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

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1 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 and by Air Force Office of Scientific Research under grant AFOSR ** Supported by Air Force Office of Scientific Research under grant AFOSR-68-l411. DEPARTMENT OF STATISTICS University of North Carolina Chapel Hill, N. C.

2 - 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 This represents x 10 = 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 signifies 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.

3 -! - 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 ( O.P ~ J.j, l'-s j, ' Is ' J ' lJ.J J.~612 Jl-L~ /.,,266.ll (:; l~ P3 IhO J P4844 I~

4 2-0( (3.0, 0.5, 0.5, 0 5, 0.:) ) Y 1-(Y) Pr (Y!Y] Y 1-(Y) Pr [ Y(Y] (J. ' b.b d.') L.~ j.o.02s: ~ ')7695 '} ~ ' ~ ';) ')14 " ~) ;; ';}.) >; Y5) ' :j Y > Y37 n y y

5 .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' ' ' ' ' '7.996' ' ' ' ' '75' ' ' '70.99' ' ': (2.8, 0.'7, 0.6, 0.5, 0.4) \) '-y. ':J p('1) }>.. C-~ ~ 'jj "j p(::; ) J... L ~'j.;, ' ' '7.0928' ' ' ' ' Z ' ' ' '70' ' '74352.'714'

6 ..j - y i') I._ ('...., 0.6, 0.5, o. Lt ) ~~. u, u. i, pc ) pr : F" y f(y) pr.. I j ~; ~ j.. ) '3.72) (1275) o -? ) SJ )1.636 ti )3) /'5 ~,;4Gu5 0_).06253u.7556<:: )~ ~ '..) Si52c;;' 1).7.0))1046.1'67U~ ) (/ i./i'r) 6.9.0) ) S:5 ::,' )61)4 '( ')6352 ' G ;: "(. 5.01~ L() 7.l) ;; ;;1'( ~4 1/ '7 '.-J.0 1 I "-J ' :! r(.).01,2094.8~n )lL~ ( n 1.-,. c c u.j ~)7697 ( ')77:;0 t' 5, ' EJ :0132 9fJ ) ) / [; ') (j17G~ h.0169> '.).o161gcj n.,,' 0'...1-) r.:j.l')~1,{_, L.,J.011\ " ,':' 1:2.:-+.01)i '[' c))

7 - 5-0/' (2.8, 0.7, 0.6, 0.5, 0.4).. y p(y) Pr [y\y] y p(y) Pr (~yj \ " ') c<: (2~~ 0.;7, 0.5, 0.5, 0.5) p::j ) p(~ l p~ C'f :"j1 :J 'PI' L...!!'J:. ":f '

8 (../ (2. b, r, : v. i, 0.5, o. :), 0.5) Con't. J 1- (, '; pr [ U:/. y 'L (... \ pr! '(~\ \,J ) ~ <I /... v 5 1.0::-' _>'Cc::;t, 5 2.0; b.01;io36 ';/1 30C; r- ~ ). 'oj.06855;; )0 ~)16b Y204b OCJ1223.6:; CU6146.:; rr r ';1':) :} ':) v0,. 05l~ l l t ) l~ lJ l~ , ) l~

9 -7- ~._- DL: (2.8, 0.7, 0.5, 0.5, 0.5) Con't. y p(y) PI' l y~yj y p(y) PI' [y,y. ' :X: (2.8, 0.6, 0.6, 0.5, 0.5) "'"'D i,', p(~ :J ).1"",/<1 "f p -( l - r" \ ~ i..,.. ~."~,j~'j~ J

10 ss -./ : C' c...u, ;. O. (), 0.6, 0.5, 0.5) y P (:i) Pr ['[- y- y P (:: ) Pr LVq ~. v ') :.J.O.o311L; )/.1 0 3Oc?)!~ ~,1 c:: /" 3 } ').2.02;,;5J6 () (4.JU ,S ') c~ L ~)6) ory'/,-_,c 9.4 t,: <:-j..,l... " ; '(26S' G.8e (883) ') S!L~ b~34L SJ ')823 4.; ~ ~ ( L~ 5 ~.0862Ge ) r: ) LI ';) c

11 ,)I : - \} - (2.8, 0.6, 0.6, 0.5, 0.5) con't. y p(y) Pr Ly~"yj y p(y) Pr Ly~'y ' Q / /: (2.5,1.2,0.5,0.4,0.4) r (J).Y... CY~:'-I j r('j ) It Cy~ 'f: ~.; "

12 - '10- :~' : (2.5, 1.2, O.S, O.b, O.b) can't y p(y) Pr[Y~ y. y pcy) PrJr-s y L.. ~ S.l Ob h?? ' h b23~ U.3h9.h199h S.h U308.43h < h S U362h ~ h58 4.2' h h h b ~ u.0885uO.66u ' h uu l1.u u hS88 6.h ul h8h h u X) h R ' h u ;: h S Oh o05638h.97587

13 (~ : (2.5, 1.2, 0.5, 0.4, 0.4) ',,-- ~ y p(j~) PrLy~i y p(y) Pr[y~yJ J ! ' u iJ D D ' )

14 1 " r:;/ (2.5,1.0,0.6,0.5,0.4) Y p(y) F(y)=pr lyty1 y p(y) F(y)=prI }28,.6818( } } } ~ ' ,8} , , a l t , ; , , ', , , } j.839} ;593 If.O.1?9C:; } l+., } If } } }

15 oi ( ) ) ) y p(y) Pr f'l5 y] y p(y) Pr [y'{yj ~-

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19

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24 - 22 -, ' I 01.. ( 2.3 j ) 0.4, 0.4 ) Y r (::/ ) PrC'!~~J '!:J ;(:/) P.. []1! ' i' ~ ' ii ~.66R sl ~ oj 5.70 lj )1 0.1)85' l ', O.AO (J"71Ti & oj 6'-'{J 0: 73700) 'O.-1-t6 tl~ ! , ot' -O~ ' O. 103~6.' A O.14516~ \ -01 " 1. LIO i -01: ~4Q bo '; ~Oi " ) J 6.90 O.5n , , O.5669Jj' -01 ' 'D. 7816"~ J ~ qO~ ; 1.1'0 U ;';"01' "0, O. 52j8~i ~ ' -01' ' ' : i' lJ04' -01 O.8131~ J 7.80 o. 4714~! Q ~90 O.IJ592-rr..,;'(rt- ' : Q ' ,, 3.30 : ~ ';39'209' -1J O J f '3'83' ~ ' , fi 89' ~ O.d ' S763} Tr 0.28'615 ":'01' , ' & -01 O.ti , , ~ a ~ SQ !:) _

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