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fl’ZV A 0 A066 272UNCLASSIFIED-FOREIGN TECHNOLOGY DIV WRIGHT—PATTERSON AFB OHIOF/G 12/1GENERALIZATION OF A . N. KOLM 060ROV’S CRITERION FOR EVALUATING T——ETC (U )SEP 77 G N NAN IYAFTD—ID (RS )T—1672—77NLn E !P r I IENDDA6-TE79!!II-

--—----- .-RS)T-].672—77FTD—ID (FOREIGN TECHNOLOG Y DIVISION GENERALIZATION OF A. N. KOLMOGOROV’S CRITERIONFOR EVALUATING THE LAW OF DISTRIBUTION BY EMPIRICALDATAby0. M. NanlyaApproved for public -release;distribution unlimited.— 11 09 1378 U. 13

.—--.-- --— . --- - - — -- — -----. - - -—--— .—-—--— FTDEDITED TR ANSLATI ONFTD—ID(RS)T— 1672—77MICROFICHE N H :15 September 1977- 77 Oo f/ q/GENERALIZATION OF A. N. KOLMOGOROV ’S CRITERIONFOR EVALUATING THE LAW OF DISTRIBUTION BY EMPIRCALDATABy :G. M. ManiyaEnglish pages:Source:6Do kiady Aka d em ii Nauk SSSR , Vol. 69, No.Lj ,l9 49, PP. 1 95.1497Country of origin: USSRTranslated by: Carol S. NackRequester: AFFDL/FBRDApproved for public release; distribution unlimitedcEsSIOII f NT OOCU AP NOUNC DJUSTIFICATIONWW1 Section Sufl ssction O0—- :—Dist.THIS TRANSLATION IS A RENDITION OP THE ORIGI.HAl. FOREIGN TEXT WITHOUT ANY ANA LYTICAL OREDITORIAL COMMENT. STATEMENTS Oft THEORIESADVOCAT EDOR IMPLIEDARETHOSEOP THE SOURCEAND DO HOT NECESSARILY REFLECT THE POSITIONON OPINION OF THE FOREIGN TECHNOLOGY DI.VISION.FTD— -. -PREPA RED BYzSII‘ I‘-.-.- -----1TRANSLATION DIVISIONFOREIGN TECHNOLOGY DIVISIONWP.AFS OHIO.Datel5 SeDtl9 77IL S)T—l 672—77.,SPtCIALA . L a id/ot-.a. . . a.-—. . . —— —— .— . .-.-----— I

- ---—.-,---- .-—-- . -— -U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATIONBlockAa6 BBr r D,, HI111E e3 ItalicA s65B pE .A 3 ,K uRI(iixJI17M MM MHHH0 a0fl nV, vSrTransliterationA,aB, b/7G, gD, dZ, z1, 1.iLit-4 ’iVWwlii, u L, 1bIUbbb e3 IC] to.9 A) ATs , tsCh , chz Y, yHU, UF, fKh , kh xU FlT, tY y0XSYSTEMSh , hShch , shchY, yE, eYu , yuYa , yaIinit ially, after vowels , and af t er b , ; e elsewhere . as ye or e.When written as ë in Russian , transliterateThe use of diacritical marks is preferred , but such marksmay be omitted when expediency dictates.* GREEKA N8NuXivG ammaDeltaryOmicron0oEZ RhoPSigmaEp BEp silonZe t ar.ALPHABETAlphaBetaEtaHIotaIThetaKappaLamb daMu6Fl.* ri00 tK)tAM AuflTPhiChipsiTu X x wOmega78 11’ 09 -aTauUpsilon--- ct--- ----—,—. a. . -- — .— . -- —-.— .— —-. ——— .-———— —.---Trans LlterationR, rS, spWiuill iqF, paC cT m X xYe , ye; E , e Zh , zhN, n0, ooC cPppY yM, mi ItalicT TK, k Block- . --— ----——.—- 1

R USS IAN AND ENGLISH TRIGONOMETRIC FUNCTIONSRussianEnglishsincostgc tgseccosecsincostancotseccscshchs inhcoshtarthcothsechcschthc thsc hcscharcarcarcarcarcarcsincostgctgSeccosecarc sharc charc thsinh cosh tanh arc ctLcoth arc scharc cschseèh csch rotlg sin cos tan 1cot sec csc curllogGRAPHICS DISCLAIMERAfl figures, graphIcs, tables , equation s , etc.merged Into th is translation were extractedfrom the best quality copy available. ii.

TDOC—1672PAGE11672GE R ER AUZA TION OP A. I. KOLNOGOROT ’S CEITIRIO N POE EVAL EJA T IE G TEE LAlOF DISTRIBUT IOR BY ENP IR ICAL DATAG. B. M aniya(Pres nted by Acade uician A. I. Kol.ogor o v on 7 OctoberLetX 1 X s1.g.asrsl a.atis.sx ;, x ; ,., X;.Xabe a set of inde pendentisv of distribution19 9)values With theP (z) . Pu rther n ore , letbe the sane series, but in orde r of their .sgnit ude s . ll call the .ap iri cal distr ibvtio n function at .p fun ctionSa(X) — - . .- -.--- ----—---- -- T-.- -- —.::: i : j

DOCa1672PAGE2I StS (z). T:. -- w ; iTL .R X , k/I St X x R 1 ,im We sill designat eDaSUp IS,(X ) pfr ) 1,D u- supf S . (x) p (x) ; .Acc ording to the known theorea proven by A. N . Kolsoq orov (1].for eich 3 .0 an rando . continuous distribution funct ion P (z)P{Da )z4J 1 2 ( I)’ * r4” ’.(1)In one of his results [2], B . V . Sairno , .stablisbes theasy.ptotic f steels fst diattibeti a D :(2)P (D -} —.1—r ’.v iVe viii g.n.ra liz. A. I . Koliogorow and I. V . Siirnov ’scoton6.sc criteria , coasid.ring the eaxinu. d eviat ion for aspecific .ction (0 C C I) . of th. groith of function P (x) .-a--——.-—

DOC—167 2PAGE3We will fiM two ra ado . valn .0,,0,) sup (S,(X).P(X)) ,D (supD,(0,, 0,) ISa(. ) P(X)I.The results we obtained can be state d in the for. of thefollowing two theoreas:Theorea 1. Let 1( x ) be the continuou s function of theIi.tribstio. of .sch of t he talep.ad it valeu x1 (i 2,., a),0, o ‘( ) u 0 )0, o ( ), 0 01 03 1.theeP{D (Or), Or)) :. ehst 0 (O,,O,;A)—VI,— a-I2 Vf ’b y,‘ ‘ 51(1,.il — ‘I‘ 0(0,,0,;)), ’hu 1’t ’1 idz,—,)dz,dz,,b — (,,,s,) 1p 1ti 2 Riri*r, z J, 1(r,,z,)— - iIz — 2Rra z, z).; A) cam also be repr.s .st.d as follows:Puactios0(S,,S,

DOC1672PAGE0’(0,, 0 ;A) He rs0(a)(x) II- r 0(m) (a) I ( ) (b) —-e—VI’ (*) (a ’) 0(a)(b’).is t ie n—tb order der ivat ive of the nor.al integral0(x) In the .o.t in terestinghav ee—” ’dz. specific cas e, when 0’(0;? ) sa‘— — e—’IJ (‘g.S,) 1— 0, we willd;d , —e— ’J” i”.‘ ) dz, dz,,whereSr.(I—Vo(I—0U)Whence we will, obtain (2) at 9—0.Theotea 2. Under th. conditions in t keor ea IIP(DI (O ’),O *)) ).n’Is ) . c1)(0,,0,;) .) ,whereup on :1 -- :Ii:::

,- - - . --- .-----— DOC—167 2PAG E— 50(0,,0,;A) s— ’l”2 .(. b.-I- 2 YI— R’ S S e— ’1”—. — “ dz ,d;—C’ ‘ ) di , dzwher eis —Ys,(1—I,)—‘We can represent function0(011 O,;A) La a different for. asfollows:G(O ,0,;A) —2 jb-I0(Ø() )(a) (b)(— 1) ’— ’e— ” E 1—ra-S a— 1 we obtaisIn part icelar , vh s. .I which was first considered by A. N. Kol.ogor ov.When —1-0 —(1) , i.e. the case0, we will have a aore co.pact andsy.a.bri ca l .zprei aio n for the li.itin g function.This .tbod aake s it possible to t he oretically solve the prob le.of the applicabil ity of the th eoretical law at th ose boun darie s where

e -?: . -DOC1672PAG E6the .aterial which is available to us is ore reliable forcompar ison.Tb. proofs of the or eas 1 and 2 are based on the theore as ofcontinuityof rando, functions and the Laplace transfor.Boscow City Pedagogical Instit ute iaeui V. P. Pote.kinReceived 7 October 1949References‘ A. Ii. K o a w o r o p o . .Olorn. d. Alt., 4. 83 (1933). ‘H. B. C M s p H o , ,Main, . c6op. ., (48), (1939).’ V. Fe ll. ,, Aim , of I’lat.h. Stat istIc s , 19, 2,177 (J948). . .; - -

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fl’z - v a0 a066 272 foreign technology div wright—patterson afb ohio f/g 12/1 generalization of a. n. kolm060rov’s criterion for evaluating t——etc(u) sep 77 g n naniya unclassified ftd—id(rs)t—1672—77 nl