I.3 Basic Definitions About Matrices 507 and rank rank rank 22 AA IAA 11 ++−− () = n. iv) If A j , j = 1, . . . , m are symmetric idempotent matrices of the same order and Σ m j =1 A j is also idempotent, the A i A j = 0 for 1 ≤ i < j ≤ m. The proof of the theorems formulated in this appendix may be found in most books of linear algebra. For example, see Birkhoff and MacLane, A Survey of Modern Algebra, 3d ed., MacMillan, 1965; S. Lang, Linear Algebra, Addison-Wesley, 1968; D. C. Murdoch, Linear Algebra for Undergraduates, Wiley, 1957; S. Perlis, Theory of Matrices, Addison-Wesley, 1952. For a brief exposition of most results from linear algebra employed in statistics, see also C. R. Rao, Linear Statistical Inference and Its Applications, Chapter 1, Wiley, 1965; H. Scheffé, The Analysis of Variance, Appendices I and II, Wiley, 1959; and F. A. Graybill, An Introduction to Linear Statistical Models, Vol. I, Chap- ter 1, McGraw-Hill, 1961. I.4 Some Theorems About Matrices and Quadratic Forms 507 508 Appendix II Noncentral t , χχ χχ χ 2 and F Distributions 508 II.1 Noncentral t -Distribution It was seen in Chapter 9, Application 2, that if the independent r.v.’s X and Y were distributed as N(0, 1) and χ 2 r , respectively, then the distribution of the r.v. TX Yr= () was the (Student’s) t-distribution with r d.f. Now let X and Y be independent r.v.’s distributed as N( δ , 1) and χ 2 r , respectively, and set ′ = () TXYr . The r.v. T ′ is said to have the noncentral t-distribution with r d.f. and noncentrality parameter δ . This distribution, as well as an r.v. having this distribution, if often denoted by t′ r: δ . Using the definition of a t′ r: δ r.v., it can be found by well known methods that its p.d.f. is given by ft rr x xt x r dx t t r r r ′ + () − () ∞ () = () ×−+ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ∈ ∫ ; ; ,. δ δ π δ exp 1 22 1 2 12 12 0 2 Γ ޒ II.2 Noncentral χχ χχ χ 2 -Distribution It was seen in Chapter 7 (see corollary to Theorem 5) that if X 1 , , X r were independent normally distributed r.v.’s with variance 1 and mean 0, then the r.v. X =Σ r j =1 X 2 j was distributed as χ 2 r . Let now the r.v.’s X 1 , , X r be indepen- dent normally distributed with variance 1 but means μ 1 , , μ r , respectively. Then the distribution of the r.v. X* =Σ r j =1 X 2 j is said to be the noncentral Appendix II Noncentral t , χχ χχ χ 2 and F Distributions II.3 Noncentral F -Distribution 509 chi-square distribution with r d.f. and noncentrality parameter δ , where δ 2 = Σ r j=1 μ 2 j . This distribution, and also an r.v. having this distribution, is often denoted by χ ′ 2 r: δ . Using the definition of a χ ′ 2 r; δ r.v., one can find its p.d.f. but it does not have any simple closed form. It can be seen that this p.d.f. is a mixture of χ 2 -distributions with Poisson weights. More precisely, one has fx Pf xx r j j rj ′ = ∞ + () = () () ≥ ∑ χ δ δδ ; ;,, 2 0 2 0 where Pe j fj j j rj rj δ δ χ δ () = () = − ++ 2 2 2 22 2 2 0 ! , , and is the p.d.f. of 1, II.3 Noncentral F -Distribution In Chapter 9, Application 2, the F-distribution with r 1 and r 2 d.f. was defined as the distribution of the r.v. F Xr Yr = 1 2 , where X and Y were independent r.v.’s distributed as χ 2 r 1 and χ 2 r 2 , respectively. Suppose now that the r.v.’s X and Y are independent and distributed as χ ′ 2 r 1 : δ and χ 2 r 2 , respectively, and set ′ =F Xr Yr 1 2 . Then the distribution of F′ is said to be the noncentral F-distribution with r 1 and r 2 d.f. and noncentrality parameter δ . This distribution, and also an r.v. having this distribution, is often denoted by F′ r 1 ,r 2 ; δ , and its p.d.f., which does not have any simple closed form, is given by the following expression: ff e c j f f f Fj j j rj rr j rr ′ − = ∞ −+ + () + () = () + () ≥ ∑ 12 2 1 12 2 2 0 1 2 1 1 2 2 1 0 ,: ; ! ,, δ δ δ δ where c rr j rj r j j = + () + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = Γ ΓΓ 1 2 1 2 1 2 01 12 12 , , , 510 Appendix II Noncentral t , χχ χχ χ 2 and F Distributions REMARKS (i) By setting δ = 0 in the noncentral t, χ 2 and F-distributions, we obtain the t, χ 2 and F-distributions, respectively. In view of this, the latter distribu- tions may also be called central t, χ 2 and F-distributions. (ii) Tables for the noncentral t, χ 2 and F-distributions are given in a reference cited elsewhere, namely, Handbook of Statistical Tables by D. B. Owen. Addison-Wesley, 1962. Tables 511 511 Appendix III Tables Table 1 The Cumulative Binomial Distribution The tabulated quantity is j k j nj n j pp = − ∑ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ( ) 0 1. p nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16 2 0 0.8789 0.7656 0.6602 0.5625 0.4727 0.3906 0.3164 0.2500 1 0.9961 0.9844 0.9648 0.9375 0.9023 0.8594 0.8086 0.7500 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 3 0 0.8240 0.6699 0.5364 0.4219 0.3250 0.2441 0.1780 0.1250 1 0.9888 0.9570 0.9077 0.8437 0.7681 0.6836 0.5933 0.5000 2 0.9998 0.9980 0.9934 0.9844 0.9695 0.9473 0.9163 0.8750 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 4 0 0.7725 0.5862 0.4358 0.3164 0.2234 0.1526 0.1001 0.0625 1 0.9785 0.9211 0.8381 0.7383 0.6296 0.5188 0.4116 0.3125 2 0.9991 0.9929 0.9773 0.9492 0.9065 0.8484 0.7749 0.6875 3 1.0000 0.9998 0.9988 0.9961 0.9905 0.9802 0.9634 0.9375 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 5 0 0.7242 0.5129 0.3541 0.2373 0.1536 0.0954 0.0563 0.0312 1 0.9656 0.8793 0.7627 0.6328 0.5027 0.3815 0.2753 0.1875 2 0.9978 0.9839 0.9512 0.8965 0.8200 0.7248 0.6160 0.5000 3 0.9999 0.9989 0.9947 0.9844 0.9642 0.9308 0.8809 0.8125 4 1.0000 1.0000 0.9998 0.9990 0.9970 0.9926 0.9840 0.9687 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 512 Appendix III Tables Table 1 (continued) p nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16 6 0 0.6789 0.4488 0.2877 0.1780 0.1056 0.0596 0.0317 0.0156 1 0.9505 0.8335 0.6861 0.5339 0.3936 0.2742 0.1795 0.1094 2 0.9958 0.9709 0.9159 0.8306 0.7208 0.5960 0.4669 0.3437 3 0.9998 0.9970 0.9866 0.9624 0.9192 0.8535 0.7650 0.6562 4 1.0000 0.9998 0.9988 0.9954 0.9868 0.9694 0.9389 0.8906 5 1.0000 1.0000 1.0000 0.9998 0.9991 0.9972 0.9930 0.9844 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 7 0 0.6365 0.3927 0.2338 0.1335 0.0726 0.0373 0.0178 0.0078 1 0.9335 0.7854 0.6114 0.4449 0.3036 0.1937 0.1148 0.0625 2 0.9929 0.9537 0.8728 0.7564 0.6186 0.4753 0.3412 0.2266 3 0.9995 0.9938 0.9733 0.9294 0.8572 0.7570 0.6346 0.5000 4 1.0000 0.9995 0.9965 0.9871 0.9656 0.9260 0.8628 0.7734 5 1.0000 1.0000 0.9997 0.9987 0.9952 0.9868 0.9693 0.9375 6 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990 0.9969 0.9922 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8 0 0.5967 0.3436 0.1899 0.1001 0.0499 0.0233 0.0100 0.0039 1 0.9150 0.7363 0.5406 0.3671 0.2314 0.1350 0.0724 0.0352 2 0.9892 0.9327 0.8238 0.6785 0.5201 0.3697 0.2422 0.1445 3 0.9991 0.9888 0.9545 0.8862 0.7826 0.6514 0.5062 0.3633 4 1.0000 0.9988 0.9922 0.9727 0.9318 0.8626 0.7630 0.6367 5 1.0000 0.9999 0.9991 0.9958 0.9860 0.9640 0.9227 0.8555 6 1.0000 1.0000 0.9999 0.9996 0.9983 0.9944 0.9849 0.9648 7 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9987 0.9961 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 0 0.5594 0.3007 0.1543 0.0751 0.0343 0.0146 0.0056 0.0020 1 0.8951 0.6872 0.4748 0.3003 0.1747 0.0931 0.0451 0.0195 2 0.9846 0.9081 0.7707 0.6007 0.4299 0.2817 0.1679 0.0898 3 0.9985 0.9817 0.9300 0.8343 0.7006 0.5458 0.3907 0.2539 4 0.9999 0.9975 0.9851 0.9511 0.8851 0.7834 0.6506 0.5000 5 1.0000 0.9998 0.9978 0.9900 0.9690 0.9260 0.8528 0.7461 6 1.0000 1.0000 0.9998 0.9987 0.9945 0.9830 0.9577 0.9102 7 1.0000 1.0000 1.0000 0.9999 0.9994 0.9977 0.9926 0.9805 8 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9980 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 0 0.5245 0.2631 0.1254 0.0563 0.0236 0.0091 0.0032 0.0010 1 0.8741 0.6389 0.4147 0.2440 0.1308 0.0637 0.0278 0.0107 2 0.9790 0.8805 0.7152 0.5256 0.3501 0.2110 0.1142 0.0547 3 0.9976 0.9725 0.9001 0.7759 0.6160 0.4467 0.2932 0.1719 4 0.9998 0.9955 0.9748 0.9219 0.8275 0.6943 0.5369 0.3770 5 1.0000 0.9995 0.9955 0.9803 0.9428 0.8725 0.7644 0.6230 6 1.0000 1.0000 0.9994 0.9965 0.9865 0.9616 0.9118 0.8281 7 1.0000 1.0000 1.0000 0.9996 0.9979 0.9922 0.9773 0.9453 8 1.0000 1.0000 1.0000 1.0000 0.9998 0.9990 0.9964 0.9893 Tables 513 Table 1 (continued) p nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16 10 9 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 11 0 0.4917 0.2302 0.1019 0.0422 0.0162 0.0057 0.0018 0.0005 1 0.8522 0.5919 0.3605 0.1971 0.0973 0.0432 0.0170 0.0059 2 0.9724 0.8503 0.6589 0.4552 0.2816 0.1558 0.0764 0.0327 3 0.9965 0.9610 0.8654 0.7133 0.5329 0.3583 0.2149 0.1133 4 0.9997 0.9927 0.9608 0.8854 0.7614 0.6014 0.4303 0.2744 5 1.0000 0.9990 0.9916 0.9657 0.9068 0.8057 0.6649 0.5000 6 1.0000 0.9999 0.9987 0.9924 0.9729 0.9282 0.8473 0.7256 7 1.0000 1.0000 0.9999 0.9988 0.9943 0.9807 0.9487 0.8867 8 1.0000 1.0000 1.0000 0.9999 0.9992 0.9965 0.9881 0.9673 9 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9983 0.9941 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 12 0 0.4610 0.2014 0.0828 0.0317 0.0111 0.0036 0.0010 0.0002 1 0.8297 0.5467 0.3120 0.1584 0.0720 0.0291 0.0104 0.0032 2 0.9649 0.8180 0.6029 0.3907 0.2240 0.1135 0.0504 0.0193 3 0.9950 0.9472 0.8267 0.6488 0.4544 0.2824 0.1543 0.0730 4 0.9995 0.9887 0.9429 0.8424 0.6900 0.5103 0.3361 0.1938 5 1.0000 0.9982 0.9858 0.9456 0.8613 0.7291 0.5622 0.3872 6 1.0000 0.9998 0.9973 0.9857 0.9522 0.8822 0.7675 0.6128 7 1.0000 1.0000 0.9996 0.9972 0.9876 0.9610 0.9043 0.8062 8 1.0000 1.0000 1.0000 0.9996 0.9977 0.9905 0.9708 0.9270 9 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9938 0.9807 10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9992 0.9968 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 13 0 0.4321 0.1762 0.0673 0.0238 0.0077 0.0022 0.0006 0.0001 1 0.8067 0.5035 0.2690 0.1267 0.0530 0.0195 0.0063 0.0017 2 0.9565 0.7841 0.5484 0.3326 0.1765 0.0819 0.0329 0.0112 3 0.9931 0.9310 0.7847 0.5843 0.3824 0.2191 0.1089 0.0461 4 0.9992 0.9835 0.9211 0.7940 0.6164 0.4248 0.2565 0.1334 5 0.9999 0.9970 0.9778 0.9198 0.8078 0.6470 0.4633 0.2905 6 1.0000 0.9996 0.9952 0.9757 0.9238 0.8248 0.6777 0.5000 7 1.0000 1.0000 0.9992 0.9944 0.9765 0.9315 0.8445 0.7095 8 1.0000 1.0000 0.9999 0.9990 0.9945 0.9795 0.9417 0.8666 9 1.0000 1.0000 1.0000 0.9999 0.9991 0.9955 0.9838 0.9539 10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9968 0.9888 11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9983 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 14 0 0.4051 0.1542 0.0546 0.0178 0.0053 0.0014 0.0003 0.0001 1 0.7833 0.4626 0.2312 0.1010 0.0388 0.0130 0.0038 0.0009 514 Appendix III Tables Table 1 (continued) p nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16 14 2 0.9471 0.7490 0.4960 0.2811 0.1379 0.0585 0.0213 0.0065 3 0.9908 0.9127 0.7404 0.5213 0.3181 0.1676 0.0756 0.0287 4 0.9988 0.9970 0.8955 0.7415 0.5432 0.3477 0.1919 0.0898 5 0.9999 0.9953 0.9671 0.8883 0.7480 0.5637 0.3728 0.2120 6 1.0000 0.9993 0.9919 0.9167 0.8876 0.7581 0.5839 0.3953 7 1.0000 0.9999 0.9985 0.9897 0.9601 0.8915 0.7715 0.6047 8 1.0000 1.0000 0.9998 0.9978 0.9889 0.9615 0.8992 0.7880 9 1.0000 1.0000 1.0000 0.9997 0.9976 0.9895 0.9654 0.9102 10 1.0000 1.0000 1.0000 1.0000 0.9996 0.9979 0.9911 0.9713 11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9935 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9991 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 15 0 0.3798 0.1349 0.0444 0.0134 0.0036 0.0009 0.0002 0.0000 1 0.7596 0.4241 0.1981 0.0802 0.0283 0.0087 0.0023 0.0005 2 0.9369 0.7132 0.4463 0.2361 0.1069 0.0415 0.0136 0.0037 3 0.9881 0.8922 0.6946 0.4613 0.2618 0.1267 0.0518 0.0176 4 0.9983 0.9689 0.8665 0.6865 0.4729 0.2801 0.1410 0.0592 5 0.9998 0.9930 0.9537 0.8516 0.6840 0.4827 0.2937 0.1509 6 1.0000 0.9988 0.9873 0.9434 0.8435 0.6852 0.4916 0.3036 7 1.0000 0.9998 0.9972 0.9827 0.9374 0.8415 0.6894 0.5000 8 1.0000 1.0000 0.9995 0.9958 0.9799 0.9352 0.8433 0.6964 9 1.0000 1.0000 0.9999 0.9992 0.9949 0.9790 0.9364 0.8491 10 1.0000 1.0000 1.0000 0.9999 0.9990 0.9947 0.9799 0.9408 11 1.0000 1.0000 1.0000 1.0000 0.9999 0.9990 0.9952 0.9824 12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9992 0.9963 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 16 0 0.3561 0.1181 0.0361 0.0100 0.0025 0.0005 0.0001 0.0000 1 0.7359 0.3879 0.1693 0.0635 0.0206 0.0057 0.0014 0.0003 2 0.9258 0.6771 0.3998 0.1971 0.0824 0.0292 0.0086 0.0021 3 0.9849 0.8698 0.6480 0.4050 0.2134 0.0947 0.0351 0.0106 4 0.9977 0.9593 0.8342 0.6302 0.4069 0.2226 0.1020 0.0384 5 0.9997 0.9900 0.9373 0.8103 0.6180 0.4067 0.2269 0.1051 6 1.0000 0.9981 0.9810 0.9204 0.7940 0.6093 0.4050 0.2272 7 1.0000 0.9997 0.9954 0.9729 0.9082 0.7829 0.6029 0.4018 8 1.0000 1.0000 0.9991 0.9925 0.9666 0.9001 0.7760 0.5982 9 1.0000 1.0000 0.9999 0.9984 0.9902 0.9626 0.8957 0.7728 10 1.0000 1.0000 1.0000 0.9997 0.9977 0.9888 0.9609 0.8949 11 1.0000 1.0000 1.0000 1.0000 0.9996 0.9974 0.9885 0.9616 12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9975 0.9894 13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9979 Tables 515 Table 1 (continued) p nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16 16 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 17 0 0.3338 0.1033 0.0293 0.0075 0.0017 0.0003 0.0001 0.0000 1 0.7121 0.3542 0.1443 0.0501 0.0149 0.0038 0.0008 0.0001 2 0.9139 0.6409 0.3566 0.1637 0.0631 0.0204 0.0055 0.0012 3 0.9812 0.8457 0.6015 0.3530 0.1724 0.0701 0.0235 0.0064 4 0.9969 0.9482 0.7993 0.5739 0.3464 0.1747 0.0727 0.0245 5 0.9996 0.9862 0.9180 0.7653 0.5520 0.3377 0.1723 0.0717 6 1.0000 0.9971 0.9728 0.8929 0.7390 0.5333 0.3271 0.1662 7 1.0000 0.9995 0.9927 0.9598 0.8725 0.7178 0.5163 0.3145 8 1.0000 0.9999 0.9984 0.9876 0.9484 0.8561 0.7002 0.5000 9 1.0000 1.0000 0.9997 0.9969 0.9828 0.9391 0.8433 0.6855 10 1.0000 1.0000 1.0000 0.9994 0.9954 0.9790 0.9323 0.8338 11 1.0000 1.0000 1.0000 0.9999 0.9990 0.9942 0.9764 0.9283 12 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987 0.9935 0.9755 13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987 0.9936 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9988 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 18 0 0.3130 0.0904 0.0238 0.0056 0.0012 0.0002 0.0000 0.0000 1 0.6885 0.3228 0.1227 0.0395 0.0108 0.0025 0.0005 0.0001 2 0.9013 0.6051 0.3168 0.1353 0.0480 0.0142 0.0034 0.0007 3 0.9770 0.8201 0.5556 0.3057 0.1383 0.0515 0.0156 0.0038 4 0.9959 0.9354 0.7622 0.5187 0.2920 0.1355 0.0512 0.0154 5 0.9994 0.9814 0.8958 0.7175 0.4878 0.2765 0.1287 0.0481 6 0.9999 0.9957 0.9625 0.8610 0.6806 0.4600 0.2593 0.1189 7 1.0000 0.9992 0.9889 0.9431 0.8308 0.6486 0.4335 0.2403 8 1.0000 0.9999 0.9973 0.9807 0.9247 0.8042 0.6198 0.4073 9 1.0000 1.0000 0.9995 0.9946 0.9721 0.9080 0.7807 0.5927 10 1.0000 1.0000 0.9999 0.9988 0.9915 0.9640 0.8934 0.7597 11 1.0000 1.0000 1.0000 0.9998 0.9979 0.9885 0.9571 0.8811 12 1.0000 1.0000 1.0000 1.0000 0.9996 0.9970 0.9860 0.9519 13 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9964 0.9846 14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9962 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 19 0 0.2934 0.0791 0.0193 0.0042 0.0008 0.0001 0.0000 0.0000 1 0.6650 0.2938 0.1042 0.0310 0.0078 0.0016 0.0003 0.0000 2 0.8880 0.5698 0.2804 0.1113 0.0364 0.0098 0.0021 0.0004 3 0.9722 0.7933 0.5108 0.2631 0.1101 0.0375 0.0103 0.0022 4 0.9947 0.9209 0.7235 0.4654 0.2440 0.1040 0.0356 0.0096 516 Appendix III Tables Table 1 (continued) p nk1/16 2/16 3/16 4/16 5/16 6/16 7/16 8/16 19 5 0.9992 0.9757 0.8707 0.6678 0.4266 0.2236 0.0948 0.0318 6 0.9999 0.9939 0.9500 0.8251 0.6203 0.3912 0.2022 0.0835 7 1.0000 0.9988 0.9840 0.9225 0.7838 0.5779 0.3573 0.1796 8 1.0000 0.9998 0.9957 0.9713 0.8953 0.7459 0.5383 0.3238 9 1.0000 1.0000 0.9991 0.9911 0.9573 0.8691 0.7103 0.5000 10 1.0000 1.0000 0.9998 0.9977 0.9854 0.9430 0.8441 0.0672 11 1.0000 1.0000 1.0000 0.9995 0.9959 0.9793 0.9292 0.8204 12 1.0000 1.0000 1.0000 0.9999 0.9990 0.9938 0.9734 0.9165 13 1.0000 1.0000 1.0000 1.0000 0.9998 0.9985 0.9919 0.9682 14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9980 0.9904 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9978 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 20 0 0.2751 0.0692 0.0157 0.0032 0.0006 0.0001 0.0000 0.0000 1 0.6148 0.2669 0.0883 0.0243 0.0056 0.0011 0.0002 0.0000 2 0.8741 0.5353 0.2473 0.0913 0.0275 0.0067 0.0013 0.0002 3 0.9670 0.7653 0.4676 0.2252 0.0870 0.0271 0.0067 0.0013 4 0.9933 0.9050 0.6836 0.4148 0.2021 0.0790 0.0245 0.0059 5 0.9989 0.9688 0.8431 0.6172 0.3695 0.1788 0.0689 0.0207 6 0.9999 0.9916 0.9351 0.7858 0.5598 0.3284 0.1552 0.0577 7 1.0000 0.9981 0.9776 0.8982 0.7327 0.5079 0.2894 0.1316 8 1.0000 0.9997 0.9935 0.9591 0.8605 0.6829 0.4591 0.2517 9 1.0000 0.9999 0.9984 0.9861 0.9379 0.8229 0.6350 0.4119 10 1.0000 1.0000 0.9997 0.9961 0.9766 0.9153 0.7856 0.5881 11 1.0000 1.0000 0.9999 0.9991 0.9926 0.9657 0.8920 0.7483 12 1.0000 1.0000 1.0000 0.9998 0.9981 0.9884 0.9541 0.8684 13 1.0000 1.0000 1.0000 1.0000 0.9996 0.9968 0.9838 0.9423 14 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9953 0.9793 15 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9989 0.9941 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9987 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 21 0 0.2579 0.0606 0.0128 0.0024 0.0004 0.0001 0.0000 0.0000 1 0.6189 0.2422 0.0747 0.0190 0.0040 0.0007 0.0001 0.0000 2 0.8596 0.5018 0.2175 0.0745 0.0206 0.0046 0.0008 0.0001 3 0.9612 0.7366 0.4263 0.1917 0.0684 0.0195 0.0044 0.0007 4 0.9917 0.8875 0.6431 0.3674 0.1662 0.0596 0.0167 0.0036 5 0.9986 0.9609 0.8132 0.5666 0.3172 0.1414 0.0495 0.0133 6 0.9998 0.9888 0.9179 0.7436 0.5003 0.2723 0.1175 0.0392 7 1.0000 0.9973 0.9696 0.8701 0.6787 0.4405 0.2307 0.0946 8 1.0000 0.9995 0.9906 0.9439 0.8206 0.6172 0.3849 0.1917 [...]... 88 89 90 Tables 529 Table 5 Critical Values for the Chi-Square Distribution Let χ 2 be a random variable having the chi-square distribution with r degrees of r freedom Then the tabulated quantities are the numbers x for which ( ) P χ r2 ≤ x = γ r 0.005 0.01 γ 0.025 0.05 0 .10 0.25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 — 0. 010 0.072 0.207... 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 526 Appendix III Tables Table 4 Critical Values for Student’s t-Distribution Let tr be a random variable having the Student’s t-distribution with r degrees of freedom Then the tabulated quantities are the numbers x for which ( ) P t r ≤ x = γ γ r 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33... 65. 410 54.776 56.061 57.342 58.619 59.892 61.162 62.428 63.691 64.950 66.206 67.459 68. 710 69.957 57.648 58.964 60.275 61.581 62.883 64.181 65.476 66.766 68.053 69.336 70.616 71.893 73.166 γ 33 34 35 36 37 38 39 40 41 42 43 44 45 532 Appendix III Tables Table 6 Critical Values for the F-Distribution Let Fr ,r be a random variable having the F-distribution with r1, r2 degrees of freedom Then the tabulated... of Statistical Tables, published by Addison-Wesley, by permission of the publishers r2 542 Appendix III Tables Table 7 Table of Selected Discrete and Continuous Distributions and Some of their Characteristics Distribution Binomial, B(n, p) (Bernoulli, B(1, p) Poisson P(λ) Probability density function Mean Variance ⎛ n⎞ f ( x) = ⎜ ⎟ p x q n− x , x = 0, 1, , n; ⎝ x⎠ 0 < p < 1, q = 1 − p np npq f (x)... 1.4312 2.0379 2. 5102 3.0053 3.7054 4.2759 0.96247 1.4320 2.0047 2.4563 2.9291 3.5971 4.1 410 0.96993 1.4159 1.9770 2.4117 2.8664 3.5082 4.0305 0.97606 1.4095 1.9532 2.3737 2.8132 3.4331 3.9374 0.98116 1.4042 1.9333 2.3421 2.7689 3.3706 3.8599 0.500 0.750 0.900 0.950 0.975 0.990 0.995 18 r2 These tables have been adapted from Donald B Owen’s Handbook of Statistical Tables, published by Addison-Wesley,... 0, 1, , min(r , m) Negative Binomial ⎛ r + x − 1⎞ x f ( x) = p r ⎜ ⎟ q , x = 0, 1, ; ⎝ x ⎠ mnr(m + n − r ) (m + n) (m + n − 1) 2 rq p rq p2 q p q ) p2 vector of expectations: vector of variances: (np1, , npk)′ (np1q1, , npkqk)′ 0 < p < 1, q = 1 − p (Geometric f ( x) = pq x , x = 0, 1, Multinomial f ( x1 , , xk ) = n! × x1! x2 ! ⋅ ⋅ ⋅ xk ! x x p1x1 p2 2 ⋅ ⋅ ⋅ pkk , x j ≥ 0 integers,... 0.995 6 r2 r2 535 Tables Table 6 (continued ) r1 γ 30 40 48 60 120 ∞ γ 1 0.500 0.750 0.900 0.950 0.975 0.990 0.995 2.1452 9.6698 62.265 250.09 100 1.4 6260.7 25044 2.1584 9.7144 62.529 251.14 100 5.6 6286.8 25148 2.1650 9.7368 62.662 251.67 100 7.7 6299.9 25201 2.1716 9.7591 62.794 252.20 100 9.8 6313.0 25253 2.1848 9.8041 63.061 253.25 101 4.0 6339.4 25359 2.1981 9.8492 63.328 254.32 101 8.3 6366.0 25465... 4.1229 1.0523 1.4 310 1.9323 2.3 410 2.7874 3.4494 4.0149 1.0582 1.4221 1.9036 2.2962 2.7249 3.3608 3.9039 0.500 0.750 0.900 0.950 0.975 0.990 0.995 12 r2 r2 540 Appendix III Tables Table 6 (continued ) r1 γ 1 2 3 4 5 6 γ 13 0.500 0.750 0.900 0.950 0.975 0.990 0.995 0.48141 1.4500 3.1362 4.6672 6.4143 9.0738 11.374 0.73145 1.5452 2.7632 3.8056 4.9653 6.7 010 8.1865 0.83159 1.5451 2.5603 3. 4105 4.3472 5.7394... 0.5275 0.6834 0.8 110 0.9003 0.9538 0.9814 0.9935 0.9981 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 0.1630 0.2835 0.4335 0.5926 0.7369 0.8491 0.9240 0.9667 0.9874 0.9960 0.9989 0.9998 1.0000 1.0000 1.0000 0.0539 0.1148 0.2122 0.3450 0.5000 0.6550 0.7878 0.8852 0.9462 0.9784 0.9927 0.9980 0.9995 0.9999 1.0000 520 Appendix III Tables Table 2 The Cumulative Poisson Distribution The tabulated quantity is k ∑e... · · · + pk = 1 Normal, N(μ, σ2) ⎡ (x − μ ) 2 exp⎢− ⎢ 2σ 2 2π σ ⎣ x ∈ ;ޒμ ∈ ,ޒσ > 0 1 f (x) = (Standard Normal, N(0, 1) f (x) = ⎛ x2 ⎞ exp⎜ − ⎟ , x ∈ޒ ⎝ 2⎠ 2π 1 f ( x) = Gamma ⎤ ⎥, ⎥ ⎦ ⎛ x⎞ 1 x α −1 exp⎜ − ⎟ , x > 0 ; Γ(α )β α ⎝ β⎠ qj = 1 − pj, j = 1, , k μ σ2 0 1) αβ αβ2 r 2r α, β > 0 Chi-square f ( x) = r −1 1 ⎛ x⎞ x 2 exp⎜ − ⎟ , x > 0; ⎝ 2⎠ ⎛ r ⎞ r2 Γ⎜ ⎟ 2 ⎝ 2⎠ r > 0 integer . a brief exposition of most results from linear algebra employed in statistics, see also C. R. Rao, Linear Statistical Inference and Its Applications, Chapter 1, Wiley, 1965; H. Scheffé, The Analysis. of Modern Algebra, 3d ed., MacMillan, 1965; S. Lang, Linear Algebra, Addison-Wesley, 1968; D. C. Murdoch, Linear Algebra for Undergraduates, Wiley, 1957; S. Perlis, Theory of Matrices, Addison-Wesley,. I.3 Basic Definitions About Matrices 507 and rank rank rank 22 AA IAA 11 ++−− () = n. iv) If A j , j = 1, . . . , m are symmetric idempotent matrices of the same order and Σ m j =1 A j is also