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

Transcription

1 Proceedings of he World Congress on Engineering and Compuer Science 07 Vol II WCECS 07, Ocober 5-7, 07, San Francisco, USA Classes of Ordinary Differenial Equaions Obained for he Probabiliy Funcions of Eponeniaed Frĕche Disribuion Hilary I. Okagbue, Abiodun A. Opanuga, Member, IAENG, Enahoro A. Owoloko and Muminu O. Adamu Absrac In his work, he differenial calculus was used o obain some classes of ordinary differenial equaions (ODE) for he probabiliy densiy funcion, quanile funcion, survival funcion, inverse survival funcion, hazard funcion and reversed hazard funcion of he eponeniaed Frĕche disribuion. The saed necessary condiions required for he eisence of he ODEs are consisen wih he various parameers ha defined he disribuion. Soluions of hese ODEs by using numerous available mehods are a new ways of undersanding he naure of he probabiliy funcions ha characerize he disribuion. The mehod can be eended o oher probabiliy disribuions and can serve an alernaive o approimaion. Inde Terms Eponeniaed, Fréche disribuion, hazard funcion, calculus, differeniaion. N I. INTRODUCTION ADARAJAH and Koz [] proposed he disribuion as an improved model over he paren Fréche disribuion. The disribuion is a sub model of eponeniaed Gumbel ype- Disribuion proposed by []. The disribuion has been applied as a regression model in modeling posiive responses [3]. Oher eponeniaed class of disribuions include: eponeniaed Weibull [4-6], eponeniaed eponenial [7], eponeniaed generalized invered eponenial disribuion [8], eponeniaed generalized inverse Gaussian disribuion [9], eponeniaed invered Weibull disribuion [0-], gamma-eponeniaed eponenial disribuion [], eponeniaed gamma disribuion [3], eponeniaed Gumbel disribuion [4], eponeniaed uniform disribuion [5], bea eponeniaed Weibull disribuion [6], eponeniaed log-logisic disribuion [7], eponeniaed Kumaraswamy disribuion [8], eponeniaed modified Weibull eension disribuion [9] and eponeniaed Pareo disribuion [0]. The aim of his research is o develop ordinary differenial equaions (ODE) for he probabiliy densiy funcion (PDF), Manuscrip received July 4, 07; revised July 9, 07. This work was sponsored by Covenan Universiy, Oa, Nigeria. H. I. Okagbue, A. A. Opanuga and E. A. Owoloko are wih he Deparmen of Mahemaics, Covenan Universiy, Oa, Nigeria. hilary.okagbue@covenanuniversiy.edu.ng abiodun.opanuga@covenanuniversiy.edu.ng alfred.owoloko@covenanuniversiy.edu.ng M. O. Adamu is wih he Deparmen of Mahemaics, Universiy of Lagos, Akoka, Lagos, Nigeria. ISBN: ISSN: (Prin); ISSN: (Online) Quanile funcion (QF), survival funcion (SF), inverse survival funcion (ISF), hazard funcion (HF) and reversed hazard funcion (RHF) of eponeniaed Frĕche disribuion by he use of differenial calculus. Calculus is a very key ool in he deerminaion of mode of a given probabiliy disribuion and in esimaion of parameers of probabiliy disribuions, amongs oher uses. The research is an eension of he ODE o oher probabiliy funcions oher han he PDF. Similar works done where he PDF of probabiliy disribuions was epressed as ODE whose soluion is he PDF are available. They include: Laplace disribuion [], bea disribuion [], raised cosine disribuion [3], Loma disribuion [4], bea prime disribuion or invered bea disribuion [5]. II. PROBABILITY DENSITY FUNCTION The probabiliy densiy funcion of he eponeniaed Frȇche disribuion is given as; ( ) f ( ) e ( e ) () he probabiliy densiy funcion of he eponeniaed Frȇche disribuion, differeniae equaion (), o obain; ( ) e ( ) ( ) e f ( ) f ( ) ( ) e ( e ) ( e ) (),,, 0. ( ) ( )e f ( ) f ( ) ( e ) (3) WCECS 07

2 Proceedings of he World Congress on Engineering and Compuer Science 07 Vol II WCECS 07, Ocober 5-7, 07, San Francisco, USA Anoher process of differeniaion is carried ou on equaion (3) o obain; ( ) ( )e f ( ) f ( ) ( e ) ( ) ( ) f( ) ( )( e ) ( e ) ( ) ( ) e ( e ) ( ) ( ) ( e ) ( ) ( )( ) e f( ) f( ) (4),,, 0. The following equaions obained from equaion (3) are needed o simplify equaion (4); ( ) ( ) ( ) ( )e f f ( ) ( e ) ( ) ( )e ( ) f ( ) f ( ) ( e ) ( ) ( ( )e ) ( e ) ( ) f( ) f ( ) ( ) ( )( e ) ( e ) ( ) f( ) f ( ) (5) (6) (7) (8) ( ) ( ) ( )e ( e ) ( ) f( ) f ( ) ( ) ( ) ( ) ( )e ( e ) ( ) f( ) f ( ) ( ) ( ) ( )e ( e ) ( ) f( ) ( ) f ( ) ( ) ( ) ( )e ( e ) ( ) f( ) f ( ) (9) (0) () () Subsiue equaions (5), (8), (0) and () ino equaion (4); ( ) ( ) f ( ) f ( ) f ( ) f( ) ( ) f( ) f ( ) ( ) f( ) f ( ) f( ) ( ) f( ) f ( ) (3),, 0,. f () e ( e ) (4) ( )e f() ( ) f() (5) ( e ), A case was considered, ha is when equaion (3) becomes; ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

3 Proceedings of he World Congress on Engineering and Compuer Science 07 Vol II WCECS 07, Ocober 5-7, 07, San Francisco, USA f( ) f ( ) f ( ) f ( ) f ( ) f( ) f( ) 3 f ( ) (6) Simplify equaion (6) o obain f ( ) ( ) f ( ) f( ) f( ) (7) ( ) f( ) 4 III. QUANTILE FUNCTION The Quanile funcion of he eponeniaed Frȇche disribuion is given as; Qp ( ) [ln( ( ) ] p (8) he Quanile funcion of he eponeniaed Frȇche disribuion, differeniae equaion (8), o obain; ( ) ( p) [ln( ( p) )] Q( p) ( ( p) ) (9),, 0,0 p. Equaion (9) can be simplified as; Q( p) ( p) [ln( ( p) )] ( p)( ( p) )[ln( ( p) )] Subsiue equaion (8) ino equaion (0) o obain; Q( p) ( p) Q( p) ( p)( ( p) )[ln( ( p) )] Equaion (8) is simplified o obain; (0) () [ln( ( p) ] () Q( p) ln( ( p) (3) Q ( p) Subsiue equaion (3) ino equaion (); Q( p) ( p) Q ( p) ( ( p) ) (4) The ordinary differenial equaions can be obained for he given values of he parameers. Some of he cases of he given parameers are given in Table. Table : Classes of differenial equaions obained for he quanile funcion of eponeniaed Frȇche disribuion for differen parameers. Ordinary Differenial Equaion pq( p) Q ( p) 0 pq( p) Q ( p) 0 3 pq( p) Q ( p) pq( p) Q ( p) 0 ( p)( p) Q( p) ( p) Q ( p) 0 4( p)( p) Q( p) ( p) Q ( p) 0 4( p)( p) Q( p) 3 ( p) Q ( p) 0 6( p)( p) Q( p) 3 ( p) Q ( p) 0 IV. SURVIVAL FUNCTION The survival funcion of he eponeniaed Frȇche disribuion is given as; S ( ) [ e ] (5) he survival funcion of he eponeniaed Frȇche disribuion, differeniae equaion (5), o obain; ( ) S( ) e ( e ) (6),,, 0. Subsiue equaion (6) ino (5); e S ( ) S() ( e ) Equaion (5) can be simplified as; S (7) ( ) e (8) S ( ) e (9) Subsiue equaions (8) and (9) ino (7); S( )( S S() ( )) (30) S () ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

4 Proceedings of he World Congress on Engineering and Compuer Science 07 Vol II WCECS 07, Ocober 5-7, 07, San Francisco, USA ( S ( ) S( )) S() (3) The ordinary differenial equaions can be obained for he given values of he parameers. Some of he cases of he given parameers are given in Table. Table : Classes of differenial equaions obained for he survival funcion of eponeniaed Frȇche disribuion for differen parameers. Ordinary differenial equaion S( ) S( ) 0 S( ) S( ) 0 3 S( ) S( ) 0 3 S( ) 8 S( ) 8 0 Q( p) p Q ( p) ( p ) (38) ( p ) Q( p) p Q ( p) 0 (39) The ordinary differenial equaions can be obained for he given values of he parameers. Some of he cases of he given parameers are given in Table 3. Table 3: Classes of differenial equaions obained for he inverse survival funcion of eponeniaed Frȇche disribuion for differen parameers. Ordinary Differenial Equaion ( p) Q( p) Q ( p) 0 ( p) Q( p) Q ( p) 0 3 ( p) Q( p) Q ( p) 0 3 8( p) Q( p) Q ( p) 0 V. INVERSE SURVIVAL FUNCTION The inverse survival funcion of he eponeniaed Frȇche disribuion is given as; Qp ( ) [ln( ] p (3) he inverse survival funcion of he eponeniaed Frȇche disribuion, differeniae equaion (3), o obain; p Q( p) ( ) [ln( p )] ( p ) (33),, 0,0 p. Equaion (33) can be simplified as; Q( p) p [ln( p )] p( p )(ln( p )) Subsiue equaion (3) ino equaion (34) o obain; Q( p) p Q( p) p( p )(ln( p )) (35) Equaion (3) is simplified o obain; (34) [ln( p )] (36) Q( p) ln( p ) (37) Q ( p) Subsiue equaion (37) ino equaion (35); VI. HAZARD FUNCTION The hazard funcion of he eponeniaed Frȇche disribuion is given as; h () ( ) e [ e ] (40) he hazard funcion of he eponeniaed Frȇche disribuion, differeniae equaion (40), o obain; ( ) ( ) ( ) e ( ) e h( ) h( ) ( ) e [ e ] [ e ] (4),,, 0. ( ) e h( ) h( ) [ e ] ( ) h ( ) h( ) h( ) (4) (43) The ordinary differenial equaions can be obained for he given values of he parameers. Some of he cases of he given parameers are given in Table 4. ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

5 Proceedings of he World Congress on Engineering and Compuer Science 07 Vol II WCECS 07, Ocober 5-7, 07, San Francisco, USA Table 4: Classes of differenial equaions obained for he hazard funcion of eponeniaed Frȇche disribuion for differen parameers. Ordinary Differenial Equaion h( ) ( ) h( ) h ( ) 0 h( ) ( ) h( ) h ( ) h( ) (3 ) h( ) h ( ) h( ) (3 8) h( ) h ( ) 0 h( ) (4 ) h( ) h ( ) 0 h( ) (4 4) h( ) h ( ) h( ) (6 4) h( ) h ( ) h( ) (6 6) h( ) h ( ) 0 VII. REVERSED HAZARD FUNCTION The reversed hazard funcion of he eponeniaed Frȇche disribuion is given as; ( ) j() (44) he reversed hazard funcion of he eponeniaed Frȇche disribuion, differeniae equaion (44), o obain; ( ) ( ) j( ) ( ) ( ) (45),,, 0. The firs order ordinary differenial equaion for he reversed hazard funcion of he eponeniaed Frȇche disribuion is given by; j( ) ( ) j( ) 0 (46) j() (47) The ODEs of all he probabiliy funcions considered can be obained for he paricular values of he disribuion. Several analyic, semi-analyic and numerical mehods can be applied o obain he soluions of he respecive differenial equaions [6-40]. Also comparison wih wo or more soluion mehods is useful in undersanding he link beween ODEs and he probabiliy disribuions. VIII. CONCLUDING REMARKS In his work, differeniaion was used o obain some classes of ordinary differenial equaions for he probabiliy densiy funcion (PDF), quanile funcion (QF), survival funcion (SF), inverse survival funcion (ISF), hazard funcion (HF) and reversed hazard funcion (RHF) of he eponeniaed Frȇche disribuions. Ineresingly, he case of RHF yielded simple ODE compared wih he oher probabiliy and reliabiliy funcions. In all, he parameers ha define he disribuion deermine he naure of he respecive ODEs and he range deermines he eisence of he ODEs. ACKNOWLEDGMENT The auhors are unanimous in appreciaion of financial sponsorship from Covenan Universiy. The consrucive suggesions of he reviewers are grealy appreciaed. REFERENCES [] S. Nadarajah and S. Koz, The eponeniaed ype disribuions, Aca Applic. Mah., vol. 9, no., pp. 97-, 006. [] I.E. Okorie, A.C. Akpana and J. Ohakwe, The Eponeniaed Gumbel ype- disribuion: properies and applicaion. In. J. Mah. Mah. Sci., Ar. nọ , 06. [3] F.F. Gündüz and A.I. Genç, The eponeniaed Fréche regression: an alernaive model for acuarial modelling purposes, J. Sa. Compu. Simul., vol. 86, no. 7, pp , 06. [4] M. Pal, M.M. Ali and J. Woo, Eponeniaed Weibull disribuion, Saisica, vol. 66, no., pp , 006. [5] G.S. Mudholkar and D.K. Srivasava, Eponeniaed Weibull family for analyzing bahub failure-rae daa, IEEE Trans. Relia., vol. 4, no., pp , 993. [6] M.M. Nassar and F.H. Eissa, On he eponeniaed Weibull disribuion, Comm. Sa. Theo. Meh., vol. 3, no. 7, pp , 003. [7] R.D. Gupa and D. Kundu, Eponeniaed eponenial family: an alernaive o gamma and Weibull disribuions, Biomerical J., vol. 43, no., pp. 7-30, 0. [8] P.E. Ogununde, A.O. Adejumo and O.S. Balogun, Saisical properies of he eponeniaed generalized invered eponenial disribuion, Appl. Mah., vol. 4, no., pp , 04. [9] A.J. Lemone and G.M. Cordeiro, The eponeniaed generalized inverse Gaussian disribuion, Sa. Prob. Le., vol. 8, no. 4, pp , 0. [0] A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova, The eponeniaed invered Weibull disribuion, Appl. Mah. Inf. Sci, vol. 6, no., pp. 67-7, 0. [] I. Elbaal and H.Z. Muhammed, Eponeniaed generalized inverse Weibull disribuion, Appl. Mah. Sci., vol. 8, no. 8, pp , 04. [] M.M. Risić and N. Balakrishnan, The gamma-eponeniaed eponenial disribuion, J. Sa. Compu. Simul., vol. 8, no. 8, pp. 9-06, 0. [3] S. Nadarajah and A.K. Gupa, The eponeniaed gamma disribuion wih applicaion o drough daa, Calcua Sa. Assoc. Bull., vol. 59, no. -, pp. 9-54, 007. [4] S. Nadarajah, The eponeniaed Gumbel disribuion wih climae applicaion, Environmerics, vol. 7, no., pp. 3-3, 006. [5] C.S. Lee and H.Y. Won, Inference on reliabiliy in an eponeniaed uniform disribuion, J. Korean Daa Info. Sci. Soc., vol. 7, no., pp , 006. [6] G.M. Cordeiro, A.E. Gomes, C.Q. da-silva and E.M. Orega, The bea eponeniaed Weibull disribuion, J. Sa. Compu. Simul., vol. 83, no., pp. 4-38, 03. [7] K. Rosaiah, R.R.L. Kanam and S. Kumar, Reliabiliy es plans for eponeniaed log-logisic disribuion, Econ. Qual. Con/, vol., no., pp , 006. [8] A.J. Lemone, W. Barreo-Souza and G.M. Cordeiro, The eponeniaed Kumaraswamy disribuion and is log- ISBN: ISSN: (Prin); ISSN: (Online) WCECS 07

6 Proceedings of he World Congress on Engineering and Compuer Science 07 Vol II WCECS 07, Ocober 5-7, 07, San Francisco, USA ransform, Braz. J. Prob. Sa., vol. 7, no., pp. 3-53, 03. [9] A.M. Sarhan and J. Apaloo, Eponeniaed modified Weibull eension disribuion, Relia. Engine. Sys. Safey, vol., pp , 03. [0] A.I. Shawky and H.H. Abu-Zinadah, Eponeniaed Pareo disribuion: differen mehod of esimaions, In. J. Conemp. Mah. Sci., vol. 4, no. 4, pp , 009. [] 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 probabiliy ploing using MATLAB, 00. [4] N. Balakrishnan and C.D. Lai, Coninuous bivariae disribuions, nd ediion, Springer 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. Opanuga and S.A. Adeosun, A semi - analyical mehod for soluions of a cerain class of second order ordinary differenial equaions, Applied Mahemaics, vol. 5, no. 3, pp , 04. [7] S.O. Edeki, A.A Opanuga and H.I Okagbue, On ieraive echniques for numerical soluions of linear and nonlinear differenial equaions, J. Mah. Compuaional Sci., vol. 4, no. 4, pp , 04. [8] A.A. Opanuga, 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 numericalanalyical mehod, Appl. Mah. Sciences, vol. 8, no. 64, pp , 04. [9] S.O. Edeki, A.A. Opanuga, H.I. Okagbue, G.O. Akinlabi, S.A. Adeosun and A.S. Osheku, A Numerical-compuaional echnique for solving ransformed Cauchy-Euler equidimensional equaions of homogenous ype. Adv. Sudies Theo. Physics, vol. 9, no., pp. 85 9, 05. [30] S.O. Edeki, E.A. Owoloko, A.S. Osheku, A.A. Opanuga, H.I. Okagbue and G.O. Akinlabi, Numerical soluions of nonlinear biochemical model using a hybrid numericalanalyical echnique, In. J. Mah. Analysis, vol. 9, no. 8, pp , 05. [3] A.A. Opanuga, S.O. Edeki, H.I. Okagbue and G.O. Akinlabi, Numerical soluion of wo-poin boundary value problems via differenial ransform mehod, Global J. Pure Appl. Mah., vol., no., pp , 05. [3] A.A. Opanuga, S.O. Edeki, H.I. Okagbue and G. O. Akinlabi, A novel approach for solving quadraic Riccai differenial equaions, In. J. Appl. Engine. Res., vol. 0, no., pp. 9-96, 05. [33] A.A Opanuga, O.O. Agboola and H.I. Okagbue, Approimae soluion of mulipoin boundary value problems, J. Engine. Appl. Sci., vol. 0, no. 4, pp , 05. [34] A.A. Opanuga, O.O. Agboola, H.I. Okagbue and J.G. Oghonyon, Soluion of differenial equaions by hree semianalyical echniques, In. J. Appl. Engine. Res., vol. 0, no. 8, pp , 05. [35] A.A. Opanuga, H.I. Okagbue, S.O. Edeki and O.O. Agboola, Differenial ransform echnique for higher order boundary value problems, Modern Appl. Sci., vol. 9, no. 3, pp. 4-30, 05. [36] A.A. Opanuga, 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. Comp. Theo. Nanosci., vol., no. 0, pp , 05. [37] A.A. Opanuga, E.A. Owoloko, H.I. Okagbue, Comparison Homoopy Perurbaion and Adomian Decomposiion Techniques for Parabolic Equaions, Lecure Noes in ISBN: ISSN: (Prin); ISSN: (Online) Engineering and Compuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., pp [38] A.A. Opanuga, E.A. Owoloko, H. I. Okagbue, O.O. Agboola, "Finie Difference Mehod and Laplace Transform for Boundary Value Problems," Lecure Noes in Engineering and Compuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., pp [39] A.A. Opanuga, H.I. Okagbue, O.O. Agboola, "Irreversibiliy Analysis of a Radiaive MHD Poiseuille Flow hrough Porous Medium wih Slip Condiion," Lecure Noes in Engineering and Compuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., pp [40] A.A. Opanuga, E.A. Owoloko, O.O. Agboola, H.I. Okagbue, "Applicaion of Homoopy Perurbaion and Modified Adomian Decomposiion Mehods for Higher Order Boundary Value Problems," Lecure Noes in Engineering and Compuer Science: Proceedings of The World Congress on Engineering 07, 5-7 July, 07, London, U.K., pp WCECS 07