An Algorithm of a Longest of Runs Test for Very Long. Sequences of Bernoulli Trials
|
|
- Myles Riley
- 5 years ago
- Views:
Transcription
1 A Algothm of a Logest of Rus Test fo Vey Log equeces of Beoull Tals Alexade I. KOZYNCHENKO Faculty of cece, Techology, ad Meda, Md wede Uvesty, E-857, udsvall, wede alexade_kozycheko@yahoo.se Abstact A ew algothm of computg statstcs of a logest of us test s poposed fo the case of equal pobablty Beoull tals pocesses. The algothm s fouded o the aalyss of the evet tee dagam, whch has show the ole of Fboacc umbes of hghe odes coutg the umbe of outcomes of teest the sample space. The poof by ducto s gve. Compaed to the classcal combatoal fomulas, the poposed algothm povdes the eo-fee exact pobabltes ad makes possble the pocessg of vey log bomal data sets up to 3 o cotempoay computes. Keywods: us tests, logest u, Beoull tals, Fboacc umbes, computg algothms Mathematcs ubject Classfcatos: 6G; 6-4; B39. Itoducto Dstbuto-fee tests fo adomess of a sample data play a mpotat ole much statstcal feece ad ae elevat to may applcatos socology, bology, psychology, egeeg etc., cludg such patcula poblems as egesso ad
2 cuve fttg. Thee s a geat body of lteatue o the subject, wothy of meto of whch ae the books by egel ad Castella, [], ad pet []. Ths pape s coceed wth the computatoal aspects of a mpotat dstbutofee us test, amely, the logest of us test of adomess appled to log pattes of bomal tals. Amogst a umbe of publcatos the us tests vestgato, t s woth metog the pape by Mood, [3], ad the moogaph by Badley, [4], who gave a detaled teatmet of us tests, as well as the appopate suvey of the woks doe 94s-6s by Olmstead, Mostelle, Gat, Bu ad Cae, et al. The exstg us tests ae based o ethe umbes of us o legths of us. The total-umbe-of-us test povdes both exact combatoal fomulas ad asymptotc oes the assumpto of omally dstbuted statstcs fo lage samples see, e.g., [4], p. 6. Howeve, fo the logest of us tests thee s o such a asymptotc theoy, ad we have to use the exact combatoal fomulas. o, the questo ases as to whethe those fomulas ae applcable fo computg the statstcs o cotempoay computes the case of lage samples, o t s ecessay to deve moe adequate theoy applcable to pocessg log samples. P. Aalyss of the classcal combatoal fomulas fo the logest of us test The covetoal appoach to devg a geeal fomula fo the pobablty?, of obtag at least oe u of legth o geate amog ethe the s o the s had bee descbed [3]. It s to be oted that ethe cludes the possblty
3 3 of both. The appoach s based o the fomula of calculatg the pobablty of a sum of adom compatble evets: P?, P P,, + P ad,, whee,,? ae umbes of us of s, s, ad of uspecfed type of elemet cotag the u, espectvely; P s the pobablty of obtag at least oe u of legth, amog the s but ot amog the s; P s the pobablty of obtag at least oe such u amog the s, but ot amog the s; P ad s the pobablty of obtag at least oe such,, u amog both the s ad the s; uppose that a sequece of tals cotas s ad s. I ths case, the pobabltes ca be computed o the followg combatoal fomulas: P, / + +, / + + P,, 3
4 / / / /,, ad P 4 These u fomulas take ad as gve. But the case of Beoull tals, whe ad ae mutually exclusve outcomes wth pobabltes p ad q espectvely of occuece o a sgle tal, t would be coveet to elmate paametes ad. The exteso whee ad ae ot fxed, so that the pobablty s completely espectve of ad depeds oly o ad p, s descbed [3]. The compoud pobablty s obtaed by takg the poduct of the bomal pobablty q p ad the pobablty beg computed o -4. The sum of that poduct ove all possble values of gves the sought-fo pobablty:,,?,?, p p P p P 5 Evdetly, t s woth whle developg the computg algothm ode to check the coectess ad to evaluate the pefomace of ths fomula. The autho has ceated the C++ pogam that computes the pobablty,?, p P usg the fomulas - 5. The code s placed Appedx A. A umbe of computg tests has bee
5 5 accomplshed, ad the aalyss of the esults has evealed two dawbacks of the fomula 5. Fst of all, t gves a systematc eo that mafests tself the expesso 4. Let us cosde, fo stace, the case of 8, 4, p.5. The computatos o the fomula 5 gve the pobablty P,.375 ad the umbe of?, p outcomes of teest N 96 the umbe of all possble outcomes equals to Howeve, the coect values ae ad 94, espectvely. The easo s the fomula 4 that gves the eoeous zeo value fo the pobablty P, ad,, wheeas the coect value s -7, whch coespods to two outcomes of teest avalable: ad. I ode to futhe checkg of the classcal fomulas, the autho developed a bute-foce algothm based o the beadth-fst seach techque. It gves the exact solutos, but has the expoetal computato tme t O ad theefoe caot be appled to samples of legth > 4. The compaso of the esults obtaed by ths algothm wth that of the classcal fomulas dscloses that the classcal fomulas gve a egula postve eo whe. 5. Ths evdetly cofms the fallacy of the fomula 4, sce t elates to the paths wth two o moe us of legth whe the fomula 4 s appled. ecodly, the computg tests have evealed a uppe lmt o the legth of bomal sequeces beg pocessed o cotempoay PCs equpped, e.g., the AMD Athlo 64x Dual Coe pocesso 46+. Ths lmt amouts to 8 fo < ad p.5. uch a estcto does ot actually allow pocessg vey log bomal sequeces whee the adequate powe of us tests could be attaed.
6 6 3. Descpto of the poposed algothm usg the Fboacc umbes, ts poof, ad pefomace The logest of us test of a Beoull tals pocess ca be aalysed usg a bay tee dagam as the stadad techque of epesetg the sample space ad coutg pobabltes, [5]. The paths wth outcomes of teest cota at least oe u of legth o geate. They all ae dcated Fg., whee sold ad dotted les mea success o, say, ad falue o of a chace expemet, coespodgly. Numbe of a tal j, C Fg.. The half of a evet tee dagam showg the umbe of the us of legth the sequece of Beoull tals of legth 8 4 B F D A E 4 7 Fboacc umbes F+-, - of ode -, j- 4 ome paths cotag a u of legth at the eds ae show completely fom the oot to a leaf e.g., ABDF, wheeas the othes ae pooled to clustes of paths havg both a tal commo explct pat that eds wth a u of legth ad a subsequet abtay sub-tee see, e.g., clustes ABC o ABDE.
7 7 We wll cosde the patcula ad most mpotat case of a Beoull tals pocess wth equal pobabltes of successes ad falues o a chace expemet p q.5. Hee, the pobabltes of all outcomes ae the same, beg equal to.5 fo Beoull tals. Hece, ode to compute the pobablty P?, of obtag at least oe u of legth o geate amog ethe the s o the s we eed to calculate the umbe of outcomes of teest. I the case 8, 4 depcted Fg., ths umbe ca be estmated as follows: 4 3 8, p N, 6 4?, 4 3 whee the fst tem,, coespods to the cluste ABC, the secod oe,, coespods to ABDE, ad so o utl the tem 7 that elates to the dvdual paths ot cludg sub-tees, as ABDF. As we ca see, the factos,,, 4, 7 fom a pat wthout zeos of the sequece of Fboacc umbes of 3 d ode. Havg aalysed the tee dagams fo othe, cases, the geeal fomula fo abtay, < s deved: N, p.5?, F + F F F F, +, + 3, whee F, s a + th Fboacc umbe of - ode. + +, +, 7 o, the fomula fo the sought-fo pobablty s deved fom by dvdg t by the umbe of all possble outcomes : F +, P + 8?,, p.5 The fomula 7 ca be poved by mathematcal ducto:
8 8. The bass: the fomula 7 holds whe. Ideed, ths case 7 gves us the coect umbe two of paths of legth that cota a u of legth : N +, p.5 F F?, +,,. The ductve step: suppose that the fomula 7 holds fo some. We eed to pove that the fomula 7 also holds whe + s substtuted fo. Let us wte dow the fomula 7 fo +: N +, p.5?, F +, + + F + F +, + 3, + + The fst summad of ths expesso gves a umbe of those outcomes of teest fo + bay tals, whch ae geeated fom the eds of all paths exstg at the th level of a tee dagam. These outcomes ae epeseted by clustes of paths at the +st level see Fg.. The secod summad gves a umbe of the sgle outcomes of teest appeag at the + st level. These ew outcomes belog to the paths havg oly oe u of the legth whch s stuated at the ed of the path. These tematg us ogate at the paths havg the same featue oly oe u of the legth at the ed of the path at levels, -,. The total umbe of these geeatg paths equals to the sum of the umbes at levels,, K, +. As we ca see fom the fomula 7 fo the th level wtte usg the Hoe scheme N +?,, p.5 F +, F + F + + F + K+ F + F KK +, +, K +, 3,,.
9 9 the abovemetoed umbes equal to the Fboacc umbes F, + j, j,. Ths meas that the acto F +3, of the secod summad equals to the sum of Fboacc umbes j F + j, ad, theefoe, s deed a Fboacc umbe of - ode by defto. That s, the ductve step s pove The aalyss of computato pefomace of the fomula 8 has bee caed out usg the C++ code gve Appedx B. Fst of all, the pogam calculates the elated Fboacc umbes placed to aay fb. I so dog the computg algothm takes to accout the e stuctue of a Fboacc umbes sequece, whch cotas a tal sub-sequece of umbes beg a powe of. eveal Fboacc umbes sequeces ae lsted Table A, the tal sub-sequeces beg selected by the gey backgoud colou. Table A. Fboacc umbes F+, of ode,,
10 The secod pat of the pogam computes the pobablty P, p.5 by the?, fomula 8. The algothm ad code ae able to make calculatos o cotempoay PCs fo vey log Beoull tals sequeces, up to 3. The esults obtaed ae depcted the Fg. ad ca be used to test fo adomess of a patte of Beoull tals wth p q. 5 ull hypothess. Pobablty of obtag at least oe u of legth o geate amog ethe the s o the s, p.5 P?, ze of the sequece of Beoull tals Fo example, let us cosde a sequece of s ad s of legth 6 cotag a u of legth 4, ad test the ull hypothess ude the sgfcace level α. 5. Cosultg the Fg., oe ca fd that the ull hypothess should be ejected. If we cease the umbe of tals up to, the chace pobablty that a sequece of Beoull tals wth p q. 5 would cota a u of 4 o moe cosecutve ethe
11 s o s s about.6, so the ull hypothess caot be ejected at the gve sgfcace level. The Fboacc umbes of hghe odes ae used othe Beoull tals elated poblems, such as the co tossg see, e.g. [6], whee the pobablty that o us of k cosecutve tals wll occu co tosses s gve by Fboacc k-step umbe kth ode. F k + /, whee k F l s a 4. ummay I the pape, a ew poweful appoach to the logest of us test s descbed, whch ca effectvely eplace the classcal combatoal fomulas the patcula, but mpotat, case of equal pobabltes Beoull tals pocesses. Ths appoach s based o a thoough aalyss of the evet tee dagam, whch suggested devg a cocse fomula fo the pobablty of obtag at least oe u of legth o geate amog ethe the s o the s. The deved fomula extesvely uses the Fboacc umbes of hghe odes. The fomula poves to be capable pocessg vey log dchotomous sequeces up to 3 as compaed to 8 fo the classcal combatoal appoach. The coectess of the esults obtaed was checked by a beadth-fst seach algothm, ad the complete cocdece has bee show. The sde esult of the pape les evealg a egula eo beg heet the classcal combatoal algothm some cases. 5. Ackowledgemets The autho would lke to thak Pof. Wej-M Huag fo hs commets ad suggestos that led to mpovemets the pape.
12 Appedx A //The C++ code developed fo computg the statstcs of the //classcal logest of us test: #clude<osteam> #clude<cmath> #clude<omap> usg amespace std; double Factoalt double t; t ; fot, ; < ; ++ t * ; etu t; double Ct, t f < etu ; double t; t ; fot, ; > -+; -- t * ; etu t/factoal; double Pobt, t s, double p.5, double q.5 double pob ; t ; fot ; < ; ++ double pob, pob, pob ; fo ; < /s; ++ pob + pow-, +*C-+, *C-*s, -; fo ; < -/s; ++ pob + pow-, +*C+, *C-*s, ; fot ; < -s+; ++ double a, a, a3, a4 ; fo ; <-/s-; ++ a + pow-, +*C, *C--*s-, -; fo ; < --+/s-; ++ a + pow-, +*C-, *C---*s-, -; fo ; < --/s-; ++ a3 + pow-, +*C, *C---*s-, -; fo ; < ---/s-; ++ a4 + pow-, +*C+, *C---*s-, ; pob + a*a + *a3 + a4; pob + pob + pob pob*powp, *powq, -; etu pob; t ma t 8, s 4; double pob Pob, s; cout.setfos::fxed; cout << " " << << " " << "s " << s << edl <<"cout "
13 3 <<setw6<<setpecso<<pob*pow,<< edl << "pob4 " << setw4<<setpecso4<<pob<<edl; etu ; Appedx B //The C++ code fo the poposed logest of us test algothm //usg the Fboacc umbes: #clude<osteam> #clude<cmath> #clude<omap> usg amespace std; double RusFbt, t s double* fb ew double[-s+]; fot ; < -s; ++ fb[] ; double p ; fs > cout << "eo" << edl; etu -; else fb[] ; fb[] ; fo ; < s- && < -s+; ++ fb[] pow,-; fo s-; < -s; ++ fot j ; j < s-; ++j fb[] + fb[-j-]; fo ; < -s; ++ p + fb[]*pow.5, s+; p * ; cout.setfos::fxed; cout << " " << << " s " << s << edl; cout << " pob4 " << setpecso4 << p << edl; delete [] fb; etu p; t ma t 8, s 4; RusFb,s; etu ;
14 4 Refeeces [] egel,., Castella, N.J., J., 988, Nopaametc tatstcs fo the Behavoal ceces, d ed. New Yok: McGaw-Hll. [] pet P., 993, Appled Nopaametc tatstcal Methods, d ed. Lodo: Chapma & Hall. [3] Mood, A. M., 94, The Dstbuto Theoy of Rus, Aals of Mathematcal tatstcs,, [4] Badley, J.V., 968, Dstbuto-Fee tatstcal Tests Eglewood Clffs, New Yok: Petce Hall. [5] Gstead C. M., ell J. L., 997, Itoducto to Pobablty, d ev. ed. Ameca Mathematcal ocety. [6]
Professor Wei Zhu. 1. Sampling from the Normal Population
AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple
More informationRECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S
Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets
More informationsuch that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1
Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9
More information( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi
Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)
More informationRecent Advances in Computers, Communications, Applied Social Science and Mathematics
Recet Advaces Computes, Commucatos, Appled ocal cece ad athematcs Coutg Roots of Radom Polyomal Equatos mall Itevals EFRAI HERIG epatmet of Compute cece ad athematcs Ael Uvesty Cete of amaa cece Pa,Ael,4487
More informationXII. Addition of many identical spins
XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.
More informationχ be any function of X and Y then
We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto,
More informationOn EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx
Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.
More informationExponential Generating Functions - J. T. Butler
Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a 2 2 + a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle
More information2.1.1 The Art of Estimation Examples of Estimators Properties of Estimators Deriving Estimators Interval Estimators
. ploatoy Statstcs. Itoducto to stmato.. The At of stmato.. amples of stmatos..3 Popetes of stmatos..4 Devg stmatos..5 Iteval stmatos . Itoducto to stmato Samplg - The samplg eecse ca be epeseted by a
More informationLecture 10: Condensed matter systems
Lectue 0: Codesed matte systems Itoducg matte ts codesed state.! Ams: " Idstgushable patcles ad the quatum atue of matte: # Cosequeces # Revew of deal gas etopy # Femos ad Bosos " Quatum statstcs. # Occupato
More information= y and Normed Linear Spaces
304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads
More informationThe Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof
MATEC Web of Cofeeces ICIEA 06 600 (06) DOI: 0.05/mateccof/0668600 The ea Pobablty Desty Fucto of Cotuous Radom Vaables the Real Numbe Feld ad Its Estece Poof Yya Che ad Ye Collee of Softwae, Taj Uvesty,
More informationModule Title: Business Mathematics and Statistics 2
CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ Semeste Eamatos 009/00 Module Ttle: Busess Mathematcs ad Statstcs Module Code: STAT 6003 School: School of Busess ogamme Ttle: Bachelo of
More informationMinimum Hyper-Wiener Index of Molecular Graph and Some Results on Szeged Related Index
Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: 359-0040 Vol Issue, Febuay - 05 Mmum Hype-Wee Idex of Molecula Gaph ad Some Results o eged Related Idex We Gao School of Ifomato Scece ad Techology,
More informationThe Exponentiated Lomax Distribution: Different Estimation Methods
Ameca Joual of Appled Mathematcs ad Statstcs 4 Vol. No. 6 364-368 Avalable ole at http://pubs.scepub.com/ajams//6/ Scece ad Educato Publshg DOI:.69/ajams--6- The Expoetated Lomax Dstbuto: Dffeet Estmato
More informationˆ SSE SSE q SST R SST R q R R q R R q
Bll Evas Spg 06 Sggested Aswes, Poblem Set 5 ECON 3033. a) The R meases the facto of the vaato Y eplaed by the model. I ths case, R =SSM/SST. Yo ae gve that SSM = 3.059 bt ot SST. Howeve, ote that SST=SSM+SSE
More informationFairing of Parametric Quintic Splines
ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}
More informationTrace of Positive Integer Power of Adjacency Matrix
Global Joual of Pue ad Appled Mathematcs. IN 097-78 Volume, Numbe 07), pp. 079-087 Reseach Ida Publcatos http://www.publcato.com Tace of Postve Itege Powe of Adacecy Matx Jagdsh Kuma Pahade * ad Mao Jha
More informationChapter 7 Varying Probability Sampling
Chapte 7 Vayg Pobablty Samplg The smple adom samplg scheme povdes a adom sample whee evey ut the populato has equal pobablty of selecto. Ude ceta ccumstaces, moe effcet estmatos ae obtaed by assgg uequal
More informationGREEN S FUNCTION FOR HEAT CONDUCTION PROBLEMS IN A MULTI-LAYERED HOLLOW CYLINDER
Joual of ppled Mathematcs ad Computatoal Mechacs 4, 3(3), 5- GREE S FUCTIO FOR HET CODUCTIO PROBLEMS I MULTI-LYERED HOLLOW CYLIDER Stasław Kukla, Uszula Sedlecka Isttute of Mathematcs, Czestochowa Uvesty
More informationBest Linear Unbiased Estimators of the Three Parameter Gamma Distribution using doubly Type-II censoring
Best Lea Ubased Estmatos of the hee Paamete Gamma Dstbuto usg doubly ype-ii cesog Amal S. Hassa Salwa Abd El-Aty Abstact Recetly ode statstcs ad the momets have assumed cosdeable teest may applcatos volvg
More informationDistribution of Geometrically Weighted Sum of Bernoulli Random Variables
Appled Mathematc, 0,, 8-86 do:046/am095 Publhed Ole Novembe 0 (http://wwwscrpog/oual/am) Dtbuto of Geometcally Weghted Sum of Beoull Radom Vaable Abtact Deepeh Bhat, Phazamle Kgo, Ragaath Naayaachaya Ratthall
More informationNon-axial symmetric loading on axial symmetric. Final Report of AFEM
No-axal symmetc loadg o axal symmetc body Fal Repot of AFEM Ths poject does hamoc aalyss of o-axal symmetc loadg o axal symmetc body. Shuagxg Da, Musket Kamtokat 5//009 No-axal symmetc loadg o axal symmetc
More information2. Sample Space: The set of all possible outcomes of a random experiment is called the sample space. It is usually denoted by S or Ω.
Ut: Rado expeet saple space evets classcal defto of pobablty ad the theoes of total ad copoud pobablty based o ths defto axoatc appoach to the oto of pobablty potat theoes based o ths appoach codtoal pobablty
More informationHyper-wiener index of gear fan and gear wheel related graph
Iteatoal Joual of Chemcal Studes 015; (5): 5-58 P-ISSN 49 858 E-ISSN 1 490 IJCS 015; (5): 5-58 014 JEZS Receed: 1-0-015 Accepted: 15-0-015 We Gao School of Ifomato Scece ad Techology, Yua Nomal Uesty,
More informationMinimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses
Mmzg sphecal abeatos Explotg the exstece of cojugate pots sphecal leses Let s ecall that whe usg asphecal leses, abeato fee magg occus oly fo a couple of, so called, cojugate pots ( ad the fgue below)
More informationCISC 203: Discrete Mathematics for Computing II Lecture 2, Winter 2019 Page 9
Lectue, Wte 9 Page 9 Combatos I ou dscusso o pemutatos wth dstgushable elemets, we aved at a geeal fomula by dvdg the total umbe of pemutatos by the umbe of ways we could pemute oly the dstgushable elemets.
More informationVECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.
Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth
More informationLecture 9 Multiple Class Models
Lectue 9 Multple Class Models Multclass MVA Appoxmate MVA 8.4.2002 Copyght Teemu Keola 2002 1 Aval Theoem fo Multple Classes Wth jobs the system, a job class avg to ay seve sees the seve as equlbum wth
More informationAIRCRAFT EQUIVALENT VULNERABLE AREA CALCULATION METHODS
4 TH ITERATIOAL COGRESS OF THE AEROAUTICAL SCIECES AIRCRAFT EQUIVALET VULERABLE AREA CALCULATIO METHODS PEI Yag*, SOG B-Feg*, QI Yg ** *College of Aeoautcs, othweste Polytechcal Uvesty, X a, Cha, ** Depatmet
More informationFUZZY MULTINOMIAL CONTROL CHART WITH VARIABLE SAMPLE SIZE
A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) FUZZY MUTINOMIA CONTRO CHART WITH VARIABE SAMPE SIZE A. PANDURANGAN Pofesso ad Head Depatmet of Compute Applcatos Vallamma Egeeg College,
More informationQuestion 1. Typical Cellular System. Some geometry TELE4353. About cellular system. About cellular system (2)
TELE4353 Moble a atellte Commucato ystems Tutoal 1 (week 3-4 4 Questo 1 ove that fo a hexagoal geomety, the co-chael euse ato s gve by: Q (3 N Whee N + j + j 1/ 1 Typcal Cellula ystem j cells up cells
More informationChapter Linear Regression
Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use
More informationFIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES
Joual of Appled Matheatcs ad Coputatoal Mechacs 7, 6(), 59-7 www.ac.pcz.pl p-issn 99-9965 DOI:.75/jac.7..3 e-issn 353-588 FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationIterative Algorithm for a Split Equilibrium Problem and Fixed Problem for Finite Asymptotically Nonexpansive Mappings in Hilbert Space
Flomat 31:5 (017), 143 1434 DOI 10.98/FIL170543W Publshed by Faculty of Sceces ad Mathematcs, Uvesty of Nš, Seba Avalable at: http://www.pmf..ac.s/flomat Iteatve Algothm fo a Splt Equlbum Poblem ad Fxed
More informationAllocations for Heterogenous Distributed Storage
Allocatos fo Heteogeous Dstbuted Stoage Vasleos Ntaos taos@uscedu Guseppe Cae cae@uscedu Alexados G Dmaks dmaks@uscedu axv:0596v [csi] 8 Feb 0 Abstact We study the poblem of stog a data object a set of
More information(b) By independence, the probability that the string 1011 is received correctly is
Soluto to Problem 1.31. (a) Let A be the evet that a 0 s trasmtted. Usg the total probablty theorem, the desred probablty s P(A)(1 ɛ ( 0)+ 1 P(A) ) (1 ɛ 1)=p(1 ɛ 0)+(1 p)(1 ɛ 1). (b) By depedece, the probablty
More informationInequalities for Dual Orlicz Mixed Quermassintegrals.
Advaces Pue Mathematcs 206 6 894-902 http://wwwscpog/joual/apm IN Ole: 260-0384 IN Pt: 260-0368 Iequaltes fo Dual Olcz Mxed Quemasstegals jua u chool of Mathematcs ad Computatoal cece Hua Uvesty of cece
More informationTHREE-PARAMETRIC LOGNORMAL DISTRIBUTION AND ESTIMATING ITS PARAMETERS USING THE METHOD OF L-MOMENTS
RELIK ; Paha 5. a 6.. THREE-PARAMETRIC LOGNORMAL DISTRIBUTION AND ESTIMATING ITS PARAMETERS USING THE METHOD OF L-MOMENTS Daa Bílová Abstact Commo statstcal methodology fo descpto of the statstcal samples
More informationLearning Bayesian belief networks
Lectue 6 Leag Bayesa belef etwoks Mlos Hauskecht mlos@cs.ptt.edu 5329 Seott Squae Admstato Mdtem: Wedesday, Mach 7, 2004 I class Closed book Mateal coveed by Spg beak, cludg ths lectue Last yea mdtem o
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationA New Approach to Moments Inequalities for NRBU and RNBU Classes With Hypothesis Testing Applications
Iteatoal Joual of Basc & Appled Sceces IJBAS-IJENS Vol: No:6 7 A New Appoach to Momets Iequaltes fo NRBU ad RNBU Classes Wth Hypothess Testg Applcatos L S Dab Depatmet of Mathematcs aculty of Scece Al-Azha
More informationPattern Avoiding Partitions, Sequence A and the Kernel Method
Avalable at http://pvamuedu/aam Appl Appl Math ISSN: 93-9466 Vol 6 Issue (Decembe ) pp 397 4 Applcatos ad Appled Mathematcs: A Iteatoal Joual (AAM) Patte Avodg Pattos Sequece A5439 ad the Keel Method Touf
More informationQuasi-Rational Canonical Forms of a Matrix over a Number Field
Avace Lea Algeba & Matx Theoy, 08, 8, -0 http://www.cp.og/joual/alamt ISSN Ole: 65-3348 ISSN Pt: 65-333X Qua-Ratoal Caocal om of a Matx ove a Numbe el Zhueg Wag *, Qg Wag, Na Q School of Mathematc a Stattc,
More informationLecture 12: Spiral: Domain Specific HLS. Housekeeping
8 643 ectue : Spal: Doma Specfc HS James C. Hoe Depatmet of ECE Caege Mello Uvesty 8 643 F7 S, James C. Hoe, CMU/ECE/CACM, 7 Houseeepg You goal today: see a eample of eally hghlevel sythess (ths lectue
More informationAtomic units The atomic units have been chosen such that the fundamental electron properties are all equal to one atomic unit.
tomc uts The atomc uts have bee chose such that the fudametal electo popetes ae all equal to oe atomc ut. m e, e, h/, a o, ad the potetal eegy the hydoge atom e /a o. D3.33564 0-30 Cm The use of atomc
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
More informationRobust Regression Analysis for Non-Normal Situations under Symmetric Distributions Arising In Medical Research
Joual of Mode Appled Statstcal Methods Volume 3 Issue Atcle 9 5--04 Robust Regesso Aalyss fo No-Nomal Stuatos ude Symmetc Dstbutos Asg I Medcal Reseach S S. Gaguly Sulta Qaboos Uvesty, Muscat, Oma, gaguly@squ.edu.om
More informationON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE
O The Covegece Theoems... (Muslm Aso) ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE Muslm Aso, Yosephus D. Sumato, Nov Rustaa Dew 3 ) Mathematcs
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationNumerical Solution of Non-equilibrium Hypersonic Flows of Diatomic Gases Using the Generalized Boltzmann Equation
Recet Advaces Flud Mechacs, Heat & Mass asfe ad Bology Numecal Soluto of No-equlbum Hypesoc Flows of Datomc Gases Usg the Geealzed Boltzma Equato RAMESH K. AGARWAL Depatmet of Mechacal Egeeg ad Mateals
More informationA. Thicknesses and Densities
10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe
More informationPermutations that Decompose in Cycles of Length 2 and are Given by Monomials
Poceedgs of The Natoa Cofeece O Udegaduate Reseach (NCUR) 00 The Uvesty of Noth Caoa at Asheve Asheve, Noth Caoa Ap -, 00 Pemutatos that Decompose Cyces of Legth ad ae Gve y Moomas Lous J Cuz Depatmet
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationHarmonic Curvatures in Lorentzian Space
BULLETIN of the Bull Malaya Math Sc Soc Secod See 7-79 MALAYSIAN MATEMATICAL SCIENCES SOCIETY amoc Cuvatue Loetza Space NEJAT EKMEKÇI ILMI ACISALIOĞLU AND KĀZIM İLARSLAN Aaa Uvety Faculty of Scece Depatmet
More informationChapter 17. Least Square Regression
The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz Pt 5 - CURVE FITTING Descbes techques
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationUniversity of Pavia, Pavia, Italy. North Andover MA 01845, USA
Iteatoal Joual of Optmzato: heoy, Method ad Applcato 27-5565(Pt) 27-6839(Ole) wwwgph/otma 29 Global Ifomato Publhe (HK) Co, Ltd 29, Vol, No 2, 55-59 η -Peudoleaty ad Effcecy Gogo Gog, Noma G Rueda 2 *
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationFault diagnosis and process monitoring through model-based case based reasoning
Fault dagoss ad pocess motog though model-based case based easog Nelly Olve-Maget a, Stéphae Negy a, Glles Héteux a, Jea-Mac Le La a a Laboatoe de Gée Chmque (CNRS - UMR 5503), Uvesté de Toulouse ; INPT-ENSIACET
More informationAPPROXIMATE ANALYTIC WAVE FUNCTION METHOD IN ELECTRON ATOM SCATTERING CALCULATIONS. Budi Santoso
APPROXIMATE ANALYTIC WAVE FUNCTION METHOD IN ELECTRON ATOM SCATTERING CALCULATIONS Bud Satoso ABSTRACT APPROXIMATE ANALYTIC WAVE FUNCTION METHOD IN ELECTRON ATOM SCATTERING CALCULATIONS. Appoxmate aalytc
More informationA GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING
TATITIC IN TRANITION-ew sees Octobe 9 83 TATITIC IN TRANITION-ew sees Octobe 9 Vol. No. pp. 83 9 A GENERAL CLA OF ETIMATOR UNDER MULTI PHAE AMPLING M.. Ahed & Atsu.. Dovlo ABTRACT Ths pape deves the geeal
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationTHE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the
More informationThe calculation of the characteristic and non-characteristic harmonic current of the rectifying system
The calculato of the chaactestc a o-chaactestc hamoc cuet of the ectfyg system Zhag Ruhua, u Shagag, a Luguag, u Zhegguo The sttute of Electcal Egeeg, Chese Acaemy of Sceces, ejg, 00080, Cha. Zhag Ruhua,
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More informationA DATA DRIVEN PARAMETER ESTIMATION FOR THE THREE- PARAMETER WEIBULL POPULATION FROM CENSORED SAMPLES
Mathematcal ad Computatoal Applcatos, Vol. 3, No., pp. 9-36 008. Assocato fo Scetfc Reseach A DATA DRIVEN PARAMETER ESTIMATION FOR THE THREE- PARAMETER WEIBULL POPULATION FROM CENSORED SAMPLES Ahmed M.
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More informationAn Unconstrained Q - G Programming Problem and its Application
Joual of Ifomato Egeeg ad Applcatos ISS 4-578 (pt) ISS 5-0506 (ole) Vol.5, o., 05 www.ste.og A Ucostaed Q - G Pogammg Poblem ad ts Applcato M. He Dosh D. Chag Tved.Assocate Pofesso, H L College of Commece,
More informationmeans the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
More informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationChapter 2: Descriptive Statistics
Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationMedian as a Weighted Arithmetic Mean of All Sample Observations
Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of
More information1. Overview of basic probability
13.42 Desg Prcples for Ocea Vehcles Prof. A.H. Techet Sprg 2005 1. Overvew of basc probablty Emprcally, probablty ca be defed as the umber of favorable outcomes dvded by the total umber of outcomes, other
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationProbability. Stochastic Processes
Pobablty ad Stochastc Pocesses Weless Ifomato Tasmsso System Lab. Isttute of Commucatos Egeeg g Natoal Su Yat-se Uvesty Table of Cotets Pobablty Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Statstcal
More informationObjectives. Learning Outcome. 7.1 Centre of Gravity (C.G.) 7. Statics. Determine the C.G of a lamina (Experimental method)
Ojectves 7 Statcs 7. Cete of Gavty 7. Equlum of patcles 7.3 Equlum of g oes y Lew Sau oh Leag Outcome (a) efe cete of gavty () state the coto whch the cete of mass s the cete of gavty (c) state the coto
More informationThe internal structure of natural numbers, one method for the definition of large prime numbers, and a factorization test
Fal verso The teral structure of atural umbers oe method for the defto of large prme umbers ad a factorzato test Emmaul Maousos APM Isttute for the Advacemet of Physcs ad Mathematcs 3 Poulou str. 53 Athes
More informationBayesian Nonlinear Regression Models based on Slash Skew-t Distribution
Euopea Ole Joual of Natual ad Socal Sceces 05; www.euopea-scece.com Vol.4, No. Specal Issue o New Dmesos Ecoomcs, Accoutg ad Maagemet ISSN 805-360 Bayesa Nolea Regesso Models based o Slash Skew-t Dstbuto
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationMA 524 Homework 6 Solutions
MA 524 Homework 6 Solutos. Sce S(, s the umber of ways to partto [] to k oempty blocks, ad c(, s the umber of ways to partto to k oempty blocks ad also the arrage each block to a cycle, we must have S(,
More informationApplication of Generating Functions to the Theory of Success Runs
Aled Mathematcal Sceces, Vol. 10, 2016, o. 50, 2491-2495 HIKARI Ltd, www.m-hkar.com htt://dx.do.org/10.12988/ams.2016.66197 Alcato of Geeratg Fuctos to the Theory of Success Rus B.M. Bekker, O.A. Ivaov
More informationf f... f 1 n n (ii) Median : It is the value of the middle-most observation(s).
CHAPTER STATISTICS Pots to Remember :. Facts or fgures, collected wth a defte pupose, are called Data.. Statstcs s the area of study dealg wth the collecto, presetato, aalyss ad terpretato of data.. The
More informationSome Notes on the Probability Space of Statistical Surveys
Metodološk zvezk, Vol. 7, No., 200, 7-2 ome Notes o the Probablty pace of tatstcal urveys George Petrakos Abstract Ths paper troduces a formal presetato of samplg process usg prcples ad cocepts from Probablty
More informationNUMERICAL SIMULATION OF TSUNAMI CURRENTS AROUND MOVING STRUCTURES
NUMERICAL SIMULATION OF TSUNAMI CURRENTS AROUND MOVING STRUCTURES Ezo Nakaza 1, Tsuakyo Ibe ad Muhammad Abdu Rouf 1 The pape ams to smulate Tsuam cuets aoud movg ad fxed stuctues usg the movg-patcle semmplct
More informationP 365. r r r )...(1 365
SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty
More information