Scietific Review ISSN(e): 41-599, ISSN(p): 41-885 Vol. 4, Issue. 4, pp: 6-, 018 URL: http://arpgweb.com/?ic=joural&joural=10&ifo=aims Academic Research Publishig Group Origial Research Aekwe's Correctios o the Negative Biomial Epasio Ope Access Abstract Uchea Okwudili Aekwe Departmet of Physics, Uiversity of Sciece ad Techology, Aleiro, Nigeria For so may years ow a lot of scietist have used the series of positive biomial epasio to solve that of Negative biomial epasio, positive fractioal biomial epasio ad Negative fractioal biomial epasio which was geerated/derived usig Maclauri series to derive the series of Negative biomial epasio, positive fractioal biomial epasio ad Negative fractioal biomial epasio just as it was used to provide aswers to positive biomial epasio but fails for All the other epasio due to a deviatio made. This Mauscript cotais the correct solutio/aswers to Negative biomial epasios with proofs through worked eamples, with other forms of solvig Negative biomial epasio just as i the case of Pascal s triagle i positive biomial epasio, i Negative biomial epasio it is called Aekwe s triagle ad other methods like the combiatio method of solvig Negative biomial epasio. Keywords: Negative biomial; Epasio; Equatio. CC BY: Creative Commos Attributio Licese 4.0 1. Itroductio Accordig to Coolidge [1] the biomial Theorem, familiar at least i its elemetary aspects to every studets of algebra, has a log ad reasoably plai history. Most people associate it vaguely i their mids with the ame of Newto s; he either iveted it or it was carved o his tomb. I some way or the other it was his theorem. Well, as a matter of fact it was t, although his work did mark a importat advace i the geeral theory. We fid the first trace of Biomial Theorem i Euclid II, 4, if a straight lie be cut at radom, the square o the whole is equal to the square o the segmets ad twice the rectagle of the segmets. if the segmets are a ad b this meas i algebraic laguage a b a b ab The correspodig formula for the square differece is foud i Euclid II, 7, if a straight lie be cut at radom, the square o the whole ad that o oe of the segmets both together, are equal to twice the rectagle cotaied by the whole ad said segmet, ad the square o the remaiig segmet,. From the formula above it is see that from the left had side that if a& b are summed up together ad the squared the results obtaied must be the same for the right had side that is a squared plus b squared plus time the product of a & b. Therefore, if the result for the Negative biomial epasio estimated from the left had side of a equatio is ot eactly equal to the Right had of the equatio just as i the case of positive biomial the we ca either get a approimate results or a wrog result. With relevat worked eamples i this Mauscript shows the eact solutio to Negative biomial ad the deviatios made usig the Maclauri series to obtai the epasio for Negative biomial. Joh [] the egative biomial distributio is iterestig because it illustrates a commo progressio of statistical thikig, ad it s viewed i terms of Coutig, Cheap geeralizatio ad Modelig over dispersio.. Methodology For so may years, there has bee a Mistake made i usig the series of positive biomial i fidig the epasio of the egative biomial. This material presets ew solutios to the series of the Negative biomial usig worked eamples. Eample.1 Epad the followig egative biomial Solutio Usig the log divisio method we ve (1+) -1 (1+) -1 = 1 1 6
Scietific Review 1 1 1 1 1 (1+) - =((1+) -1 ) = = 1 = 1 1 1 1 1 1 =1 1 (1+) -1 =1 1 (1+) - = 1 1 (1 ) Studyig the series above it is see that the epasio for egative biomial is i the form of b (1+) - = a + 1 + c d... ad ot (1 ) (1 ) (1+) - = a + b +c +d +. As used mistakely by may scietist over so may years. From the series of egative biomial the coefficiets ca be determied. Proof: If b c d 1 a... 1 (1 ) (1 ) (1) Puttig =0 i equatio (1) we ve a 1 () Differetiatig equatio (1) w.r.t we ve 1 b c d... 4 (1 ) (1 ) (1 ) Multiplyig although equatio () by (1+) we ve +1 c d 1 b. (1 ) (1 ) Puttig =0 i equatio (4) we ve b (5) Differetiatig equatio (4) w.r.t we ve.1c.d 1 (1 ). (1 ) (1 ) Multiplyig although equatio (6) by (1+) we ve +.d 1 (1 )! c. (1 ) Puttig =0 i equatio (7) we ve 1! c ().. (4).. (6).. (7) ( 1) c (8)! Differetiatig equatio (7) w.r.t we ve +1..1d 1 ( )(1 )... (9) (1 ) Multiplyig although equatio (9) by (1+) we ve + 1 (1 ) 7
+ 1 ( )(1 )! d... (10) Puttig =0 i equatio (10) we ve 1 ( )! d d 1 ( ) (11)! Scietific Review Substitutig equatio (), (5), (8) ad (11) ito equatio (1) we ve ( 1) (1 ) 1 1!(1 ) ( 1)( )!(1 ) Puttig... i equatio (1) we ve ( 1) (1 ) 1 1!(1 ) ( 1)( )!(1 ) Where OR (1 )... (1) (1) (1 ) (1 )( ) 1... 1!(1 )!(1 ) Note (14) I the geeral case such as ( ) a ( a) 1 1 a From which 1 a a we ve the epasio to be give as ca be epaded usig either equatio (1) or (14) ad the multiplied through by. Eample.. Fid the epasio of the series (1 ) ad hece compute its umerical value from the epasio whe =1 ad whe = respectively. Solutio Usig equatio (1) above we ve ( ) ( 1) ( 1)( ) (1 ) 1... 1!(1 )!(1 ) ( 1) ( 1)( ) 1... 1!(1 )!(1 ) 1 (1 ) (1 ) (1 ) 1... Whe =1 we ve 1 1 (1) (1) (1) (1 1) () 1... 8 () () 81 6 1 1 (1 ) 8 8 = = ( ) ( )( 1) 1... 1!(1 )!(1 ) = 1 1 1 8 4 8 8
Scietific Review Whe =1 (1 ) 8 1 Also whe = i the same way we ve (1 ) 1 64.. The Negative Triagle of the Biomial Epasio Cosiderig the followig epasios (1+) 0 = 1-1 (1 ) 1 1 - (1 ) 1 1 (1 ) - (1 ) 1 1 (1 ) (1 ) From the epasio above we ve Figure-1. Aekwe's Triagle The egative triagle of biomial epasio was formed from the series of egative biomial epasio, just like that of the positive triagle of biomial epasio ow kow as the Pascal s triagle. Note that the egative biomial epasio is foud the same way the positive biomial is foud oly that i the egative biomial there is traspositio (redistributio) of charges/sigs. Figure-. Traspositio of the Positive Biomial Epasio After traspositio we ve 9
Scietific Review Figure-. Aekwe's Triagle Note that after traspositio we've the triagle of the egative biomial as the Aekwe s Triagle. [-7] For easy uderstadig ote from the charge distributio that all odd pricipal diagoals are positive while all eve diagoals are egative i.e. all odd positio i a epasio takes positive sig while all eve positio after traspositio takes egative sig. I.e. after traspositio Figure-4. Positios of the Negative Biomial Epasio OR Note that i egative biomial epasio we ve 0
a 1 a ( a) 1 1 Scietific Review a a ( 1) 1 a 1 a! 1 1 a ( 1)( )... a! 1 1 a ( 1) a ( 1)( ) a 1... a!( a)!( a) 1 a ( 1) a ( a) ( a)! ( a) ( 1)( ) a! ( a)... Where I terms of the egative biomial epasio we ve the applicatio of k(costats) from the egative triagle of biomial epasio as 1 1 a ( a) ( a) K1 K Eample.. Fid the epasio of (+a) - from the egative triagle of biomial epasio. Solutio From the egative triagle of biomial epasio we ve the coefficiet as (1-1) (K 1 =1, K =-, & K =1) o substitutio ito equatio (y) above we ve 1 a a ( a)... ( a) ( a) Eample.4. Fid the epasio of (+a) - from the egative triagle of biomial epasio. Solutio From the egative triagle we ve the coefficiet as (1-1) 1 a a a ( a)... ( a) ( a) ( a).. Epasio of the Negative Biomial Usig the Combiatio Method Lookig at the series geerated above i equatio (X), the Series ca be writte i terms of combiatio give by () 1 a K... ( a) (y) 1
0 1! a ( a) 0! 0! a Scietific Review 1 1! a 1! a... 1! 1! a!! a Eample.5. Epad the followig egative biomial usig the combiatio method (+1) - Solutio 0 1! 1 ( 1) 0! 0! 1 1 1! 1 1! 1 1! 1! 1!! 1 = 1 1 1 1 1 (1) (1) 1 1 = 1 1 ( 1) ( 1) Fid the epasio of (+1) - Solutio 0 1! 1 ( 1) ( ) 0! 0! 1 1 1! 1 1! 1 ( ) 1! 1! 1 ( )!! 1 1 1 1 = 1 4 4 ( 1) 4 ( 1) 1 1 1 1 ( 1) 4 ( 1) 4 ( 1)... Real Life Applicatio of Biomial Theorem The biomial theorem has a lot of Applicatios. [8, 9] Some of the applicatios i real life Situatios are: Computig I computig areas, biomial theorem has bee very Useful such as i distributio of IP addresses. With Biomial theorem, the automatic distributio of IP Addresses is ot oly possible but also the Distributio of virtual IP addresses. Ecoomy Ecoomists used biomial theorem to cout probabilities that deped o umerous ad very distributed variables to predict the way the ecoomy will behave i the et few years. To be able to come up with realistic predictios, biomial theorem is used i this field. Architecture Architecture idustry i desig of ifrastructure, allows egieers, to calculate the magitudes of the projects ad thus deliverig accurate estimates of ot oly the costs but also time required to costruct them. For cotractors, it is a very importat tool to help esurig the costig projects is competet eough to deliver profits..4. Applicatios of Biomial Epasio i Physics The biomial Epasio has other Applicatios i physics amogst which we've its Applicatios i 1. Gravitatioal time dilatio.. Kietic eergy. Electric quadrupole field. 4. Relativity factor gamma 5. Kiematic time dilatio..
Scietific Review Other Applicatios of Biomial Epasios are i: [10] Agriculture i Solvig Problems i Geetics.. Coclusio From the worked eamples doe above we ve see the correct solutio to Negative biomial epasio ad leart how to solve Negative biomial usig Aekwe s triagle ad combiatio methods together with Negative biomial epasio methods of solvig Negative biomial epasio. Refereces [1] Coolidge, J. L., 1949. The story of the biomial theorem. Harvard Uiversity. [] Joh, D., 009. "Cook Negative biomial distributio." [] Godma, A., Talbert, J. F., ad Ogum, G. E. O., 1984. Additioal mathematics for West Africa. Harlow: Logma. [4] Egbe, E., Odili, G. A., ad Ugbebor, O. O., 000. Further mathematics published by Africaa. Gbagada, Lagos: FEB Publishers. [5] Stroud, K. A. ad Deter, J. B., 01. Egieerig mathematics. 7th ed. Covetry Uiversity, UK.: Palgrave. [6] Tuttuh, A. M. R., Sivasubramaiam, S., ad Adegoke, R., 016. "Further mathematics project." [7] Uchea, O. A., 017. The foudatio to mathematical scieces. Germay: Lambert Academic Publishig. [8] 01. "Real Life Applicatio of Biomial Theorem." Available: https://irsyadirifa.wordpress.com [9] Carl, R. N., 015. "Biomial epasio." Available: http://hyperphysics.phy-astr.gsu.edu/hbase/alg.html [10] CAI Xiuqig, L. J., ZHUANG Nasheg,, 016. "Applicatio of the biomial theorem i solvig problems i geetics."