Classes of Ordinary Differential Equations Obtained for the Probability Functions of Frĕchet Distribution

Transcription

1 Proceedings of he World Congress on Engineering and Comuer Science 07 Vol I WCECS 07, Ocober 5-7, 07, San Francisco, USA Classes of Ordinary Differenial Equaions Obained for he Probabiliy Funcions of Frĕche Disribuion Hilary I. Okagbue, Member, IAENG, Pelumi E. Ogununde, Abiodun A. Oanuga, Enahoro A. Owoloko Absrac In his aer, he differenial calculus was used o obain some classes of ordinary differenial equaions (ODE) for he robabiliy densiy funcion, quanile funcion, survival funcion, inverse survival funcion, hazard funcion and reversed hazard funcion of he Frĕche disribuion. The saed necessary condiions required for he eisence of he ODEs are consisen wih he various arameers ha defined he disribuion. Soluions of hese ODEs by using numerous available mehods are a new ways of undersanding he naure of he robabiliy funcions ha characerize he disribuion. The mehod can be eended o oher robabiliy disribuions and can serve an alernaive o aroimaion. Inde Terms Quanile funcion, Frĕche disribuion, reversed hazard funcion, calculus, differeniaion, robabiliy densiy funcion. F I. INTRODUCTION RȆCHET disribuion is one of he mosly alied disribuions in ereme value heory. Deailed informaion abou he disribuion can be obained from he early works of [] and [] and book wrien by [3]. Differen mehods of esimaion of he arameers of he disribuion were discussed eensively by [4]. Some of he alicaions are as follows: Zaharim e al. [5] used he disribuion o model wind seed daa, Harlow [6] worked on he usefulness of he disribuion in modeling engineering roblems, Nadarajah and Koz [7] discussed eensively on he alicaion of he Fréche random variables o sociological models. Vovoras and Tsokos [8] used he disribuion o model and analyze reciiaion daa. Deails on he alicaion of he disribuion in modeling eremal daa can be found in [9]. Many auhors and researchers have roosed modificaions or develoed generalizaions of he disribuion. Some of hem are: bea Fréche disribuion [0] [], Kumaraswamy Fréche disribuion [], ransmued Fréche disribuion [3], ransmued eoneniaed Fréche disribuion [4], gamma eended Fréche disribuion [5], Marshall Olkin Fréche disribuion [6], ransmued Marshall Olkin Fréche This work was sonsored by Covenan Universiy, Oa, Nigeria. H. I. Okagbue, P. E. Ogununde, A. A. Oanuga and E. A. Owoloko are wih he Dearmen of Mahemaics, Covenan Universiy, Oa, Nigeria. hilary.okagbue@covenanuniversiy.edu.ng elumi.ogununde@covenanuniversiy.edu.ng abiodun.oanuga@covenanuniversiy.edu.ng alfred.owoloko@covenanuniversiy.edu.ng disribuion [7], Weibull Fréche disribuion [8], siarameer Fréche disribuion [9], bea eonenial Fréche disribuion [0] The aim of his research is o develo ordinary differenial equaions (ODE) for he robabiliy densiy funcion (PDF), Quanile funcion (QF), survival funcion (SF), inverse survival funcion (ISF), hazard funcion (HF) and reversed hazard funcion (RHF) of Fréche disribuion by he use of differenial calculus. Calculus is a very key ool in he deerminaion of mode of a given robabiliy disribuion and in esimaion of arameers of robabiliy disribuions, amongs oher uses. The research is an eension of he ODE o oher robabiliy funcions oher han he PDF. Similar works done where he PDF of robabiliy disribuions was eressed as ODE whose soluion is he PDF are available. They include: Lalace disribuion [], bea disribuion [], raised cosine disribuion [3], Loma disribuion [4], bea rime disribuion or invered bea disribuion [5]. II. PROBABILITY DENSITY FUNCTION The robabiliy densiy funcion of he Frȇche disribuion is given as; ( ) ( ) e f () To obain he firs order ordinary differenial equaion for he robabiliy densiy funcion of he Frȇche disribuion, differeniae equaion (), o obain; ( ) e ( ) f ( ) f ( ) ( ) e ( ) f ( ) f ( ) The condiion necessary for he eisence of equaion is,, 0. The firs order ordinary differenial equaions can be obained for he aricular values of he arameers. The few cases are summarized in he Table. () (3) ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

2 Proceedings of he World Congress on Engineering and Comuer Science 07 Vol I WCECS 07, Ocober 5-7, 07, San Francisco, USA Table : Classes of differenial equaions obained for he robabiliy densiy funcion of he Frȇche disribuion for differen arameers. ordinary differenial equaion f ( ) ( ) f ( ) 0 3 f ( ) (3 ) f ( ) 0 f ( ) ( ) f ( ) 0 3 f ( ) (3 8) f ( ) 0 Equaion (3) is differeniaed in an aem o obain ordinary differenial equaions ha are no deenden on he owers of he arameers. ( ) ( ) 4 f ( ) f ( ) 3 ( ) + f( ) (4) The condiion necessary for he eisence of equaion (4) is,, 0. The following equaions obained from equaion (3) are needed in he simlificaion of equaion (4); f( ) ( ) f ( ) f( ) f ( ) f( ) f ( ) f( ) 3 f ( ) f ( ) f ( ) (5) (6) (7) (8) ( ) ( ) (9) f 4 f ( ) ( ) ( ) ( ) f 4 f ( ) ( ) ( ) (0) () Subsiue equaions (5), (8) and () ino equaion (4) o obain; ( ) f( ) f ( ) f ( ) f ( ) f ( ) f( ) f( ) f ( ) () f ( ) ( ) f ( ) ( ) f ( ) f( ) (3) f ( ) The second order ordinary differenial equaion for he robabiliy densiy funcion of he Frȇche disribuion is given by; ( ) ( ) ( ) f f f ( ) f ( ) f ( ) ( ) f ( ) 0 f () e (4) (5) f () ( ) e (6) III. QUANTILE FUNCTION The Quanile funcion of he Frȇche disribuion is given as; Q( ) ln (7) To obain he firs order ordinary differenial equaion for he Quanile funcion of he Frȇche disribuion, differeniae equaion (7), o obain; Q( ) ln Subsiue equaion (7) ino (8); Q( ) Q( ) ln (8) (9) The condiion necessary for he eisence of equaion is, 0,0. Equaion (7) can also be wrien as; ln Q ( ) Subsiue equaion (0) ino (9); (0) Q( ) Q ( ) Q ( ) Q( ) () Q( ) Q ( ) 0 () The firs order ordinary differenial equaions can be obained for he aricular values of he arameers obained from equaion (). The few cases are summarized in Table. Table : Classes of differenial equaions obained for he quanile funcion of he Frȇche disribuion for differen arameers. ordinary differenial equaion Q( ) Q ( ) 0 Q( ) Q ( ) 0 ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

3 Proceedings of he World Congress on Engineering and Comuer Science 07 Vol I WCECS 07, Ocober 5-7, 07, San Francisco, USA 3 3 Q( ) Q ( ) 0 3 Q( ) Q ( ) Q( ) Q ( ) Q( ) Q ( ) Q( ) Q ( ) Q( ) Q ( ) Q( ) Q ( ) 0 Equaion (8) is differeniaed in an aem o obain ordinary differenial equaions ha are no deenden on he owers of he arameers. ( ) ln Q ( ) Q ( ) ( ) ln (3) The condiion necessary for he eisence of equaion is, 0,0. Q ( ) Q( ) ln Equaion (9) can also be wrien as; Q( ) Q( ) ln Subsiue equaion (5) ino (4); ( ) Q( ) Q ( ) Q( ) Q( ) (4) (5) (6) The second order ordinary differenial equaion for he Quanile funcion of he Frȇche disribuion is given by; Q Q Q Q Q ( ) ( ) ( ) ( ) ( ) ( ) 0 Q(0.) (7) (8) Q(0.) (9) (.306) Some cases can be considered such as: When, equaions(7)-(9) become; Q( ) Q ( ) Q ( ) Q( ) Q( ) 0 (30) Q(0.).306 (3) 0 Q(0.) (.306) (3) IV. SURVIVAL FUNCTION The survival funcion of he Frȇche disribuion is given as; S ( ) e (33) To obain he firs order ordinary differenial equaion for he survival funcion of he Frȇche disribuion, differeniae equaion (33), o obain; S ( ) e ( ) ( ) e (34) S (35) The condiion necessary for he eisence of equaion is,, 0. Equaion (33) can also be wrien as; e S ( ) (36) Subsiue equaion (36) ino equaion (35); ( ) S ( ) ( S( ) ) (37) The firs order ordinary differenial equaions can be obained for he aricular values of he arameers obained from equaion (37). The few cases are summarized in Table 3. Table 3: Classes of differenial equaions obained for he survival funcion of he Frȇche disribuion for differen arameers. Ordinary differenial equaion S( ) S( ) 0 S( ) S( ) 0 S( ) 3 S( ) 3 0 S( ) S( ) 0 S( ) 8 S( ) 8 0 S( ) 8 S( ) 8 0 S( ) 3 S( ) 3 0 S( ) 4 S( ) 4 0 S( ) 8 S( ) 8 0 Equaion (35) is differeniaed in an aem o obain ordinary differenial equaions ha are no deenden on he owers of he arameers. ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

4 Proceedings of he World Congress on Engineering and Comuer Science 07 Vol I WCECS 07, Ocober 5-7, 07, San Francisco, USA ( ) e ( ) S( ) S ( ) ( ) e (38) The condiion necessary for he eisence of equaion is,, 0. ( ) S ( ) S( ) ( ) S( ) S( ) Equaion (37) can also be wrien as; S() S( ) Subsiue equaion (4) ino equaion (40); S( ) ( ) S( ) S( ) S( ) (39) (40) (4) (4) The second order ordinary differenial equaion for he survival funcion of he Frȇche disribuion is given by; S S S S S ( ( ) ) ( ) ( ) ( )( ( ) ) ( ) 0 (43) S() e (44) S() e (45) Some cases can be considered such as: When, equaions(43)-(45) become; ( S( ) ) S( ) S ( ) ( S( ) ) S( ) 0 (46) S() e (47) S() e (48) V. INVERSE SURVIVAL FUNCTION The inverse survival funcion of he Frȇche disribuion is given as; Q( ) ln (49) To obain he firs order ordinary differenial equaion for he inverse survival funcion of he Frȇche disribuion, differeniae equaion (49), o obain; Q( ) ( ) ln Subsiue equaion (49) ino (50); Q( ) Q( ) ( ) ln (50) (5) The condiion necessary for he eisence of equaion is, 0,0. Equaion (49) can also be wrien as; ln Q ( ) Subsiue equaion (5) ino (5); (5) Q( ) Q ( ) Q ( ) Q( ) ( ) ( ) (53) ( ) Q( ) Q ( ) 0 (54) The firs order ordinary differenial equaions can be obained for he aricular values of he arameers obained from equaion (54). The few cases are summarized in Table 4. Table 4: Classes of differenial equaions obained for he inverse survival funcion of he Frȇche disribuion for differen arameers. ordinary differenial equaion ( ) Q( ) Q ( ) 0 ( ) Q( ) Q ( ) 0 3 3( ) Q( ) Q ( ) 0 3 ( ) Q( ) Q ( ) 0 3 8( ) Q( ) Q ( ) ( ) Q( ) Q ( ) ( ) Q( ) Q ( ) ( ) Q( ) Q ( ) ( ) Q( ) Q ( ) 0 VI. HAZARD FUNCTION The hazard funcion of he Frȇche disribuion is given as; h () h () ( ) e e ( ) e (55) (56) To obain he firs order ordinary differenial equaion for he hazard funcion of he Frȇche disribuion, differeniae equaion (56), o obain; ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

5 Proceedings of he World Congress on Engineering and Comuer Science 07 Vol I WCECS 07, Ocober 5-7, 07, San Francisco, USA ( ) ( ) ( ) h (e ) ( ) e (e ) h( ) (57) The condiion necessary for he eisence of equaion is,, 0. ( ) ( ) e h( ) h( ) (e ) ( ) h( ) h( )e h( ) Equaion (56) can be simlified as; ( ) ( ) h (58) (59) () e (60) h( ) h( ) Subsiue equaion (60) ino equaion (59); ( ) ( ) h ( ) h( ) h( ) (6) h( ) h ( ) ( ( )) h( ) 0 (6) Differeniaion is carried ou again in order o obain an ordinary differenial equaion ha does no conain he owers of he arameers.; ( ) h ( ) h( )e h( ) ( ) h( )e e h( ) h( ) (63) The condiion necessary for he eisence of equaion is,, 0. The following equaions obained from equaion (59) are needed o simlify equaion (63); h( ) ( ) h ( )e (64) h ( ) ( ) h ( )e (65) e h( ) (66) Subsiue equaions (64) and (66) ino equaion (63); h ( ) ( ) h( ) h( ) h( ) h( ) h( ) h h equaion (67) becomes; When, h ( ) h( ) ( ) ( ) h( ) h( ) h( ) h h ( ) ( ) VII. REVERSED HAZARD FUNCTION (67) (68) The reversed hazard funcion of he Frȇche disribuion is given as; ( ) j() (69) ( ) ( ) j( ) ( ) ( ) (70) The condiion necessary for he eisence of equaion is,, 0. Subsiue equaion (69) ino equaion (70) o obain; ( ) j( ) j( ) (7) The firs order ordinary differenial equaion for he reversed hazard funcion of he Frȇche disribuion is given by; j( ) ( ) j( ) 0 (7) j() (73) The ODEs of all he robabiliy funcions considered can be obained for he aricular values of he disribuion. Several analyic, semi-analyic and numerical mehods can be alied o obain he soluions of he resecive differenial equaions [6-40]. Also comarison wih wo or more soluion mehods is useful in undersanding he link beween ODEs and he robabiliy disribuions. VIII. CONCLUDING REMARKS In his work, differeniaion was used o obain some classes of ordinary differenial equaions for he robabiliy densiy funcion (PDF), quanile funcion (QF), survival funcion (SF), inverse survival funcion (ISF), hazard funcion (HF) and reversed hazard funcion (RHF) of he Frȇche disribuion. The work was simlified by he alicaion of simle algebraic rocedures. Ineresingly, he ODE of he RHF yielded simle resul comared wih ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

6 Proceedings of he World Congress on Engineering and Comuer Science 07 Vol I WCECS 07, Ocober 5-7, 07, San Francisco, USA ohers. In all, he arameers ha define he disribuion deermine he naure of he resecive ODEs and he range deermines he eisence of he ODEs. ACKNOWLEDGMENT The auhors are unanimous in areciaion of financial sonsorshi from Covenan Universiy. The consrucive suggesions of he reviewers are grealy areciaed. REFERENCES [] M. Fréche, Sur la Loi des Erreurs d Observaion, Bullein Sociéé Mahé. Moscou, vol. 33,. 5-8, 94. [] M. Fréche, :Sur la loi de robabilié de l'écar maimum, Ann. Soc. Polon. Mah., vol. 6,. 93-, 98. [3] S. Koz and S. Nadarajah, Ereme Value Disribuions: Theory and Alicaions, Imerial College Press, London. ISBN: , 00. [4] A.N. Joshi and D.K. Ghosh, Freche disribuion : Parameric esimaion and is comarison by differen mehods of esimaion, In. J. Agric. Sa. Sci., vol., no., , 05. [5] A. Zaharim, S.K. Najid, A.M. Razali and K. Soian, Analysing Malaysian wind seed daa using saisical disribuion, Proc. WSEAS In. Conf. Energy and Environ., Cambridge, 009. [6] D.G. Harlow, Alicaions of he Fréche disribuion funcion, In. J. Maer. Prod. Technol. Vol. 7, , 00. [7] S. Nadarajah and S. Koz, Sociological models based on Fréche random variables, Qual. Quan., vol. 4, , 008. [8] D. Vovoras and C.P. Tsokos, Saisical analysis and modeling of reciiaion daa, Nonl. Anal. Theo. Meh. Al., vol. 7, no.,. e69-e77, 009. [9] K. Müller and W.D. Richer, Modelling eremal daa, J. Sa. Comu. Simul., vol. 87, no. 5, , 07. [0] S. Nadarajah and A.K. Gua, The bea Fréche disribuion, Far Eas J. Theor. Sa. vol. 4,. 5 4, 004. [] W. Barreo-Souza, G.M. Cordeiro and A.B. Simas, Some resuls for bea Fréche disribuion, Comm. Sa. Theo. Meh., vol. 40, , 0. [] M.E. Mead and A.R. Abd-Elawab, A noe on Kumaraswamy Fréche disribuion, Aus. J. Basic Al. Sci. vol. 8, , 04. [3] M.R. Mahmoud and R.M. Mandouh, On he ransmued Fréche disribuion, J. Al. Sci. Res., vol. 9, , 03. [4] I. Elbaal, G. Asha and V. Raja, Transmued eoneniaed Fréche disribuion: Proeries and alicaions, J. Sa. Al. Prob., vol. 3, , 04. [5] R.V. da Silva, T.A.N. de Andrade, D.B.M. Maciel, R.P.S. Camos and G.M. Cordeiro, A new lifeime model: The gamma eended Fréche disribuion, J. Sa. Theo. Al., vol., , 03. [6] E. Krishna, K.K. Jose, T. Alice and M.M. Risic, The Marshall Olkin Fréche disribuion, Comm. Sa. Theo. Meh., vol. 4, , 03. [7] A.Z. Afify, G.G. Hamedani, I. Ghosh and M.E. Mead, The ransmued Marshall Olkin Fréche disribuion: Proeries and alicaions, In. J. Sa. Prob., vol. 4,. 3 48, 05. [8] A.Z. Afify, H.M. Yousof, G.M. Cordeiro, E.M.M. Orega and Z.M. Nofal, The Weibull Fréche disribuion and is alicaions, J. Al. Sa., vol. 43, no. 4, , 06. [9] H.M. Yousof, A.Z. Afify, A.E.H.N. Ebraheim, G.G. Hamedani and N.S. Bu, On si-arameer fréche disribuion: Proeries and alicaions, Pak. J. Sa. Oer. Res., vol., no.,. 8-99, 06. [0] M.E. Mead, A.Z. Afify, G.G. Hamedani and I. Ghosh, The bea eonenial Fréche disribuion wih alicaions, Ausrian J. Sa., vol. 46, no.,. 4-63, 07. [] N.L. Johnson, S. Koz and N. Balakrishnan, Coninuous univariae disribuions, Wiley New York. ISBN: , 994. [] W.P. Elderon, Frequency curves and correlaion, Charles and Edwin Layon. London, 906. [3] H. Rinne, Locaion scale disribuions, linear esimaion and robabiliy loing using MATLAB, 00. [4] N. Balakrishnan and C.D. Lai, Coninuous bivariae disribuions, nd ediion, Sringer New York, London, 009. [5] N.L. Johnson, S. Koz and N. Balakrishnan, Coninuous Univariae Disribuions, Volume. nd ediion, Wiley, 995. [6] S.O. Edeki, H.I. Okagbue, A.A. Oanuga and S.A. Adeosun, A semi - analyical mehod for soluions of a cerain class of second order ordinary differenial equaions, Alied Mahemaics, vol. 5, no. 3, , 04. [7] S.O. Edeki, A.A Oanuga and H.I Okagbue, On ieraive echniques for numerical soluions of linear and nonlinear differenial equaions, J. Mah. Comuaional Sci., vol. 4, no. 4, , 04. [8] A.A. Oanuga, S.O. Edeki, H.I. Okagbue, G.O. Akinlabi, A.S. Osheku and B. Ajayi, On numerical soluions of sysems of ordinary differenial equaions by numerical-analyical mehod, Al. Mah. Sciences, vol. 8, no. 64, , 04. [9] S.O. Edeki, A.A. Oanuga, H.I. Okagbue, G.O. Akinlabi, S.A. Adeosun and A.S. Osheku, A Numerical-comuaional echnique for solving ransformed Cauchy-Euler equidimensional equaions of homogenous ye. Adv. Sudies Theo. Physics, vol. 9, no.,. 85 9, 05. [30] S.O. Edeki, E.A. Owoloko, A.S. Osheku, A.A. Oanuga, H.I. Okagbue and G.O. Akinlabi, Numerical soluions of nonlinear biochemical model using a hybrid numerical-analyical echnique, In. J. Mah. Analysis, vol. 9, no. 8, , 05. [3] A.A. Oanuga, S.O. Edeki, H.I. Okagbue and G.O. Akinlabi, Numerical soluion of wo-oin boundary value roblems via differenial ransform mehod, Global J. Pure Al. Mah., vol., no., , 05. [3] A.A. Oanuga, S.O. Edeki, H.I. Okagbue and G. O. Akinlabi, A novel aroach for solving quadraic Riccai differenial equaions, In. J. Al. Engine. Res., vol. 0, no.,. 9-96, 05. [33] A.A Oanuga, O.O. Agboola and H.I. Okagbue, Aroimae soluion of mulioin boundary value roblems, J. Engine. Al. Sci., vol. 0, no. 4, , 05. [34] A.A. Oanuga, O.O. Agboola, H.I. Okagbue and J.G. Oghonyon, Soluion of differenial equaions by hree semi-analyical echniques, In. J. Al. Engine. Res., vol. 0, no. 8, , 05. [35] A.A. Oanuga, H.I. Okagbue, S.O. Edeki and O.O. Agboola, Differenial ransform echnique for higher order boundary value roblems, Modern Al. Sci., vol. 9, no. 3,. 4-30, 05. [36] A.A. Oanuga, S.O. Edeki, H.I. Okagbue, S.A. Adeosun and M.E. Adeosun, Some Mehods of Numerical Soluions of Singular Sysem of Transisor Circuis, J. Com. Theo. Nanosci., vol., no. 0, , 05. [37] A.A. Oanuga, H.I. Okagbue, O.O. Agboola, "Irreversibiliy Analysis of a Radiaive MHD Poiseuille Flow hrough Porous Medium wih Sli Condiion," Lecure Noes in Engineering and Comuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., [38] A.A. Oanuga, E.A. Owoloko, H.I. Okagbue, Comarison Homooy Perurbaion and Adomian Decomosiion Techniques for Parabolic Equaions, Lecure Noes in Engineering and Comuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., [39] A.A. Oanuga, E.A. Owoloko, H. I. Okagbue, O.O. Agboola, "Finie Difference Mehod and Lalace Transform for Boundary Value Problems," Lecure Noes in Engineering and Comuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., [40] A.A. Oanuga, E.A. Owoloko, O.O. Agboola, H.I. Okagbue, "Alicaion of Homooy Perurbaion and Modified Adomian Decomosiion Mehods for Higher Order Boundary Value Problems," Lecure Noes in Engineering and Comuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07