21034 10.5966 13.816 15.202 18.421 3 6.25139 7.81473 9.34840 11.3449 12.8381 16.266 17.730 21.108 4 7.77944 9.48773 11.1433 13.2767 14.8602 18.467 19.997 23.513 5 9.23635 11.0705 12.8325 15.0863 16.7496 20.515 22.105 25.745 6 10.6446 12.5916 14.4494 16.8119 18.5476 22.458 24.103 27.856 7 12.0170 14.0671 16.0128 18.4753 20.2777 24.322 26.018 29.877 8 13.3616 15.5073 17.5346 20.0902 21.9550 26.125 27.868 31.828 9 14.6837 16.9190 19.0228 21.6660 23.5893 27.877 29.666 33.720 10 15.9871 18.3070 20.4831 23.2093 25.1882 29.588 31.420 35.564 11 17.2750 19.6751 21.9200 24.7250 26.7569 31.264 33.137 37.367 12 18.5494 21.0261 23.3367 26.2170 28.2995 32.909 34.821 39.134 13 19.8119 22.3621 24.7356 27.6883 29.8194 34.528 36.478 40.871 14 21.0642 23.6848 26.1190 29.1413 31.3193 36.123 38.109 42.579 15 22.3072 24.9958 27.4884 30.5779 32.8013 37.697 39.719 44.263 16 23.5418 26.2962 28.8454 31.9999 34.2672 39.252 41.308 45.925 17 24.7690 27.5871 30.1910 33.4087 35.7185 40.790 42.879 47.566 18 25.9894 28.8693 31.5264 34.8053 37.1564 42.312 44.434 49.189 19 27.2036 30.1435 32.8523 36.1908 38.5822 43.820 45.973 50.796 20 28.4120 31.4104 34.1696 37.5662 39.9968 45.315 47.498 52.386 21 29.6151 32.6705 35.4789 38.9321 41.4010 46.797 49.011 53.962 22 30.8133 33.9244 36.7807 40.2894 42.7956 48.268 50.511 55.525 23 32.0069 35.1725 38.0757 41.6384 44.1813 49.728 52.000 57.075 24 33.1963 36.4151 39.3641 42.9798 45.5585 51.179 53.479 58.613 25 34.3816 37.6525 40.6465 44.3141 46.9278 52.620 54.947 60.140 26 35.5631 38.8852 41.9232 45.6417 48.2899 54.052 56.407 61.657 27 36.7412 40.1133 43.1944 46.9630 49.6449 55.476 57.858 63.164 28 37.9159 41.3372 44.4607 48.2782 50.9933 56.892 59.30,0 64.662 29 39.0875 42.5569 45.7222 49.5879 52.3356 58.302 60.735 66.152 30 40.2560 43.7729 46.9792 50.8922 53.6720 59.703 62.162 67.633 40 51.8050 55.7585 59.3417 63.6907 66.7659 73.402 76.095 82.062 50 63.1671 67.5048 71.4202 76.1539 79.4900 86.661 89.560 95.969 60 74.3970 79.0819 83.2976 88.3794 91.9517 99.607 102.695 109.503 70 85.5271 90.5312 95.0231 100.425 104.215 112.317 115.578 122.755 80 96.5782 101.879 106.629 112.329 116.321 124.839 128.261 135.783 90 107.565 113.145 118.136 124.116 128.299 137.208 140.782 148.627 100 118.498 124.342 129.561 135.807 140.169 149.449 153.167 161.319 Appendix C. Percentage Points of the t-Distribution; Values of t in Terms of A and v [4.12] v\A 0.2 0.5 0.8 0.9 0.95 0.98 0.99 0.995 0.998 0.999 0.9999 0.99999 0.999999
1 0.325 1.000 3.078 6.314 12.706 31.821 63.657 127.321 318.309 636.619 6366.198 63661.977 636619.772 2 0.289 0.816 1.886 2.920 4.303 6.965 9.925 14.089 22.327 31.598 99.992 316.225 999.999 3 0.277 0.765 1.638 2.353 3.182 4.541 5.841 7.453 10.214 12.924 28.000 60.397 130.155 4 0.271 0.741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610 15.544 27.771 49.459 5 0.267 0.727 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869 11.178 17.897 28.477
6 0.265 0.718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959 9.082 13.555 20.047 7 0.263 0.711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408 7.885 11.215 15.764 8 0.262 0.706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041 7.120 9.782 13.257 9 0.261 0.703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781 6.594 8.827 11.637 10 0.260 0.700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587 6.211 8.150 10.516 11 0.260 0.697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437 5.921 7.648 9.702 12 0.259 0.695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318 5.694 7.261 9.085 13 0.259 0.694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221 5.513 6.955 8.604 14 0.258 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140 5.363 6.706 8.218 15 0.258 0.691 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073 5.239 6.502 7.903 16 0.258 0.690 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015 5.134 6.330 7.642 17 0.257 0.689 1.333 1.740 2.110 2.567 2.898 3.223 3.646 3.965 5.044 6.184 7.421 18 0.257 0.688 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922 4.966 6.059 7.232 19 0.257 0.688 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883 4.897 5.949 7.069 20 0.257 0.687 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850 4.837 5.854 6.927 21 0.257 0.6B6 1.323 1.721 2.0BO 2.51B 2.831 3.135 3.527 3.B19 4.7B4 5.769 6.802 22 0.256 0.686 1.321 1.717 2.074 2.50B 2.819 3.119 3.505 3.792 4.736 5.694 6.692 23 0.256 0.6B5 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.76B 4.693 5.627 6.593 24 0.256 0.6B5 1.318 1.711 2.064 2.492 2.797 3.090 3.467 3:745 4.654 5.566 6.504 25 0.256 0.6B4 1.316 1.708 2.060 2.485 2.7B7 3.078 3.450 3.725 4.619 5.511 6.424
26 0.256 0.684 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707 4.5B7 5.461 6.352 "0> 0.256 0.684 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690 4.55B "0 27 5.415 6.2B6 <> 0.256 0.683 1.701 2.048 2.467 2.763 3.047 3.408 3.674 4.530 t:l 2B 1.313 5.373 6.225 p,. 29 0.256 0.683 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659 4.506 5.335 6.170 ~. 0.256 1.697 2.042 2.457 2.750 3.030 4.482 30 0.683 1.310 3.385 3.646 5.299 6.119 n 40 0.255 0.681 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551 4.321 5.053 5.768 60 0.254 0.679 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460 4.169 4.825 5.449 120 0.254 0.677 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373 4.025 4.613 5.158 ao 0.253 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291 3.891 4.417 4.892 "., -w 414 Appendix D
Appendix D. Percentage Points of the F-Distribution; Values of F in Terms of Q, VJ, V2 [4.12]
Q(FIJlI.I'J)=0.5 Wi 1 2 3 4 5 6 8 12 15 20 30 60 ... 1 1.00 1.50 1.71 1.82 1.89 1.94 2.00 2.07 2.09 2.12 2.15 2.17 2.20 2 0.667 1.00 1.13 1.21 1.25 1.28 1.32 1.36 1.38 1.39 1.41 1.43 1.44 3 0.585 0.881 1.00 1.06 1.10 1.13 1.16 1.20 1.21 1.23 1.24 1.25 1.27 4 0.549 0.828 0.941 1.00 1.04 1.06 1.09 1.13 1.14 1.15 1.16 1.18 1.19 5 0.528 0.799 0.907 0.965 1.00 1.02 1.05 1.09 1.10 1.11 1.12 1.14 1.15 6 0.515 0.780 0.886 0.942 0.977 1.00 1.03 1.06 1.07 1.08 1.10 1.11 1.12 7 0.506 0.767 0.871 0.926 0.960 0.983 1.01 1.04 1.05 1.07 1.08 1.09 1.10 8 0.499 0.757 0.860 0.915 0.948 0.971 1.00 1.03 1.04 1.05 1.07 1.08 1.09 9 0.494 0.749 0.852 0.906 0.939 0.962 0.990 1.02 1.03 1.04 1.05 1.07 1.08 10 0.490 0.743 0.845 0.899 0.932 0.954 0.983 1.01 1.02 1.03 1.05 1.06 1.07 11 0.486 0.739 0.840 0.893 0.926 0.948 0.977 1.01 1.02 1.03 1.04 1.05 1.06 12 G.484 0.735 0.835 0.888 0.921 0.943 0.972 1.00 1.01 1.02 1.03 1.05 1.06 13 G.481 0.731 0.832 0.885 0.917 0.939 0.967 0.996 1.01 1.02 1.03 1.04 1.05 14 0.479 0.729 0.828 0.881 0.914 0.936 0.964 0.992 1.00 1.01 1.03 1.04 1.05 15 0.478 0.726 0.826 0.878 0.911 0.933 0.960 0.989 1.00 1.01 1.02 1.03 1.05 16 G.476 0.724 0.823 0.876 0.908 0.930 0.958 0.986 0.997 1.01 1.02 1.03 1.04 17 0.475 0.722 0.821 0.874 0.906 0.928 0.955 0.983 G.995 1.01 1.02 1.03 1.04 18 0.474 0.721 0.819 0.872 0.904 0.926 0.953 G.981 0.992 1.00 1.02 1.03 1.04 19 G.473 0.719 0.818 0.870 0.902 0.924 0.951 0.979 0.990 1.00 1.01 1.02 1.04 20 0.472 0.718 0.816 0.868 0.900 0.922 0.950 0.977 0.989 1.00 1.01 1.02 1.03 21 G.471 0.716 0.815 0.867 0.899 0.921 0.948 0.976 0.987 0.998 1.01 1.02 1.03 22 0.470 0.715 0.814 0.866 0.898 0.919 0.947 0.974 0.986 0.997 1.01 1.02 1.03 23 0.470 0.714 0.813 0.864 0.896 0.918 G.945 0.973 0.984 0.996 1.01 1.02 1.03 24 0.469 0.714 0.812 0.863 0.895 0.917 0.944 0.972 0.983 0.994 1.01 1.02 1.03 25 G.468 0.713 0.811 0.862 0.894 0.916 0.943 0.971 0.982 0.993 1.00 1.02 1.03 26 0.468 0.712 0.810 0.861 0.893 0.915 0.942 0.970 0.981 0.992 1.00 1.01 1.03 27 0.467 0.711 0.809 0.861 0.892 0.914 0.941 0.969 0.980 0.991 1.00 1.01 1.03 28 0.467 0.711 0.808 0.860 0.892 0.913 0.940 0.968 G.979 G.990 1.00 1.01 1.02 29 0.466 0.710 0.808 0.859 0.891 0.912 0.940 0.967 0.978 0.990 1.00 1.01 1.02 30 0.466 0.709 0.807 0.858 0.890 0.912 0.939 0.966 0.978 0.989 1.00 1.01 1.02 40 0.463 0.705 0.802 0.854 0.885 0.907 0.934 0.961 0.972 0.983 0.994 1.01 1.02 60 0.461 0.701 0.798 0.849 0.880 0.901 G.928 0.956 0.967 0.978 0.989 1.00 1.01 120 0.458 0.697 0.793 0.844 0.875 0.896 0.923 0.950 0.961 0.972 0.983 0.994 1.01 0.455 0.693 0.789 0.839 0.870 0.891 0.918 0.945 0.956 0.967 0.978 0.989 1.00 Q(FIl'J..I'J)-0.25 ~\I'J. 1 2 3 4 5 6 8 12 15 20 30 60 1 5.83 7.50 8.20 8.58 8.82 8.98 9.19 9.41 9.49 9.58 9.67 9.76 9.85 2 2.57 3.00 3.15 3.23 3.28 3.31 3.35 3.39 3.41 3.43 3.44 3.46 3.48 3 2.02 2.28 2.36 2.39 2.41 2.42 2.44 2.45 2.46 2.46 2.47 2.47 2.47 4 1.81 2.00 2.05 2.06 2.07 2.08 2.08 2.08 2.08 2.08 2.08 2.08 2.08 5 1.69 1.85 1.88 1.89 1.89 1.89 1.89 1.89 1.89 1.88 1.88 1.87 1.87 6 1.62 1.76 1.78 1.79 1.79 1.78 1.78 1.77 1.76 1.76 1.75 1.74 1.74 7 1.57 1.70 1.72 1.72 1.71 1.71 1.70 1.68 1.68 1.67 1.66 1.65 1.65 8 1.54 1.66 1.67 1.66 1.66 1.65 1.64 1.62 1.62 1.61 1.60 1.59 1.58 9 1.51 1.62 1.63 1.63 1.62 1.61 1.60 1.58 1.57 1.56 1.55 1.54 1.53 10 1.49 1.60 1.60 1.59 1.59 1.58 1.56 1.54 1.53 1.52 1.51 1.50 1.48 11 1.47 1.58 1.58 1.57 1.56 1.55 1.53 1.51 1.50 1.49 1.48 1.47 1.45 12 1.46 1.56 1.56 1.55 1.54 1.53 1.51 1.49 1.48 1.47 1.45 1.44 1.42 13 1.45 1.55 1.55 1.53 1.52 1.51 1.49 1.47 1.46 1.45 1.43 1.42 1.40 14 1.44 1.53 1.53 1.52 1.51 1.50 1.48 1.45 1.44 1.43 1.41 1.40 1.38 15 1.43 1.52 1.52 1.51 1.49 1.48 1.46 1.44 1.43 1.41 1.40 1.38 1.36 16 1.42 1.51 1.51 1.50 1.48 1.47 1.45 1.43 1.41 1.40 1.38 1.36 1.34 17 1.42 1.51 1.50 1.49 1.47 1.46 1.44 1.41 1.40 1.39 1.37 1.35 1.33 18 1.41 1.50 1.49 1.48 1.46 1.45 1.43 1.40 1.39 1.38 1.36 1.34 1.32 19 1.41 1.49 1.49 1.47 1.46 1.44 1.42 1.40 1.38 1.37 1.35 1.33 1.30 20 1.40 1.49 1.48 1.47 1.45 1.44 1.42 1.39 1.37 1.36 1.34 1.32 1.29 21 1.40 1.48 1.48 1.46 1.44 1.43 1.41 1.38 1.37 1.35 1.33 1.31 1.28 22 1.40 1.48 1.47 1.45 1.44 1.42 1.40 1.37 1.36 1.34 1.32 1.30 1.28 23 1.39 1.47 1.47 1.45 1.43 1.42 1.40 1.37 1.35 1.34 1.32 1.30 1.27 24 1.39 1.47 1.46 1.44 1.43 1.41 1.39 1.36 1.35 1.33 1.31 1.29 1.26 25 1.39 1.47 1.46 1.44 1.42 1.41 1.39 1.36 1.34 1.33 1.31 1.28 1.25 26 1.38 1.46 1.45 1.44 1.42 1.41 1.38 1.35 1.34 1.32 1.30 1.28 1.25 27 1.38 1.46 1.45 1.43 1.42 1.40 1.38 1.35 1.33 1.32 1.30 1.27 1.24 28 1.38 1.46 1.45 1.43 1.41 1.40 1.38 1.34 1.33 1.31 1.29 1.27 1.24 29 1.38 1.45 1.45 1.43 1.41 1.40 1.37 1.34 1.32 1.31 1.29 1.26 1.23 30 1.38 1.45 1.44 1.42 1.41 1.39 1.37 1.34 1.32 1.30 1.28 1.26 1.23 40 1.36 1.44 1.42 1.40 1.39 1.37 1.35 1.31 1.30 1.28 1.25 1.22 1.19 60 1.35 1.42 1.41 1.38 1.37 1.35 1.32 1.29 1.27 1.25 1.22 1.19 1.15 120 1.34 1.40 1.39 1.37 1.35 1.33 1.30 1.26 1.24 1.22 1.19 1.16 1.10 1.32 1.39 1.37 1.35 1.33 1.31 1.28 1.24 1.22 1.19 1.16 1.12 1.00 Compiled from E. S. Pearson, H. O. Hartley (eds.): Biometrika tables jor statisticians, vol. I (Cambridge Univ. Press, Cambridge, England, 1954) (with permission). Appendix D 415
Appendix D (cont.)
Q(Flpl,J.\J)~0.1 v2r1 1 2 3 4 5 6 8 12 15 20 30 60 39.86 49.50 53.59 55.83 57.24 58.20 59.44 60.71 61.22 61.74 62.26 62.79 63.33 8.53 9.00 9.16 9.24 9.29 9.33 9.37 9.41 9.42 9.44 9.46 9.47 9.49 5.54 5.46 5.39 5.34 5.31 5.28 5.25 5.22 5.20 5.18 5.17 5.15 5.13 4.54 4.32 4.19 4.11 4.05 4.01 3.95 3.90 3.87 3.84 3.82 3.79 3.76 4.06 3.78 3.62 3.52 3.45 3.40 3.34 3.27 3.24 3.21 3.17 3.14 3.10 6 3.78 3.46 3.29 3.18 3.11 3.05 2.98 2.90 2.87 2.84 2.80 2.76 2.72 7 3.59 3.26 3.07 2.96 2.88 2.83 2.75 2.67 2.63 2.59 2.56 2.51 2.47 8 3.46 3.11 2.92 2.81 2.73 2.67 2.59 2.50 2.46 2.42 2.38 2.34 2.29 9 3.36 3.01 2.81 2.69 2.61 2.55 2.47 2.38 2.34 2.30 2.25 2.21 2.16 10 3.29 2.92 2.73 2.61 2.52 2.46 2.38 2.28 2.24 2.20 2.16 2.11 2.06 11 3.23 2.86 2.66 2.54 2.45 2.39 2.30 2.21 2.17 2.12 2.08 2.03 1.97 12 3.18 2.81 2.61 2.48 2.39 2.33 2.24 2.15 2.10 2.06 2.01 1.96 1.90 13 3.14 2.76 2.56 2.43 2.35 2.28 2.20 2.10 2.05 2.01 1.96 1.90 1.85 14 3.10 2.73 2.52 2.39 2.31 2.24 2.15 2.05 2.01 1.96 1.91 1.86 1.80 15 3.07 2.70 2.49 2.36 2.27 2.21 2.12 2.02 1.97 1.92 1.87 1.82 1.76 16 3.05 2.67 2.46 2.33 2.24 2.18 2.09 1.99 1.94 1.89 1.84 1.78 1.72 17 3.03 2.64 2.44 2.31 2.22 2.15 2.06 1.96 1.91 1.86 1.81 1.75 1.69 18 3.01 2.62 2.42 2.29 2.20 2.13 2.04 1.93 1.89 1.84 1.78 1.72 1.66 19 2.99 2.61 2.40 2.27 2.18 2.11 2.02 1.91 1.86 1.81 1.76 1.70 1.63 20 2.97 2.59 2.38 2.25 2.16 2.09 2.00 1.89 1.84 1.79 1.74 1.68 1.61 21 2.96 2.57 2.36 2.23 2.14 2.08 1.98 1.87 1.83 1.78 1.72 1.66 1.59 22 2.95 2.56 2.35 2.22 2.13 2.06 1.97 1.86 1.81 1.76 1.70 1.64 1.57 23 2.94 2.55 2.34 2.21 2.11 2.05 1.95 1.84 1.80 1.74 1.69 1.62 1.55 24 2.93 2.54 2.33 2.19 2.10 2.04 1.94 1.83 1.78 1.73 1.67 1.61 1.53 25 2.92 2.53 2.32 2.18 2.09 2.02 1.93 1.82 1.77 1.72 1.66 1.59 1.52 26 2.91 2.52 2.31 2.17 2.08 2.01 1.92 1.81 1.76 1.71 1.65 1.58 1.50 27 2.90 2.51 2.30 2.17 2.07 2.00 1.91 1.80 1.75 1.70 1.64 1.57 1.49 2B 2.89 2.50 2.29 2.16 2.06 2.00 1.90 1.79 1.74 1.69 1.63 1.56 1.48 29 2.89 2.50 2.28 2.15 2.06 1.99 1.89 1.78 1.73 1.6B 1.62 1.55 1.47 30 2.B8 2.49 2.28 2.14 2.05 1.98 1.88 1.77 1.72 1.67 1.61 1.54 1.46 40 2.84 2.44 2.23 2.09 2.00 1.93 1.83 1.71 1.66 1.61 1.54 1.47 1.38 60 2.79 2.39 2.18 2.04 1.95 1.87 1.77 1.66 1.60 1.54 1.48 1.40 1.29 120 2.75 2.35 2.13 1.99 1.90 1.82 1.72 1.60 1.55 1.48 1.41 1.32 1.19 2.71 2.30 2.08 1.94 1.85 1.77 1.67 1.55 1.49 1.42 1.34 1.24 1.00
Q(FIV1,v2)~0.05 VN1 1 2 3 4 5 6 8 12 15 20 30 60 1 161.4 199.5 215.7 224.6 230.2 234.0 238.9 243.9 245.9 248.0 250.1 252.2 254.3 2 18.51 19.00 19.16 19.25 19.30 19.33 19.37 19.41 19.43 19.45 19.46 19.48 19.50 3 10.13 9.55 9.28 9.12 9.01 8.94 8.85 8.74 8.70 8.66 8.62 8.57 8.53 4 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.91 5.86 5.80 5.75 5.69 5.63 5 6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.68 4.62 4.56 4.50 4.43 4.% 6 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.00 3.94 3.87 3.81 3.74 3.67 7 5.59 4.74 4.35 4.12 3.97 3.87 3.73 3.57 3.51 3.44 3.38 3.30 3.23 8 5.32 4.46 4.07 3.84 3.69 3.58 3.44 3.28 3.22 3.15 3.08 3.01 2.93 9 5.12 4.26 3.86 3.63 3.48 3.37 3.23 3.07 3.G1 2.94 2.86 2.79 2.71 10 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.91 2.85 2.77 2.70 2.62 2.54 11 4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.79 2.72 2.65 2.57 2.49 2.40 12 4.75 3.89 3.49 3.26 3.11 3.00 2.85 2.69 2.62 2.54 2.47 2.38 2.30 13 4.67 3.81 3.41 3.18 3.03 2.92 2.77 2.60 2.53 2.46 2.38 2.30 2.21 14 4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.53 2.46 2.39 2.31 2.22 2.13 15 4.54 3.68 3.29 3.06 2.90 2.79 2.64 2.48 2.40 2.33 2.25 2.16 2.07 16 4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.42 2.35 2.28 2.19 2.11 2.01 17 4.45 3.59 3.20 2.96 2.81 2.70 2.55 2.38 2.31 2.23 2.15 2.06 1.96 18 4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.34 2.27 2.19 2.11 2.02 1.92 19 4.38 3.52 3.13 2.90 2.74 2.63 2.48 2.31 2.23 2.16 2.07 1.98 1.88 20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.28 2.20 2.12 2.04 1.95 1.84 21 4.32 3.47 3.07 2.84 2.68 2.57 2.42 2.25 2.18 2.10 2.01 1.92 1.81 22 4.30 3.44 3.05 2.82 2.66 2.55 2.40 2.23 2.15 2.07 1.98 1.89 1.78 23 4.28 3.42 3.03 2.80 2.64 2.53 2.37 2.20 2.13 2.05 1.96 1.86 1. 76 24 4.26 3.40 3.01 2.78 2.62 2.51 2.36 2.18 2.11 2.03 1.94 1.84 1.73 25 4.24 3.39 2.99 2.76 2.60 2.49 2.34 2.16 2.09 2.01 1.92 1.82 1.71 26 4.23 3.37 2.98 2.74 2.59 2.47 2.32 2.15 2.07 1.99 1.90 1.80 1.69 27 4.21 3.35 2.96 2.73 2.57 2.46 2.31 2.13 2.06 1.97 1.88 1.79 1.67 28 4.20 3.34 2.95 2.71 2.56 2.45 2.29 2.12 2.04 1.96 1.87 1.77 1.65 29 4.18 3.33 2.93 2.70 2.55 2.43 2.28 2.10 2.03 1.94 1.85 1.75 1.64 30 4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.09 2.01 1.93 1.84 1. 74 1.62 40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.00 1.92 1.84 1.74 1.64 1.51 60 4.00 3.15 2.76 2.53 2.37 2.25 2.10 1.92 1.84 1.75 1.65 1.53 1.39 120 3.92 3.07 2.68 2.45 2.29 2.17 2.02 1.83 1.75 1.66 1.55 1.43 1.25 3.84 3.00 2.60 2.37 2.21 2.10 1.94 1.75 1.67 1.57 1.46 1.32 1.00 416 Appendix D
Appendix D (cont.)
Q(F\V1,J2)=0.025 v2\"1 1 2 3 4 5 6 8 12 15 20 30 60 1 647.8 799.5 864.2 899.6 921.8 937.1 956.7 976.7 984.9 993.1 1001 1010 1018 2 38.51 39.00 39.17 39.25 39.30 39.33 39.37 39.41 39.43 39.45 39.46 39.48 39.50 3 17.44 16.04 15.44 15.10 14.88 14.73 14.54 14.34 14.25 14.17 14.08 13.99 13.90 4 12.22 10.65 9.98 9.60 9.36 9.20 8.98 8.75 8.66 8.56 8.46 8.36 8.26 5 10.01 8.43 7.76 7.39 7.15 6.98 6.76 6.52 6.43 6.33 6.23 6.12 6.02 6 8.81 7.26 6.60 6.23 5.99 5.82 5.60 5.37 5.27 5.17 5.07 4.96 4.85 7 8.07 6.54 5.89 5.52 5.29 5.12 4.90 4.67 4.57 4.47 4.36 4.25 4.14 8 7.57 6.06 5.42 5.05 4.82 4.65 4.43 4.20 4.10 4.00 3.89 3.78 3.67 9 7.21 5.71 5.08 4.72 4.48 4.32 4.10 3.87 3.77 3.67 3.56 3.45 3.33 10 6.94 5.46 4.83 4.47 4.24 4.07 3.85 3.62 3.52 3.42 3.31 3.20 3.08 11 6.72 5.26 4.63 4.28 4.04 3.88 3.66 3.43 3.33 3.23 3.12 3.00 2.88 12 6.55 5.10 4.47 4.12 3.89 3.73 3.51 3.28 3.18 3.07 2.9& 2.85 2.72 13 6.41 4.97 4.35 4.00 3.77 3.60 3.39 3.15 3.05 2.95 2.84 2.72 2.60 14 6.30 4.86 4.24 3.89 3.66 3.50 3.29 3.05 2.95 2.84 2.73 2.61 2.49 15 6.20 4.77 4.15 3.80 3.58 3.41 3.20 2.9& 2.86 2.76 2.64 2.52 2.40 16 6.12 4.69 4.08 3.73 3.50 3.34 3.12 2.89 2.79 2.68 2.57 2.45 2.32 17 6.04 4.62 4.01 3.66 3.44 3.28 3.06 2.82 2.72 2.62 2.50 2.38 2.25 18 5.98 4.56 3.95 3.61 3.38 3.22 3.01 2.77 2.67 2.56 2.44 2.32 2.19 19 5.92 4.51 3.90 3.56 3.33 3.17 2.96 2.72 2.62 2.51 2.39 2.27 2.13 20 5.87 4.46 3.86 3.51 3.29 3.13 2.91 2.68 2.57 2.46 2.35 2.22 2.09 21 5.83 4.42 3.82 3.48 3.25 3.09 2.87 2.64 2.53 2.42 2.31 2.18 2,04 22 5.79 4.38 3.78 3.44 3.22 3.05 2.84 2.60 2.50 2.39 2.27 2.14 2.00 23 5.75 4.35 3.75 3.41 3.18 3.02 2.81 2.57 2.47 2.36 2.24 2.11 1.97 24 5.72 4.32 3.72 3.38 3.15 2.99 2.78 2.54 2.44 2.33 2.21 2.08 1.94 25 5.69 4.29 3.69 3.35 3.13 2.97 2.75 2.51 2.41 2.30 2.18 2.05 1.91 26 5.66 4.27 3.67 3.33 3.10 2.94 2.73 2.49 2.39 2.28 2.16 2.03 1.88 27 5.63 4.24 3.65 3.31 3.08 2.92 2.71 2.47 2.36 2.25 2.13 2.00 1.85 28 5.61 4.22 3.63 3.29 3.06 2.90 2.69 2.45 2.34 2.23 2.11 1.98 1.83 29 5.59 4.20 3.61 3.27 3.04 2.88 2.67 2.43 2.32 2.21 2.09 1.9& 1.81 30 5.57 4.18 3.59 3.25 3.03 2.87 2.65 2.41 2.31 2.20 2.07 1.94 1.79 40 5.42 4.05 3.46 3.13 2.90 2.74 2.53 2.29 2.18 2.07 1.94 1.80 1.64 60 5.29 3.93 3.34 3.01 2.79 2.63 2.41 2.17 2.06 1.94 1.82 1.67 1.48 120 5.15 3.80 3.23 2.89 2.67 2.52 2.30 2.05 1.94 1.82 1.69 1.53 1.31 5.02 3.69 3.12 2.79 2.57 2.41 2.19 1.94 1.83 1.71 1.57 1.39 1.00 Q(F\v,yz) =0.01 V2\"1 1 2 3 4 5 6 8 12 15 20 30 60 1 4052 4999.5 5403 5625 5764 5859 5982 6106 6157 6209 6261 6313 6366 2 98.50 99.00 99.17 99.25 99.30 99.33 99.37 99.42 99.43 99.45 99.47 99.48 99.50 3 34.12 30.82 29.46 28.71 28.24 27.91 27.49 27.05 26.87 26.69 26.50 26.32 26.13 4 21.20 18.00 16.69 15.98 15.52 15.21 14.80 14.37 14.20 14.02 13.84 13.65 13.46 5 16.26 13.27 12,06 11.39 10.97 10.67 10.29 9.89 9.72 9.55 9.38 9.20 9.02 6 13.75 10.92 9.78 9.15 8.75 8.47 8.10 7.72 7.56 7.40 7.23 7.06 6.88 7 12.25 9.55 8.45 7.85 7.46 7.19 6.84 6.47 6.31 6.16 5.99 5.82 5.65 8 11.26 8.65 7.59 7.01 6.63 6.37 6.03 5.67 5.52 5.36 5.20 5.03 4.86 9 10.56 8.02 6.99 6.42 6.06 5.80 5.47 5.11 4.96 4.81 4.65 4.48 4.31 10 10.04 7.56 6.55 5.99 5.64 5.39 5.06 4.71 4.56 4.41 4.25 4.08 3.91 11 9.65 7.21 6.22 5.67 5.32 5.07 4.74 4.40 4.25 4.10 3.94 3.78 3.60 12 9.33 6.93 5.95 5.41 5.06 4.82 4.50 4.16 4.01 3.86 3.70 3.54 3.36 13 9.07 6.70 5.74 5.21 4.86 4.62 4.30 3.96 3.82 3.66 3.51 3.34 3.17 14 8.86 6.51 5.56 5.04 4.69 4.46 4.14 3.80 3.66 3.51 3.35 3.18 3.00 15 8.68 6.36 5.42 4.89 4.56 4.32 4.00 3.67 3.52 3.37 3.21 3.05 2.87 16 8.53 6.23 5.29 4.77 4.44 4.20 3.89 3.55 3.41 3.26 3.10 2.93 2.75 17 8.40 6.11 5.18 4.67 4.34 4.10 3.79 3.46 3.31 3.16 3.00 2.83 2.65 18 1l.29 6.01 5.09 4.58 4.25 4.01 3.71 3.37 3.23 3.08 2.92 2.75 2.57 19 8.18 5.93 5.01 4.50 4.17 3.94 3.63 3.30 3.15 3.00 2.84 2.67 2.49 20 8.10 5.85 4.94 4.43 4.10 3.87 3.56 3.23 3.09 2.94 2.78 2.61 2.42 21 8.02 5.78 4.87 4.37 4.04 3.81 3.51 3.17 3.03 2.88 2.72 2.55 2.36 22 7.95 5.72 4.82 4.31 3.99 3.76 3.45 3.12 2.98 2.83 2.67 2.50 2.31 23 7.88 5.66 4.76 4.26 3.94 3.71 3.41 3.07 2.93 2.78 2.62 2.45 2.26 24 7.82 5.61 4.72 4.22 3.90 3.67 3.36 3.03 2.89 2.74 2.58 2.40 2.21 25 7.77 5.57 4.68 4.18 3.85 3.63 3.32 2.99 2.85 2.70 2.54 2.36 2.17 26 7.72 5.53 4.64 4.14 3.82 3.59 3.29 2.96 2.81 2.66 2.50 2.33 2.13 27 7.68 5.49 4.60 4.11 3.78 3.56 3.26 2.93 2.78 2.63 2.47 2.29 2.10 28 7.64 5.45 4.57 4.07 3.75 3.53 3.23 2.90 2.75 2.60 2.44 2.26 2.06 29 7.60 5.42 4.54 4.04 3.73 3.50 3.20 2.87 2.73 2.57 2.41 2.23 2.03 30 7.56 5.39 4.51 4.02 3.70 3.47 3.17 2.84 2.70 2.55 2.39 2.21 2.01 40 7.31 5.18 4.31 3.83 3.51 3.29 2.99 2.66 2.52 2.37 2.20 2.02 1.80 60 7.08 4.98 4.13 3.65 3.34 3.12 2.82 2.50 2.35 2.20 2.03 1.84 1.60 120 6.85 4.79 3.95 3.48 3.17 2.96 2.66 2.34 2.19 2.03 1.86 1.66 1.38 6.63 4.61 3.78 3.32 3.02 2.80 2.51 2.18 2.04 1.88 1.70 1.47 1.00 Appendix E. A Crib Sheet of Statistical Parameters and Their Errors
This summary is meant as a quick aid in attacking practical problems of statistical estimation. All but one of the topics (error in correlation r) are derived and developed in the text, along with examples of their use.
To be estimated Sampling expression Rms error Assumptions about data or relative error
Mean
Weighted mean .x:= I,w n xnlI, Wn a" = a VLw~/(L wn)2 {xn -
Regression & = ([X7] [X])-I [X]T Y Covariance of errors {Yn - (Yn)} are i.i.d. Gaussian, '1:)» '1:) coefficients {()(n} [X] is r~= 0'2 ([XT] [X])-I 0'2. .. where the matrix same variance Means (Yn) are ::s of factors, y is the data not necessarily equal. ~ tt1 Principal component Solution to eigenvalue 0')./ An = V 2/(N - I) [C] is the sample covariance eigenvalues {An} equation [C] Un = AnUn matrix. Errors L1C (i,j) are small and multivariate normal.
* In forming Sr, S~use N- 1 in place of (N - 1)-1 Appendix F. Synopsis of Statistical Tests
The aim of a statistical test is to arrive at a conclusion about some data, with a certain level of confidence. The mechanism is rejection of an initial hypothesis, on the grounds that it was simply too improbable to be true. The test hypothesis is therefore chosen to be the opposite of what it is desired to conclude, oddly enough. The following tests are derived, developed and applied in the text.
Name Aim or Test Test Statistic Assumption How to use Table Hypothesis about data
Student To infer the spread * to= VN -I Llx/So {x n} are i.i.d. Use Appendix C. For t-test ± Llx about the sam- normal, given row v = N - I,
pie mean ~\'of prob- a unknown. find column for A = 0.95 able values of the (say), for confidence true mean
Snedecor To test two sets of * 10 ==Sri S~.Make/o > I by proper As preceding. Use Appendix D. Each F-test data {xn}, Lvn} for naming of the two data sets. table is identified by its )- the hypothesis that Q or confidence level for g"0 they have the same rejecting the hypothesis 0- variance, based upon of equality. For given ~. "Ij an observed ratio table Q, find the tabu• for their sample lated Ffor VI ==NI -I variances. and V2 ==N 2 - I. The table whose F approxi• mates/o most closely identifies the confidence level Q. M Chi-square To test a set ofoc- X5 ==L (mi-N PYIN Pi The N trials are Use Appendix B. For a test curances {m;} for i= I independent and given row V ==M -I, consistency with a N ~ 4, so that find the column whose chance hypothesis the central limit tabulated X2 comes {Pi}' Nonconsistency theorem holds closest to the computed means "significance." for each RV mi. X5 at the left. The column identifies the confidence level (X for rejecting the hypothesis of chance {Pi}'
* In forming the S2 quantities, use N- I in place of (N _1)-1 Appendix F (cont.)
Name Aim or Test Test Statistic Assumption How to use Table Hypothesis about data
_ N - M - I (el - e) R-test To test a regression R 0-- K -- e Data {Yn} suffer Use Appendix D, with model for the possi- additive noise VI ==K and bility that its last {un}, where V2== N - M - 1. Each K factors are in- Un= N (0,0"2). table is identified by its significant. Q or confidence level for rejecting the hypothesis of insignificance of the factors. For a given table Q, find the tabulated F for the given VI, V2.The table whose F most close- ly approximates Ro iden- tifies the required con- fidence level Q. ( ,.2 ) 1/2 Correlation To test two data to ==VN - 2 --2 {XII}' {Yn} are Use Appendix C. For test samples {xn}, {Yn} for 1-,. i.i.d. joint nor- given row V ==N - 2, find any degree of correla- mal, with un- the column whose tabu- >- "0 tion. The hypothesis known correla- lated item is closest to "0 <> is that RV's {x,,}, {y,,} tion (subject of computed to at the left. =:; &. do not correlate, even the test). Column value A is one >< though their sample minus the confidence 'T1 correlation coefficient level for rejecting the
r is finite. .j:>. hypothesis of no corre- tv lation...... References
Chapter 1 l.l S. Brandt: Statistical and Computational Methods in Data Analysis, 2nd rev. ed. (North-Holland, Amsterdam 1976) 1.2 A. Papoulis: Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York 1965) 1.3 E. Parzen: Modern Probability Theory and Its Applications (Wiley, New York 1966) 1.4 B. Saleh: Photoelectron Statistics, Springer Series in Optical Sciences, Vol. 6 (Sprin• ger, Berlin, Heidelberg, New York 1978) 1.5 E. L. O'Neill: Introduction to Statistical Optics (Addison-Wesley, Reading, MA 1963)
Chapter 2 2.1 A. Kolmogorov: Foundations of the Theory of Probability (Chelsea, New York 1950); Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Berlin, Heidelberg, New York 1977) 2.2 A. Papoulis: Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York 1965) 2.3 C. E. Shannon: Bell Syst. Tech. 1. 27,379-423,623-656 (1948) 2.4 R. A. Fisher: Proc. Cambridge Philos. Soc. 22,700-725 (1925) 2.5 R. V. L. Hartley: Bell Syst. Tech. J. 7, 535 (1928) 2.6 w. H. Richardson: J Opt. Soc. Am. 62, 55-59 (1972) 2.7 R. A. Howard: Dynamic Probability Systems (Wiley, New York 1971)
Additional Reading Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. I (Wiley, New York 1966) Moran, P. A. P.: An Introduction to Probability Theory (Clarendon, Oxford 1968) Parzen, E.: Modern Probability Theory and Its Applications (Wiley, New York 1966) Reza, F.: An Introduction to Information Theory (McGraw-Hill, New York 1961) Yu, F. T. S.: Optics and Information Theory (Wiley, New York 1976)
Chapter 3 3.1 E. L. O'Neill: Introduction to Statistical Optics (Addison-Wesley, Reading, MA 1963) 3.2 C. Dainty (ed.): Laser Speckle and Related Phenomena, 2nd ed., Topics in Applied Physis, Vol. 9 (Springer, Berlin, Heidelberg, New York 1982) 3.3 G. N. Plass et al.: Appl. Opt. 16,643-653 (1977) 3.4 C. Cox, W. Munk: 1. Opt. Soc. Am. 44,838-850 (1954) 3.5 1. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 3.6 B. R. Frieden: 1. Opt. Soc. Am. 57, 56 (1967) 3.7 S. Q. Duntley: 1. Opt. Soc. Am. 53,214-233 (1963) 3.8 D. M. Green, 1. A. Swets: Signal Detection Theory and Psychophysics (Wiley, New York 1966) 424 References
3.9 J. A. Swets (ed.): Signal Detection and Recognition by Human Observers (Wiley, New York 1964) 3.10 C. H. Slump, H. A. Ferwerda: Optik 62, 98 (1982)
Additional Reading Clarke, L. E.: Random Variables (Longman, New York 1975) Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York 1966) Papoulis, A: Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York 1965) Pfeiffer, R. E.: Concepts of Probability Theory, 2nd rev. ed. (Dover, New York 1978)
Chapter 4 4.1 R. M. Bracewell: The Fourier Transform and Its Applications (McGraw-Hill, New York 1965) 4.2 J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 4.3 B. R. Frieden: In Picture ProceSSing and Digital Filtering, 2nd ed., ed. by T. S. Huang, Topics Appl. Phys., Vol. 6 (Springer, Berlin, Heidelberg, New York 1975) 4.4 A Einstein: Ann. Phys. 17, 549 (1905) 4.5 R. Barakat: Opt. Acta 21,903 (1974) 4.6 E. L. O'Neill: Introduction to Statistical Optics (Addison-Wesley, Reading, MA 1963) 4.7 B. Tatian: J. Opt. Soc. Am. 55, 1014 (1965) 4.8 H. Lass: Elements of Pure and Applied Mathematics (McGraw-Hill, New York 1957) 4.9 E. Parzen: Modern Probability Theory and Its Applications (Wiley, New York 1966) 4.10 A G. Fox, T. Li: Bell Syst. Tech. J. 40,453 (1961) 4.11 R. W. Lee, J. C. Harp: Proc. IEEE 57, 375 (1969) 4.12 M. Abramowitz, I. A Stegun (eds.): Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC 1964) 4.13 J. W. Strohbehn: In Laser Beam Propagation in the Atmosphere, ed. J. W. Strohbehn, Topics in Applied Physics, Vol. 25 (Springer, Berlin, Heidelberg, New York 1978)
Additional Reading Araujo, A, E. Gine': The Central Limit Theoremfor Real and Banach Valued Random Vari• ables (Wiley, New York 1980) Lukacs, E.: Characteristic Functions, 2nd ed. (Griffin, London 1970) Moran, P. A P.: An Introduction to Probability Theory (Clarendon, Oxford 1968)
Chapter 5 5.1 L. I. Goldfischer: 1. Opt. Soc. Am. 55,247 (1965) 5.2 J. C. Dainty: Opt. Acta 17,761 (1970) 5.3 D. Korff: Opt. Commun. 5, 188 (1972) 5.4 J. C. Dainty (ed.): Laser Speckle and Related Phenomena, 2nd ed., Topics in Applied Physics, Vol. 9 (Springer, Berlin, Heidelberg, New York 1982) 5.5 J. W. Goodman: Proc. IEEE 53,1688 (1965) 5.6 K. Miyamoto: "Wave Optics and Geometrical Optics in Optical Design," in Progress in Optics, Vol. I, ed. by E. Wolf (North-Holland, Amsterdam 1961) 5.7 J. J. Burke: J. Opt. Soc. Am. 60, 1262 (1970) 5.8 W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 2 (Wiley, New York 1966) p. 50 References 425
5.9 J. W. Strohbehn: In Laser Beam Propagation in the Atmosphere, ed. 1. W. Strohbehn, Topics in Applied Physics, Vol. 25 (Springer, Berlin, Heidelberg, New York 1978) 5.10 R. L. Phillips and L. C. Andrews: J. Opt. Soc. Am. 72,864 (1982)
Additional Reading See the readings for Chapters 2 and 3
Chapter 6 6.1 E. L. O'Neill: Introduction to Statistical Optics (Addison-Wesley, Reading, MA 1963) 6.2 J. R. Klauder, E. C. G. Sudarshan: Fundamentals of Quantum Optics (Benjamin, New York 1968) 6.3 M. Born, E. Wolf: Principles of Optics (Macmillan, New York 1959) 6.4 R. Hanbury-Brown, R. Q. Twiss: Proc. R. Soc. London A 248,20 I (1958) 6.5 R. E. Burgess: Disc. Faraday Soc. 28, 151 (1959) 6.6 R. Shaw, J. C. Dainty: Image Science (Academic, New York 1974) 6.7 S. Chandrasekhar: Rev. Mod. Phys. 15, I (1943)
Additional Reading Barrett, H. H., W. Swindell: Radiological Imaging, Vol. I (Academic, New York 1981) Chow, Y. S., H. Teicher: Probability Theory. Independence, Interchangeability, Martingales (Springer, Berlin, Heidelbe(g, New York 1978) Goodman,1. W: Statistical Optics (Wiley, New York, 1985) Moran, P. A. P.: An Introduction to Probability Theory (Clarendon, Oxford 1968) Saleh, B. E. A., M. C. Teich: Proc. IEEE 70,229 (1982) Thomas, 1. B.: An Introduction to Statistical Communication Theory (Wiley, New York 1969)
Chapter 7 7.1 1. J. DePalma, J. Gasper: Phot. Sci. Eng. 16,181 (1972) 7.2 G. N. Plass, G. W. Kattawar, 1. A. Guinn: Appl. Opt. 14,1924 (1975) 7.3 J. C. Dainty (ed.): Laser Speckle and Related Phenomena, 2nd ed., Topics Appl. Phys. Vol. 9 (Springer, Berlin, Heidelberg, New York 1982) 7.4 H. Fujii, J. Vozomi, T. Askura: J. Opt. Soc. Am. 66, 1222 (1976) 7.5 R. C. Gonzalez, P. Wintz: Digital Image Processing (Addison-Wesley, Reading, MA 1977) 7.6 B. R. Frieden: Appl. Opt. 4, 1400 (1965)
Additional Reading Binder, K. (ed.): Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7 (Springer, Berlin, Heidelberg, New York 1978) Brandt, S.: Statistical and Computational Methods in Data Analysis, 2nd rev. ed. (North• Holland, New York 1976) Marchuk, G. 1., G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, B. S. Elepov: Monte Carlo Methods in Atmospheric Optics, Springer Series in Optical Sciences, Vol. 12 (Springer, Berlin, Heidelberg, New York 1979) Pratt, W. K.: Digital Image Processing (Wiley, New York 1978)
Chapter 8 8.1 R. E. Hufnagel, N. R. Stanley: J. Opt. Soc. Am. 54,52 (1964) 8.2 P. Hariharan: App\. Opt. 9, 1482 (1970) 426 References
8.3 1. C. Dainty (ed.): Laser Speckle and Related Phenomena, 2nd ed., Topics Appl. Phys. Vol. 9 (Springer, Berlin, Heidelberg, New York 1982) 8.4 A M. Schneiderman, P. F. Kellen, M. G. Miller: 1. Opt. Soc. Am. 65, 1287 (1975) 8.5 a. D. Korff: 1. Opt. Soc. Am. 63,971 (1973) 8.5b. D. Korff: Opt. Commun. 5, 188 (1972) 8.6 K. T. Knox: 1. Opt. Soc. Am. 66, 1236 (1976) 8.7 A Papoulis: Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York 1965) 8.8 C. W. Helstrom: 1. Opt. Soc. Am. 57,297 (1967) 8.9 1. L. Harris: In Evaluation of Motion-Degraded Images. NASA Spec. Publ. 193, 131 (1968) 8.10 P. B. Fellgett, E. H. Linfoot: Phil os. Trans. R. Soc. London A 247,369 (1955) 8.11 B. R. Frieden: 1. Opt. Soc. Am. 60, 575 (1970) 8.12 1. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968)
Additional Reading Barrett, H. H., W. Swindell: Radiological Imaging, Vols. 1 and 2 (Academic, New York 1981) Blanc-Lapierre, A, R. Fortet: Theory of Random Functions, Vol. I (Gordon and Breach, New York 1967) Cramer, H., M. R. Leadbetter: Stationary and Related Stochastic Processes (Wiley, New York 1967) Thomas, 1. B.: An Introduction to Statistical Communication Theory (Wiley, New York 1969) Wax, N. (ed.): Selected Papers on Noise and Stochastic Processes (Dover, New York 1954) The journal Stochastic Processes and their Application (North-Holland, Amsterdam 1973-) for reference on specific problems
Chapter 9 9.1 B. R. Frieden: In Picture Processing and Digital Filtering, 2nd ed., ed. by T. S. Huang, Topics Appl. Phys., Vol. 6 (Springer, Berlin, Heidelberg, New York 1979) 9.2 P. G. Hoel: Introduction to Mathematical Statistics (Wiley, New York 1956) pp.205-207 9.3 1. S. Gradshteyn, I. M. Ryzhik: Tables of Integrals, Series, and Products (Academic, New York 1965) p. 337; p. 284 9.4 B. R. Frieden: 1. Opt. Soc. Am. 64,682 (1974)
Additional Reading Justusson, B. I.: In Two-Dimensional Digital Signal Processing II: Transforms and Median Filters ed. by T. S. Huang, Topics in Applied Physics, Vol. 43 (Springer, Berlin, Hei• delberg, New York 1981) Kendall, M. G., A Stuart: The Advanced Theory of Statistics, Vol. I (Griffin, London 1969) Mises, R. von: Mathematical Theory of Probability and Statistics (Academic, New York 1964)
Chapter 10 10.1 E. T. Jaynes: IEEE Trans. SSC-4,227 (1968) 10.2 E. L. Kosarev: Comput. Phys. Commun. 20,69 (1980) 10.3 J. MacQueen, J. Marschak: Proc. Natl. Acad. Sci. USA 72,3819 (1975) 10.4 R. Kikuchi, B. H. Soffer: 1. Opt. Soc. Am. 67, 1656 (1977) 10.5 J. P. Burg: "Maximum entropy spectral analysis," 37th Annual Soc. Exploration Geophysicists Meeting, Oklahoma City, 1967 References 427
10.6 S. J. Wernecke, L. R. D'Addario: IEEE Trans. C-26, 352 (1977) 10.7 B. R. Frieden: Proc. IEEE 73,1764 (1985) 10.8 S. Goldman: Information Theory (Prentice-Hall, New York 1953) 10.9 A. Papoulis: Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York 1965) 10.10 D. Marcuse: Engineering Quantum Electrodynamics (Harcourt, Brace and World, New York 1970) 10.11 L. Mandel, E. Wolf: Rev. Mod. Phys. 37,231 (1965)
Additional Reading Baierlein, R.: Atoms and Information Theory (Freeman, San Francisco 1971) Boyd, D. W., J. M. Steele: Ann. Statist. 6,932 (1978) Brillouin, L.: Science and Information Theory (Academic, New York 1962) Frieden, B. R.: Compo Graph. Image Proc. 12,40 (1980) Kullbach, S.: Information Theory and Statistics (Wiley, New York 1959) Sklansky, J., G. N. Wassel: Pattern Classifiers and Trainable Machines (Springer, Berlin, Heidelberg, New York 1981) Tapia, R. A., J. R. Thompson: Nonparametric Probability Density Estimation (Johns Hop• kins University Press, Baltimore 1978)
Chapter 11 11.1 A. Papoulis: Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York 1965) 11.2 M. Abramowitz, I. A. Stegun (eds.): Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC 1964) 11.3 F. B. Hildebrand: Introduction to Numerical Analysis (McGraw-Hill, New York 1956) 11.4 G. L. Wied, G. F. Bahr, P. H. Bartels: In Automated Cell Identification and Cell Sort- ing, ed. by G. L. Wied, G. F. Bahr (Academic, New York 1970) pp. 195-360 Additional Reading Breiman, L.: Statistics: With a View Toward Applications (Houghton-Mifflin, Boston 1973) Hogg, R. v., A. T. Craig: Introduction to Mathematical Statistics, 3rd ed. (Macmillan, Lon- don 1970) Kendall, M. G., A. Stuart: The Advanced Theory of Statistics, Vol. 2 (Griffin, London 1969) Mises, R. von: Mathematical Theory of Probability and Statistics (Academic, New York 1964)
Chapter 12 12.1 Student: Biometrika 6, 1 (1908) 12.2 M. Abramowitz, I. A. Stegun (eds.): Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC 1964) 12.3 C. Stein: Proceedings Third Berkeley Symposium on Mathematics, Statistics, and Probability (University of California Press, Berkeley, CA 1955) p. 197
Additional Reading Brandt, S.: Statistical and Computational Methods in Data Analysis, 2nd. rev. ed. (North• Holland, New York 1976) Hogg, R. V., A. T. Craig: Introduction to Mathematical StatisticS, 3rd ed. (Macmillan, Lon• don 1970) Mendenhall, W., R. L. Scheaffer: Mathematical Statistics with Applications (Duxbury, North Scituate, MA 1973) Kendall, M. G., A. Stuart: The Advanced Theory of Statistics, Vol. 2 (Griffin, London 1969) 428 References
Chapter 13 13.1 F. Snedecor: Statistical Methods, 5th ed. (Iowa State University Press, Ames 1962) 13.2 M. Abramowitz, I. A Stegun (Eds.): Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC 1964) 13.3 P. H. Bartels, J. A Subach: "Significance Probability Mapping, and Automated Inter• pretation of Complex Pictorial Scenes," in Digital Processing of Biomedical Images, ed. by K. Preston, M. Onoe (Plenum, New York 1976) pp. 101-114 Additional Reading Brandt, S.: Statistical and Computational Methods in Data Analysis, 2nd rev. ed. (North• Holland, New York 1976) Breiman, L.: Statistics: With a View Toward Applications (Houghton-Mifflin, Boston 1973) Kempthome, 0., L. Folks: Probability, Statistics, and Data Analysis (Iowa State University Press, Ames 1971)
Chapter 14 14.1 L. Breiman: Statistics: With a View Toward Applications (Houghton Mifflin, Boston 1973) Chap. 10 14.2 M. G. Kendall, A Stuart: The Advanced Theory of Statistics, Vol. I (Charles Griffin, London 1969) Additional Reading Bjerhammar, A: Theory of Errors and Generalized Matrix Inverses (Elsevier Scientific, Am• sterdam 1973) Mendenhall, W., R. L. Scheaffer: Mathematical Statistics with Applications (Duxbury, North Scituate, MA 1973)
Chapter 15 15.1 H. Hotelling: J. Educ. Psych. 24,417,498 (1933) 15.2 J. L. Simonds: J. Phot. Sci. Eng. 2,205 (1958) , 15.3 R. C. Gonzalez, P. Wintz: Digital Image Processing (Addison-Wesley, Reading, MA 1977) pp. 310-314 15.4 M. G. Kendall, A Stuart: The Advanced Theory of Statistics, Vol. 3 (Charles Griffin, London 1968) 15.5 M. A Girshick: Ann. Math. Statist. 10,203 (1939) Additional Reading Wilkinson, J. H.: The Algebraic Eigenvalue Problem (Clarendon, Oxford 1965)
Chapter 16 16.1 J. Weber: Historical Aspects of the Bayesian Controversy (University of Arizona, Tuc• son 1973) 16.2 A Papoulis: Probability, Random Variables and Stochastic Processes (McGraw-Hili, New York 1965) 16.3 E. Parzen: Modern Probability Theory and Its Applications (Wiley, New York 1960) 16.4 B. R. Frieden: Found. Phys. 16, 883 (1986) Additional Reading Good, I. J.: "The Bayesian Influence, or How to Sweep Subjectivism Under the Carpet," in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, Vol. II, ed. by W. L. Harper, C. A Hooker (Reidel, Boston 1976) Jaynes, E. T.: "Confidence Intervals vs Bayesian Intervals", in preceding reference References 429
Chapter 17
17.1 B. R. Frieden: Phys. Rev. A41, 4265 (1990) 17.2 H. L. Van Trees: Detection, Estimation and Modulation Theory, Part I (Wiley, New York 1968) 17.3 B. R. Frieden: Appl. Opt. 26,1755 (1987) 17.4 R. Kikuchi, B. H. Soffer: 1. Opt. Soc. Am. 67, 1656 (1977) 17.5 B. R. Frieden: Am. 1. Phys. 57, 1004 (1989) 17.6 B. R. Frieden: Opt. Lett. 14, 199 (1989) 17.7 T. D. Milster et at.: 1. Nucl. Med. 31, 632 (1990) 17.8 A. 1. Starn: Inform. and Control 2, 101 (1959) Subject Index
Absorptance 162,165,168,174, of pupil 63, 184 337-338,341-342 of spread function 130 Additivity property triangular 195 disturbance functions 218 zero-length 195 probabilities 11,39,135 Axioms of Kolmogoroff 7, 10, 11 infonnation 24 Airline problem 14,22,32,136,137 "bag of coins" paradigm 365,370 Alchemy 3 Ballistic galvanometer problem 128 Alphabet, probability of letters 286 Band-limited function 86 "and," intuitive use 7, 12,27 Bayes' probability law Ankle bones, first dice 3 see Probability law Aperture 314,315 Bayes'rule 19,27-29,290,364,365, averaging, in speckle 112, 116, 369 258,307 Bayes'theorem 367 quantum d.oJ. in 283 Bayesian estimation 264,363-372, size, for accurate transmittance 383-399 242-243,261,308 Beam transformation 175 Apodization 238 Bernoulli trials 134-161,239,243,257, Arithmetic average 78, 234 269,288,365 Astrology 3 Beta function 258 Astronomy, optical 190,282 Bias binary star application 191 see also Estimators, biased/unbiased M.E. restoration of galaxy 284-285 definition 289 median window use 256 in estimate 251,253,270,272- which star is larger? 326 274,282,284,373 Astronomy, radio 64,65,285,317 experimental 237 intensity interferometry 146 -150 least, aspect of M.E. estimate 279 Atmospheric turbulence 46,53,63, property of normal law 279 72,92,96-98,182,184 unknown 368,370,371 Lee and Harp model 92 values 274,275,365 long-tenn 190,191,197,218 Bicyclist problem 156 multiplicative model 98 Binary phase blob model 192-194 communication link 33, 121, 261 short-term 190,191,197,218 feature extraction 324-325 single scatter model 132 message 21, 121 stochastic model 185, 186 RV 79 superposition model 231-232 trials 134-161,275,289,297, Autocorrelation 19 299,300,365,369 "bo xcar" 19 5 Binomial coefficient 181, 226, 340 coefficient 136 function 181, 205, 225 -226, 268 theorem 73 of image spectrum 197 "bit," unit of infonnation 20 of natural scenes 196 Boltzmann distribution 280, 343 432 Subject Index
Bose factor 58 Classical viewpoint 245,264,363-372 Bose-Einstein degeneracy 275, 283 Coed problem 18 Bose-Einstein probability law Coefficient, regression 329,330,337 see Probability law Coherence time 66, 145, 283, 397 Brownian motion 79, 155, 279 Coherent processor 190 Burgess variance theorem 157 College dropout problem 156 Communication channel 19 Camera, scintillation 393 - 395 laser 120-124 Cascaded "mental" 137-139, ISS events 157-158 photon position 23 optics 90, 96 Communication theory 3,70 Central limit theorem 53,87-94,98, Complex random variable 38 107,108,151,185,219,236,297,310 Computed tomography 29 derivation 87-88 Conditional probability 17-19 in frequency space 89 density 38 generalizations 89 Confidence limits 233,243,264 Chance 3 chi-square test 304 Chance hypothesis 294, 299 for estimated probability 243-245, Channel 3,19,28 364-368 capacity 214,215 for surface quality 249-250 Characteristic function 70, 72, 76, 96, F-test 322 184,186,220,247,270 R-test 336-337 atmospheric speckle 187 t-test 309,314,315,317,319,346 binomial 73, 139 Conviction, user's 272-274 cascaded systems 1 57 -1 58 convolution 68,75,76,112,116,131, Cauchy 80 171,203,221-223,254,396 chi-square 115,116,248 multiple 79, 87, 90, 91 exponential 74 theorem 75,76 Gaussian 74, 76,77 Correlation geometric 80 amplitude 64,65 median window 258 coefficient 55, 149,334,347 n independent RV's 79 in counts 149 negative binomial 293 in experimental outputs 235 Neyman type A 158 in phase 186, 188 normal 74 in real and imaginary parts 204 normal, two-dimensional 75 significance measure 334,345-346, Poisson 72,73 386 sum ofn RV's 81,82,87 speckle 109 sum of two bivariate RV's 82 temporal 147 truncated RV 79 Cost function 374,383-391 two-dimensional 74,75,80 absolute 385, 387, 388 uniform 74 quadratic 385-388 chi-square step 385, 386, 388 probability1aw 112,115-117,126, symmetric convex 388-391 127,247,249,251,252,296 "counter" mechanism 151,297 statistic 295,385 Covariance chi-square test of significance 294-306, coefficient 55 385 matrix 340, 343, 348, 350, 360, for image detection 303 -305 361 for "intelligence" detection 305 Cramer-Rao error for malignancy detection 305-306 bound 378-382,408 one-, two-tail 296,298 Crime rate problem 32 Subject Index 433
Cutoff frequency 64,86,96,193,195, role in prescribed probability law 209 163-165,170 sampled 86 Data from principal components Disturbance function 353-354 see Stochastic process Data sparsity 270,272,279,291,307, 364 see also Evidence Earthquake problem 154 Degeneracy factor Edge "content" 210 Bernoulli 135,288,293 Edge spread function 48, 86, 141, 210, multinomial 270,275,282,283, 254,255 286,291,293 Efficient estimator; see Estimators photon 275,283,286 Eigenvalue equation 348,349 Degrees of freedom (d.o.f.) Karhunen-Loeve 205,268-269 chi-square law 116,117,248,249, Empirical method of physics 166,363 261,298,299,304,311,318, Empirical prior knowledge 320,335,336 see Prior knowledge F-distribution 321 Emulsion see Photographic emulsion t-distribution 312,314,346 Ensemble 178,202 quantum 275, 283, 286, 397 Ensemble average 178,203,205 sampling theory 96 over objects 188 Delta modulation 97 over particle number 222 DeMoivre-Laplace law 150-154,244, over spread functions 190-192 245 over turbulent pupils 185,186 Detection probability 145,155 Entropy see also Estimators Deterministic limit 3,4,46,62, 108, conditional 44,211-213 120,184,192,206,217,220,234 of data 213,214 Deterministic parameter 374 definition 43 Diffraction maximum 44,53,264,272-282, introduction 62-66 284-286 limit 41, 60 of noise 213 particle viewpoint 169 "equal weights" hypothesis 273,274, pattern, estimate of slit position 276,280,281 from 377-378 Equivocation 19 transfer function 63, 64, 186, 193, erf (error function) 194 see Probability law Diffuser, optical 106-109,172,228 Ergodic property 202-203,205 Dimensional reduction 354-355 Error bound, Cramer-Rao 378-382 Dirac delta function 40, 46, 47, 60, 63, Error in estimate 65,66,134,183,207,225 see Variance of error in sample estimate describes random noise 201, 204 Estimates from sample data of describes strong conviction 273 mean 234-239,382 representations 47 median 255-256,383 sifting property 40,46,82, 183, optical object 275,282-285,289 207,225 principal component eigenvalues Dirichlet integral 289 353,358,383 Discrete vs continuous statistics 47 probability law 264-286, 399-408 Distinguishable events 135 regression coefficient 333,383 "distribution-free" property of a classical signal-to-noise ratio 250-254, estimate 236,363 382 Distribution function single probability 382 see also Probability law, cumulative variance 245-249,335,382 definition 48 weighted mean 237-239,382 434 Subject Index
Estimation, Bayesian 363-399 F-test of equal variances 320-326, 385 Estimators Fl, F2 enhancement filters 260 biased 253 Factors Burg 284, 285, 291 orthogonalized 349-350 efficient 381-383 regression 328,330,335,337,343, Karhunen-Loeve 268, 269 345,347,349,350 least-squares(LS) 57:266,327-349, "fair" 354,383,386 estimate 274 maximum a posteriori MAP 34, exam (student's view) 306 272, 388, 391-399 trial 13,14,134,136,152,154, maximum cross entropy 292 273,299,305,371,372 maximum entropy 264,272-293, Film see Photographic emulsion 363, 397 Film identification problem 318, 319 maximum likelihood 233, 240, 264, Finite support 83,85,205,225,287 268,270-293, 331-333, 343, "fixed" coin test 299 364-366, 372, 376-378, 380, 382, Fisher information see Information, 392 Fisher median window 254-256 Fluctuations, stellar minimum cross entropy 275,289 intrinsic 97 minimum entropy 289 turbulence-induced 98,183-188 minimum Fisher information (MFI) Forced choice detection 67-69 399-408 Forecasting problem 33 minimum mean-square error (mmse) Fourier 206-208,210,233 methods 70-98 nonparametric 264 series 83 orthogonal expansion 264-269, Fourier transform 63-66,70,72, 75, 278,279 82,83,85,91,149,167,179,180, parametric 264 188,189,196,206,209,219,225, unbiased 62,234,236,237,245,246, 231 253,259-261, 335, 375-378, finite 178, 179 381-383, 392 theorem 224 weighted Burg 291 Fraunhofer integral 63 weighted least-squares 343 Frequencyoccurrence 10, 15, 18, 26 Euler-Lagrange equation 208,277, Frequency space-representation 403 of Fisher I 407 Events 7,8,12,16-19 Fresnel kernel 91 binary 21 Functions of random variables 99-133 certain 9-12,38 general case 104 complementary 10, 14 multiple root case 103 complex number 32 for prescribed probability law disjoint 9-12,14,15,22,39 163-165 independent 21,22 unique root case 100-102 joint 12-14,17 Functions of random variables, Markov 30-32,35-36 applications space 8, II, 13-15,40 atmospheric turbulence 132, 133 ballistic galvanometer 128 Evidence geometrical spot diagram 117 -120 compelling 291,364,368,372 holography 125, 126 meagre 241,291,364 imaging equation 101-102 Expected value 40 laser link 120-124 Experiment 7, 8, 15 laser ranging 125 Exponential integral (Ei) 67 lens-maker's equation 124 Subject Index 435
object restoration 129 Hurter-Driffield H-D curve photographic H-D law 126 principal components of 350-35 I , probability law for SIN 251-252 355 probability law of student (t-distri- probabilistic origin 159, 160 bution) 311,312 Huyghen's principle 59,63,65, 107, probability law of Snedecor 108,228 (F-distribution) 320,321 verified probabilistically 127, 128 Snell's law 125 Hypothesis see Test hypothesis speckle averaging 11 2-116 square-law detection j 03 "swimming pool" effect 128,129 ''if'' notation 17 Ignorance 3, 308 maximum statistical 273-275,289 Gaussian probability law measured by entropy 278 see Probability law universal state sought 274 Gedanken measurement, experi• i.i.d. random variables 235,245,250, ment 400-401 257,263 Geometrical imaging equation 99, 101 Image Geometrical spot diagrams 117 -120 optical 16-18,27,40,59,62,77, formed 'with arbitrary accuracy 120 90,195,206,211,251,282 Glitter patterns 57 plane 16,17,27,42,43,106,117, Grain sensitization 159 119,120,124,125,129,175, Gram-Charlier expansion 56, 266, 279 184,197 Granularity, photographic 139,308 processing 174,237,238,303-305 checkerboard model 139-141, restoration 62, 77, 129, 197, 203, 226,242-243 206-211,216-218,238,260, layered model 159-160 282-285, 395-399 Monte Carlo model 165-166 simulation 169-171, 173 . poker chip (circular grain) model smoothing 238,254-256,314 226-228 underexposed 174 Guessing, how to correct psychometric Incoherent object 64 curve for 26 Independence, linear vs statistical 335 Independence, statistical 21, 44, 235, 295 Hanbury-Brown Twiss effect 149 Independent messages 21 Hanning filter 260 Indirect method of proof, statistical 294 "hartley," unit of information 33 Indistinguishable events 269, 283 Heisenberg uncertainty 43,60,261 Information, Fisher 19 derivation 408 and lower error bound 379 Helmholtz wave equation, derivation 404 probability law estimation 399-408 Hermite polynomials 56,266 Information, Shannon 3, 19,20 Hermite-Gauss functions 267 for bivariate RV's 60 Histogram 78, 202 channel capacity 214 equalization 174-175 commutativity of 30 in estimating probability law 265, continuous limit 39 267 density 217 in forming prior probability 372 for independent messages 24 in Monte Carlo calculation 162, as limiter of restoration quality 217 171,174-175 in the optical image 211-216 test for malignancy 305-306 self- 24 Holography 125,126,132 trans- 43,211,213,214 Human eye problem 160 vs resolution 216 436 Subject Index
Informationless message 23 Likelihood equation 376, 378, 381 Inner scale of turbulence 180 law 375-378, 380, 382, 391, 397 Inoculation problem 32 Linear processor 206-209 Input 199,211 see also Estimates from sample data Insufficient data 3 see also tradeoff with resolution 239, 254, Data sparsity, Evidence, meagre 258 ''intelligence,'' data test for 305 Linear regression 57,327-349 Intensity interferometry 146-150 Lyapunov's condition 89 Inverse filter 209, 210, 217 Isoplanatism vs statistical stationarity 77 "majority rule" detection strategy 138 MAP equation 391 - 392, 394, 398 Marginal probability 12,18 Jacobian 104,110,111,118,131,132, continuous 38, 54 162 Markov events 30 Joint random variables 41 probability of winning ultimately Joint probability 17, 19, 41 (random walk on finite interval) J oint probability laws 35-36 see Probability law listing product sales prediction 30-32 Matrix 129,335 ' covariance 340 Karhunen-Loeve expansion 205-206, diagonal 340,350 268-269 eigenvalue problem 348,349,351, Kinetic energy, role in MFI principle 402 352 Klein-Gordon equation, derivation 404 inverse 333, 378 Knox-Thompson approach 197, 199 multiplication 339 Kolmogoroff's axioms 7, 10, 11 positive definite 55 Kronecker delta 235 transpose 55, 333 "maximum a posteriori" see MAP Maximum entropy estimate I'H6pital's rule, use 142,263 see Estimators Labeyrie speckle interferometry Maximum entropy, a poem 265 190-194,197 Maximum Entropy Revisited, a poem missing phase problem 191-192 286 resolution limit 192 -194 Maxwell-Boltzmann law, derivation Lady Luck 3 405 Lagrange constraint 210,276,292 Mean value 41 Laplace definition of probability 13 of algebraic expression 48, 49 Laplace rules of succession 368-371 estimate 78 Laser beam of MW output 258 measurement of wave-front variance of nearest-neighbor distances 161 261 of phase in turbulence 97 transformation 175 of a phasor 61 Laser resonator 91 ofa sum 44 Laser speckle see Speckle required, how to attain 93, 94 Lawoflargenumbers 10,15,16-18, t-test on 307-319 26,29,78,118,162,169,239,265, Means, test for equality of 318 291 Median 254 derivation 239-240 Median, sample second 367 see Estimates from sample data Least-squares fit see Estimators Median window (MW) 254-260, 263 Lens system 16 edge preservation 256 Subject Index 437
statistics 257 -260 Newton-Raphson relaxation method use in astronomy 256 164,278 Mental telepathy 137-139 Noise 23,62,93,199-206,255,360 MFI principle 402 additive 178,199-203,206,207, applications 402-408 212,237,330-332,339,340, Minimum mean-square error (mmse) 344 see Estimators boost in output 239,260 Mirror, surface scratches, stochastic entropy 213 model 231 ergodic property 202-203 Models see Statistical models gain in restoration 130 Modified Bessel function 132 Gaussian see Probability law Modulation transfer function (MTF) 189 in least-squares problem 327, offilm 166,167,189 330-332,337,340-342 Moments 41 Johnson 331 central 41 multiplicative 60,337 as data constraints 271 Poisson see Probability law generation 71 random 201-204,210,303,304 multidimensional 74, 75 roundoff 331 Moments of probability laws shot 218-221,223-226 Bayes' 366 signal-dependent 60, 324 binomial 61,73,80,139 the ultimate limiter of information Cauchy 59 216 chi-square 115,116 white see Power spectrum exponential 52,53,61,74 Normal approximation to binomial law F-distribution 321,322 150-154 Gaussian 55 Normallaw see Probability law geometric 58,80 Normalization II, 12, 48 MW'd speckle pattern Nutting formula 159,160 258-259 normal,one-dimensional 54,55,61, 74,80 normal, two-dimensional 55 Observable 7 normal, n-dimensional 55,56 Observer, role of 3 Poisson 50 Occam's razor 105,275 SNR law 252-254 Ocean, propagation through 167, 168 sinc-squared 42,60 Optical amplitude 63,71,91,96,98, sum of n normal R V's 81 107,132 sum of n Poisson R V's 81 Optical fibre, absorptance 337-343 sum ofn general RV's 81,82 Opticalobject 16,18,40,45,51,64, t-distribution 312, 313 289 uniform 51 bimodal 173 Mona Lisa problem 275,289 class law 282, 283, 289 Monte Carlo calculation 52,93, estimate (restoration) 62,77,129, 162-176,303,304 197,203,206-211,216-218, Monte Carlo calculation, applications 238,260,282-285 emulsion 165-167 featureless 284 photon image formation 169-171 parametrized 192 remote sensing 167 -169 phase retrieval problem 191-192, "significance" of a factor 346 197-199 speckle patterns 171-1 72 plane 18 "mood", test for 301-303 spectrum 189,191,197,209,218 Mutual coherence function 66, 148, 149 Optical passband 212,216 438 Subject Index
Optical phase 52-54,71,92,93,107, Poisson approximation to binomial law 172,178,184,185,190,231 142-145 retrieval problem 191-192, Positivity 77,265,270 197-199 Posterior probability law 383-385, 391 Opticalpupil 54,63,107,118,178, Power spectrum 178,180,183,206, 182,184,185,190,231 209,210,214-216 Optical Sciences Center 1 colored 179, 180 Optical transfer function 42,43,62,63, D.C. 179,180 72,85,86,90,96,178,189,196, high-frequency 179, 180 228-229 Kolmogoroff 180 for atmospheric turbulence ocean wave 179, 180 183-188,197,344 transfer theorems for 188 -192 phase effect on infonnation 216 two-dimensional 196 three-dimensional diffraction 64 un-nonnalized 194, 196, 208, 213 Optical turbulence white 179,180,182,183,189, see Atmospheric turbulence 210 "or," intuitive use 7 Principal components analysis 349-361 Orthogonal functions 205, 266, 268, Prior knowledge 267, 270-272, 269,347,351,352 278-280 Orthogonalized factors 349-350 empirical 275,289-291 Outliers, d.o.f. 85 negative 275,289 Outcome 7,8,14 physical 291, 292 Output 199,211 strongest conviction 273 weakest conviction 274 Prior probability 271,282, 363, 364, 367, Parabolic cylinder function 252 368,370,372,383,384,391-393,397 Parameter estimation 374-399 Probabilistic perspective 1 Partition law 25,28,31,35,272,369 Probability of correct decision 68, 122, continuous 38,40,128-131 124,138,139,155 Parseval's theorem 207, 210 Probability law 37,38 Peakedness 56 atmospheric 98,122-124,132-133 Phantom thyroid 395 Bayes' or Beta 366,372,373 Phase see Optical phase bimodal 173 Phase blob model 192-194,232 binomial 51,61,80,134,136,140, Phase error 62 142,144,243,288,300,306 Phase space 281 Boltzmann 280-281 Phasor 96 Bose-Einstein 58,291, 292 Photographic density 33,159-160,219 Cauchy 59,80,127,263,288 Photographic emulsion 60, 143, 155, chi-square 112,115-117,126,127, 163,165-167,219,226-228, 247,249,251,252,298,311,320, 350-351,355 321,335, 336, 376-377, 391 Photographic granularity combined Poisson sources 146 see Granularity computation of 83 -86 Photomultiplier "clumping" of photons cosine bell 164, 170, 174,303 157-158 Cox-Munk or ocean wave 57-58, Photon statistics 144-150,291-293 228,231 Plane wave 65, 98, 132 cumulative or distribution function Point spread function 18, 41,45, 62, 48,61,68,163-165,170,257, 63,72,77,85,91,96, 102, 112, 296 129,131,160,166,167,169,170, error function (erD 95, 124, 184,187,190,206,231,232,282, 152-154,244,306,368,374-375 344, 345, 396 estimate of, 265-293, 399-408 Subject Index 439
for estimated probability 4, 243, Rice-Nakagami 132, 133 271-274, 365-368 scintillation 122-124 exponential 52, 58, 61, 172, 258, 392 sinc-squared or slit diffraction F-distribution 321-323,336, 41-43,59,60,64,160,161,263 378-380 skewed Gaussian 56,57 Gaussian 53,56,58,87-92,96,97, Student 250, 251, 254, 346, 378 110,116,127, 130, 176,185,201, sum of discrete and continuous RV's 213,247,261,262,279,297,298, 79 310-315,317,330,332,339, sum of two bivariate R V's 82 344, 374-375, 378, 380 sum of two Cauchy R V's 80 Gaussian, one-sided 397 sum of two RV's 75,76 geometric 58, 80, 159,293 sum ofn RV's 79,81,82,87,88 Laplace rule of succession 368-370 sum ofn squared RV's 113-115 "Laplace-Bayes" 3 72 triangular 173,371 "Laplace-triangular" 3 71 turbulence, multiple scattering 98 laser speckle amplitude 110 turbulence, single scattering 132 laser speckle intensity Ill, 132, 171 uniform 51,61,93, 109, 118, 119, laser speckle phase III 163,165,167,170-172,176,219, laser speckle averaged intensity 115 221, 225-227, 287, 289, 290, 303, log-normal 98,122,123,339,377,391 363, 366, 368, 371, 393 Lorentz see Cauchy Probability of a probability m-distribution 133 see Probability law for estimated median 257 - 258 probability multinomial 257,269-270,290 Processing window 237 nearest-neighbor 161 Product improvement test 319 negative binomial 133, 283, 293, 396 Product theorem 79, 83, 96 normal, one dimension 53,56,69, Psychometric curve 67 78,80,81,87,89,93-97,107, Pupil see Optical pupil 129,132,151,153,165,173, 185,212,214,219,236,249, Quantum optics probability law 58, 251,252,278,279,287,297, 291-292 298,320,330,332,335,343, Quantum oscillator problem 262 346, 367, 375, 377, 392 Quantum particle 17,83,282-283 normal, two dimensions 53,60,93, 110,148,172,186 R-test of factors for "significance" normal, n dimensions 55, 130,343, 336,386 360 Radiance vs angle at satellite 167-169 parabolic cylinder 251-252 Radiometer calibration errors 343 Pascal see geometric Random coefficients 205 picket fence 47,60 Random event 3,4 Poisson 50, 53, 61, 69, 80, 81, 96, Random noise see Noise 142-150,158,176,201,219,221, Random number 18 281,375, 382, 383, 391 parameter 374, 383-399 prescribed 163-165 Random objects used for MTF quotient of chi-square and determination 188 chi-square RV's 320-321 Random sample see Sample quotient of Gaussian and square Random variable RV 37-133 see also root of chi-square RV 311-312 Probability law, listings by name of quotient of normal and square root RV, listings by property of RV of chi-square RV 251-252 Random walk 79,150,151,155 Rayleigh 133, 161, 391, 393 on finite interval 35 rectangular see Uniform model 79 440 Subject Index
RANF computer function 52,93,94, Sharpness 210-211 163,165,171,172,263 Shift theorem 72, 88, 248, 249 Ray trace transformation 118 Shot effect 144-147 Rectangle function (Rect) 51, 52 Shot noise process application 60,61, 89, 90, 96, 118, see Stochastic process 11~ 170,176,196,210,221,224, Signal-to-noise ratio SIN 111-112, 223 227,232,254,274,289 binomial law 139 Redundancy 155 laser speckle 112 Regression analysis 57,327-349 laser speckle averaged 116 Relative error 236, 241, 248, 261, laser speckle MW'd 258-260 263,360,361,382 shot noise process 224, 225 Relative occurrence 271,282 Significance probability mapping Reliability 324-325 of components 59, 158, 159 Significance test 294-306 of forecast 20 Significant Remote sensing 167 -169 data 234,294-296,299,300 Restoration see Optical object estimate factors 331,334-337,345,346,386 Restoring filter, optimum 206-210, information 20 212,217 optical image 303 -304 vs information content 217 radio image 305 Risk 386, 389-391 sinc function 41 applications 47, 74 in sampling theorem 84-86, 195 Sample 234 Skewness 56,58 correlation coefficient 345-346, Slit diffraction law 41-43,59,60,64, 382 160,161,263 covariance matrix 351 Speckle mean 234-239,254,263,267, atmospheric 46,183-188,190-194 307,310,314,315,344,351, interferometry 190-194 380,382 laser 99,105-112,133,173,183, median 255-260,383 228,258-260,263,281,293 principal components 350-361,383 reduction 112-116, 197 -199, 307 regression coefficients 333-334, simulation 171-173 358,359,361,383 used to determine MTF 189 signal-to-noise SIN 250-254, 382 Spot diagram variance 116, 245 -250, 261, 294, see Geometrical spot diagrams 353,382 Spray-painted surface, model for 231 Sampling theorems 83-87,96 Spread function Satellite-ground communication see Point spread function 120-124 Square law detector 99 "scatter spots" I 07, 108, 113, 172 Standard deviation 41, 254 Scattered radiation 66,162,165, 168, see also Variance 173,174 Star field model 231 Schrodinger wave equation, derivation Stationarity 402-403 strict sense 77, 181, 182, 185, 204 Schwarz inequality 379, 381 weak or wide sense 181, 201 , 204 Selwyn root-area law 141,226,331 Statistical Semiclassical theory of radiation 404 data analysis 2 Sensitometry see Statistical models independence 21,44,60,75,79, and Hurter-Driffield H-D curve 134,200,212,218,219,236, Shannon information see Information, 237,261,267,269,297,298, Shannon 310,330,332 Subject Index 441
inference 308,309 spray-painted surface, superposition modeling 105, 184, 185 process 231 Statistical models for physical effects: star field image, superposition atmospheric MTF, regression model process 231 344 star positions, uniform randomness atmospheric speckle, phase blob model 196 model 192-194 "swimming pool" effect, super• atmospheric speckle, stochastic position process 228, 230 model 185-186 Statistical models for sample estimates film sensitometry, principal compo• correlation coefficient 382 nents model 350-355 Laplace rule of succession 368-369, film sensitometry, regression model 371-373 328-330,333-334,336-337, mean 234-236,382 346 median 257-258,383 Fisher information model 399-408 principal components 350 -3 5 5,383 granularity, checkerboard model probability 239,243,382 139-141,226,242-243 probability law 265-269,270-282 granularity, Monte Carlo model regression coefficients 327-331,383 165-166 signal-to-noise 250-25 1,382 granularity, multiple layers model variance 245-247,382 159-160 weighted mean 237,382 granularity, poker chip model Statistical tests 226-228 chi-square test of "significant" data image formation, photon model 294-306,385 169-171,395-397 F-test of equal variances 320-326, image restoration, regression model 385 344 R-test of "significant" factors 336, information in image, stochastic 386 model 212 Synopsis sheet of 419-421 lens design, regression model t-test of accuracy in a mean 311, 345 314-316,384 mirror surface scratches, super• t -test of "correlation" of data 346, position process 231 386 multispectral images, principal com• t-test of equal means 317-319,384 ponents model 356-358 Statistics, Bayesian vs Classical 234, noise, regression model 330-331, 363-373 338,339 Stigmatic imagery 102, 131 optical fibre transmission, regression Stirling approximation 240,263,273, model 337-342 290,292 optimum restoring filter, stochastic Stochastic component 178 model 207-208 Stochastic process 2, 177 -23 2 physically maximum likely object disturbance function 218, 224, 226, restoration, Bose-Einstein model 227,229 282-285 image formation 395-397 quantum optics probability law, role in linear regression 330 Bose-Einstein model 291 shifted 181 radiometer calibration, regression shot noise 221-223,225-226 model 343 superposition 218-232 remote sensing, Monte Carlo model "swimming pool" effect 228-232 167-168 Strehl intensity 345 2 shot noise, superposition process Structure constant, refractive index eN 221-223,225-226 180 442 Subject Index
Student probability law Thermodynamics and entropy 276 see Probability law and Fisher information 400-402, 407 Student t-test see Statistical tests Three-dimensional optical transfer Subjective probability law 363 function 64 Sufficient statistic 299-301, 377 Threshold intensity Summation model 327-330 in binary detection 121, 122 to model film sensitization 159 t-test see Statistical tests "tipping" problem 33 Taylor series 73,87,88, 270 Total probability law 25 Test hypothesis 294,303,305,309, see also Partition law 311,314,315,320-324,336-337, Tracking, airplane 394-395 346,364 Transfer function Test statistic, general, properties of see also Optical transfer function 300-301 for power spectrum 189-192 Tests on physical data Transfer theorem 75,76,79 see also Statistical models for autocorrelation 197 binary communication link 261 Transformation of a RV binary outcomes, for "significance" see Functions of RV's 299,301-303 Transient current 219,224 chi-square test 294-306,385 Transmittance, film 25, 140, 155, 159 data sets, for "probable" correlation see also Photographic emulsion 345,346 aperture size for accuracy require• F-test 320-326,385 ment 242 failure rates for judgment on Trapping state 36 improvement 3 19 Trial 7,8,17,369 image data, for common origin 323 see also Bernoulli trials image data, for "intelligence" 305 Triangle function 42 image data, for "significant" correlation 195,228-229 structure 303-305 MTF 210 medical diagnostic 20,29,34, "tuning" a restoration 210 305-306 Turbulence see Atmospheric turbulence optical flat, for judgment on "twinkle," why stars 124 improvement 325 optical mirror, for estimating Unbiased estimate see Estimators "roughness" 249-250 Underwater irradiance 66 plastic lenses, to estimate discard Universal rate 261 constants, as sample means 236 R-test of "significant" factors 336, estimator, search for 274 386 state of prior ignorance 274 scanning densitometer, accuracy Un sharp masking 77, 78, 80 requirement 261 star size, relative, from intensity van Cittert-Zernike theorem 64, 149 fluctuations 326 Variance 41 Strehl intensity, for critical design cascaded events 157-158 parameters 345 Cramer-Rao inequality 378-380,408 Student t-test 307-320,346, for intensity on pool bottom 230, 384 232 thin film thickness, with accuracy intrinsic, of stellar irradiance 97 requirement 260 laser speckle, averaged 115 turbulent wavefront variance, laser speckle intensity 112 accuracy requirement 261 maximum 66,342 Thermal light field 58, 148,291,375 minimum 66 Subject Index 443
as missing datum 308 probability 240-241,382 in MW speckle output 259 probability law 267,382 Neyman type A 158 regression coefficient 339-341, 383 of noise in restored output 130 restored point object 344 of particle velocities 127 signal-to-noise 253,382 of phase in turbulence 97 variance 247-248,382 of photographic granularity 141 weighted mean 238,382 of random noise 201 Variances, test for equality of 320,322, required, how to attain 93,94 323 for shot noise process 223 Variation, coefficient of 223 as source intensity 96, 110 Vote, when "decisive" 301-303 for superposition process 222-223 in two-stage amplifier IS 7 -15 8 Variance of error in sample estimate Wavefront variance 261 absorptance of fibre 337-341,342 weather forecasting problem 156 correlation coefficient 337-342, White noise 183 382 Whittaker-Shannon interpolation 85 mean 234-235,267,307,382 "widgets" 265 median 259,383 Wiener-Helstrom filter 208-211 noise in least-squares problem 335 Winning a protracted contest, principal component eigenvalue probability of 35 360,383 Witchcraft 3