TABLES OF DISTRIBUTIONS OF QUADRATIC FORMS IN CENTRAL NORMAL VARIABLES II. Institute of Statistics Mimeo Series No. 557.

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TABLES OF DISTRIBUTIONS OF QUADRATIC FORMS IN CENTRAL NORMAL VARIABLES II N. L. Johnson* and Samuel Kotz** University of North Carolina Chapel Hill, N. C. Temple University Philadelphia, Pa. Institute of Statistics Mimeo Series No. 557 December 1967 * Supported by National Institute of General Medical Sciences under grant 5ROIGM-12868 and by Air Force Office of Scientific Research under grant AFOSR-68 1415. ** Supported by Air Force Office of Scientific Research under grant AFOSR-68-l411. DEPARTMENT OF STATISTICS University of North Carolina Chapel Hill, N. C.

- i - Introduction These tables are an extension of the tables published in [lj. They give values of the probability density function and cumulative distribution function of the random variable. where the U's are mutually independent unit normal variables satisfying the condition a 1 +a 2 +a 3 + a 4 + a 5 = 5 Methods of calculation and range of tabulation are described in [lj. The format of the tables is not quite uniform, as reduction to five significant figures was provided from a number of different sources. In some cases, the figures are given in form.17021 02-2 This represents.17021 x 10 =.0017021. In these sets of figures, and in a few others, a dot above the last figure shown indicates that it should be increased by one. For example.22109 signifies.22110. Our acknowledgements to D. W. Boyd and to Mrs. Carole Newman have already been made in [lj. We would also like to thank Mrs. Rosalie Kotz for her help in preparing this set of tables. REFERENCE [lj Johnson, N. L. and Kotz, Samuel (1967). Tables of distributions of quadratic forms in central normal variables. University of North Carolina Mimeo Series No. 543.

-! - Distribution of Positive Definite Quadratic Forms n = 5 \ '-...-' el: (3.0, 0 5, 0 5, 0.5; 0 5 ) "---- y p(y) l'r[y,s:d y p(y) Pr[Y~yJ 0.1.0089364.00036011 4.4.10887.58778 0.2.023266.0019513 4.5.10577.59851 0.3.03935(1.0050669 4.6.10273.60894 0.4.055815.0098346 4.7.099765.61906 0.5.071874.016225 4.8.096874.62889 0.6.087088.024179 4.9.094056.63844 0.7.10119.033603 5 0.091314.64771 O.P.11404.044374 5 1.088648.65671 0.9.12557.056370 5.2.086056.66544 1.0.13577.069442 5.3.083539.67392 1.1.1 ~465.083476 5.4.081097.68.215 1.2.15226.098331 5 5.078727.69014 1.3.15868.11389 5.6.076430.69790 1.4.16397.13003 5.7.074203.70543 1.5.16821.14665 5.8.072046 7127J.j, 1.6 17 l'-s. 8.16364 5 9.069957.71984 1.7.17387.18091 6.0.067934 72671.j, 1.8.17546.19839 6.1.065976.73343 1.9.17632.21598 6.2.064080.73993 2.0.17653.23363 6.3.062246.74625 2.1.17616.25127 6.4.060471.75238 2.2.1'1528.26885 6.5.058755.75835 2 3.17394.28631 6.6.057094.76414 2. Is..17221.30362 6.7.055487.76977 2.5.1'7013.32074 6.8.053933.77524 2.6.16776.3.3764 6.9.052429 780 55 2.7.16514 35 1.J.29 7 0.050975.78572 2.8.16230.37066 7.1.049568 79075 2.9.15929.386'74 7.2.048206.79564 3.0.15614.40251 7.3.046890.80039 3 1.15288.41796 7.4.045615.80502 3 2.lJ.J.953.43309 7 5.044382.80952 3.3.1J.~612 Jl-L~7 87 7.6.043189.81390 3.4.1/.,,266.ll-6231 7.7.042034.81816 3 5.13918.47640 7.8.040916.82230 3.(:;.13570.l~9014 7.9.039834.82634 3 7.13222.50354 8.0.038785.83027 3 9.12875 51659 8.1.037770.83 1 +10 3.9.12532 52929 8.2.036787.837P3 IhO.12193.54166 8.3.035835.84146 1-1-.1.11858 55368 8.4.034912.84499 1.J..2.11528.56537 8.5.034018.P4844 I~. 3.11205.57674 8.6.033151.85180

2-0( (3.0, 0.5, 0.5, 0 5, 0.:) ) Y 1-(Y) Pr (Y!Y] Y 1-(Y) Pr [ Y(Y] (J. '7.032311.65507 17.4.0047401.97545 b.b.0314959.85820 17.6.0045536.97637 d.').030706.86137 17.8.004 37L.~ 97727 j.o.02s:939.86440 18.0.0042036.97013 9 1.02~196.66736 18.2.0040393.')7695 '}.2.028474.07024 18.4.0038818.97974 ~ 3.027774.87305 18.6.003'7308.98050 ~1. 4.027095.87580 18.8.0035859.98123 ';) 5.026435.87647 19 0.0034469.981')14 " ~). 0.025794.88109 19 2.0033136.98261 ;;.7.025172.88364 19.4.0031857.98326 9.6.024567.68612 19.6.0030629.98389 ';}.).023980.68855 19.8.0029451.9844>; 10.0.023409.89092 20.0.0028320.98507 10.1.022854.89323 20.2.0027234.98562 10.2.022315.8Y5)+9 20.4.0026192.98615 10. 1+.021281.89985 20.6.0025191.98667 10.6.020302.90401' 20.8.0024231.98716 10.0.01:j376.90797 21.0.0023308.98764 11.0.01849Y.91176 21.2.0022421.98810 11.2.017667.91537 21.4.0021570.98854 11.4.01687> 91883 21.6.0020753.98896 11.6.016130 92213 21.8.0019967. 98Y37 n.8.015420 92528 22.0.0019212.98976 12.0.014745. 92830 22.2.0018487.99011.. 12.2.014103.93118 22.4.0017790.99050 12.4.013493.933y4 22.6.0017120.99085 12.6.012912.93658 22.8.0016476.99118 12.8.012359 93911 23.0.0015858.99151 13.0.011833.94153 23 2.0015263.99182 13.2.011331 94385 23.4.0014691 99212 13.4.010853.94606 23.6.0014142.99240 13.6.010398.94819 23.8.0013613.99268 13.8.0099631 95022 24.0.0013105 99295 14.0.0095484.95217 24.2.0012616 99321 14.2.0091526.95404 24.4.0012147.99345 14.4.0087747.95584 24.6.0011695.99369 14.6.0084137 95755 24.8.0011260.99392 14.8.008068y 95920 25 0.0010842.99414 15 0.0077394.96078 25.2.0010440.99435 15 2.0074244.96230 25.4.0010054.99456 15.4.0071233.96375 25.6.00096817.99476 15.6.0068353.96515 25.8.00093238.99495 15.8.0065598.96649 26.0.00089795 99513 16.0.0062962.96777 26.2.00086482.99531 16.2.0060439.96901 26.4.00093295 99548 16.4.0058024 97019 26.6.00080228.99564 16.6.0055712 97133 26.8.00077275.99580 16.8.0053499.97242 27.0.00074435 99595 17.0.0051378.97347 27.2.00071702.99610 17.2.0049347.97448 27.4.00069071.99624

.3 - DI. (3.0, 0.5,0.5, 0.5, 0.5) y p(y) Pr [y.,;y] Y p(y) pr [y~y1 "-- 2'7.6.00066539.9963'7 31.4.00032909.99819 27.8.00064101.99650 31.6.00031'720.99825 28.0.00061'755.99663 31.8.000305'75.99831 28.2.0005949'7.996'75 32.0.00029472.9983'7 28.4.0005'7323.99687 32.2.00028409.99843 28.6.00055230.99698 32.4.00027386.99849 28.8.00053215.99709 32.6.00026400.99854 29 0.00051275 99719 32.8.00025450.99859 29 2.00049408 99729 33 0.00024534.99864 29.4.00047609 99'739 33 2.00023652.99869 29.6.00045878.99748 33.4.00022803.998'74 29.8.00044210 99'75'7 33.6.00021984.99878 30.0.00042605.99766 33.8.00021195.99883 30.2.00041058.997'74 34.0.00020435.9988'7 30.4.000395'70.99'782 34.2.00019702.99891 30.6.00038135 99790 34.4.00018997.99895 30.8.00036754.99798 34.6.00018316.99898 31.0.00035424.99805 34.8.0001'7661.99902 31.2.00034143.99812 31.4 0': (2.8, 0.'7, 0.6, 0.5, 0.4) \) '-y. ':J p('1) }>.. C-~ ~ 'jj "j p(::; ) J... L ~'j.;, 0.1.0080096.00031811 3.1.15393.40145 0.2.020934.001'7424 3 2.15105.416'70 0.3.035552.0045619 3.3.14807. 43165 0.4.050617.0088722 3.4.14501.44631 0 5.065441.014680 3 5.14190.46066 0.6.079611.021942 3.6.13874.47469 0.'7.0928'79.030569 3.'7.13556.48840 0.8.10510.040483 3.8.13237.50180 0 9.11619.051556 3.9.12918.51488 1.0.12614.063682 4.0.12600.52764 1.1.13494.076744 4.1.12283.54088 1.2.14263.090635 4.2.11970 55220 1.3.14924.10523 4.3.11659.56402 1.4.15485.12045 4.4.11353.57553 1.5.15950.13617 4.5.11051.586'73 1.6.16326.15232 4.6.10754.59763 1.7.16621.16880 4.7.10463.60824 1.8.16840.18553 4.8.10176.61856 1.9.16990.20245 4.9.098960.62859 2.0.17077.21949 5.0.096217.63835 2.1.17108.23659 5 1.093535.64'784 2.2.17088.25369 5 2..090915.65'706 2.3.17023.Z7075 5.3.088359.66602 2.4.16917.28772 5.4.085866.67473 2 5.167'75.30457 5 5.08343'7.68320 2.6.16602.32126. 5.6.0810'72.69142 2.7.16401 33777 5 7.078769.69941 2.8.16177.35406 5.8.076529.'70'718 2 9.15932 3'7011 5 9.0'74352.'714'72 3 0.15670.38592 6.0.072235.72205

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