JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord THE EXONENTIAL DISTRIBUTION AND THE ALICATION TO MARKOV MODELS Usma Yusu Abubakar Deparme o Mahemacs/Sascs Federal Uversy o Techology, Ma, Ngera. Emal: abbkruy@yahoo.com Absrac Ths paper dscusses he characerscs o he expoeal dsrbuo ad he relaed dsrbuo ucos cludg gamma, webull ad logormal he relaes some o her properes o he applcao o Markov models. Oe o he major properes s orgeuless, he cosequece o hs s ha Markov ad saoary assumpos mply ha he mes bewee eves mus be egave-expoeally dsrbued. To make a decso o he applcao o Markov model o ay process real le suao, s advsed ha should be ed o he orm o he egave expoeal desy ucos whch mples ha he mos lkely mes are close o zero, ad very log mes are creasgly ulkely. Tha s, he mos lkely values are cosdered o be clusered abou he mea, ad large devaos rom he mea are vewed as creasgly ulke. I hs characersc o he egave expoeal dsrbuo seems compable wh he applcao oe has md he a Markov model may o be approprae. Keywords; Expoeal dsrbuo, Radom varables, Memory-less, Markov models, Saoary assumpo, Applcao. Iroduco I makg mahemacal models or a real- world pheomeo s always ecessary o make cera smplyg assumpos so as reder he mahemacs racable. Oe smplyg assumpo ha s oe made whe modelg wh Markov prcple s o assume ha cera radom varables are expoeally dsrbued. The reaso or hs s ha expoeal dsrbuo s boh relavely easy o work ad s oe a good approxmao o he acual dsrbuo Ross(1989). A mpora smplyg assumpo makg Markov cha models s ha he me akes o make a raso (radom varable) be descrbed by egaveexpoeal dsrbuo. I some o he applcaos Abubakar(007) ulzed boh he expoeal ad Webull respecvely o descrbe he wag me he saes o sem-markov model or leprosy reame. Also Abubakar(010) cosdered as a radom varable he me akes or suda savaah o be rasormed o sahel savaah usg Webull dsrbuo uco a sem-markov model or desercao. I s hereore o eres hs paper o exame he expoeal dsrbuo uco ad s applcao o modelg Markov processes. The expoeal dsrbuo The probably desy uco o he radom varable T havg he expoeal dsrbuo s () = Kohlas(198). The dsrbuo has as a parameer. also deermes he shape o he dsrbuo. The mea o he expoeal dsrbuo s E( T) e e o Subsug w = = = w/ ad dw = he egrad w w w w gves e dw w 1 e e dw 1. Cosder e E( T ) Le w = = =w/ ad dw = So ha E( T ) w = e o e w w w w e 0 0 w e dw dw w 0 e The varace o he expoeal dsrbuo s hereore gve by T ) - (E(T)) = / The mea s 1/λ ad he varace s (1/λ). Thus he mea ad varace are o separaely adjusable, as oe may requely desre. 9
JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord The survvor uco S() s gve by S() = (T > ) ad s he probably ha a equpme has survved up o me. Suppose F() s he cumulave dsrbuo uco o he radom varable T. The S() = 1 - F() For he expoeal dsrbuo 0 F( ) ( T ) 0 ( s) ds = e s ds =1 - e - Thereore S() = e - The alure rae or he hazard rae, h assocaed wh he radom varable T s gve by h() = ()/S() For he expoeal dsrbuo, he hazard rae s gve by h() = Ths explas ha he alure rae or a expoeal radom varable s cosa. Le us cosder he codoal probably ( < T < + / T > ). Tha s, he probably ha he equpme wll al durg he ex me us, gve ha survved a me. Usg he deo o codoal probably, we have ( < T < + / T > ) = ( < T < + ) (T > ) s ds ( ) p( T ) ( ) s( ) Where < < + Fgure1 plos hs uco or hree values o λ. Noce ha he uco erceps he vercal axs a λ, ha dmshes mooocally o zero (asympocally), ad ha he rae o covergece s proporoal o λ. The oal area uder he curve s, o course, always equal o 1, as mus be or ay desy uco. () Fgure1. The graph o Expoeal. Mos applcaos are based o s memory-less propery, whe he measureme varable T has a me dmeso. Ths propery reers o he pheomeo whch he hsory o he pas eves does o luece he probably o occurrece o prese or uure eves. Accordg o Ross(1989) A radom varable X s sad o be whou memory, or memory-less, 30
JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord {X > s + X > } = p{x > s} or all s, (1) I we hk o X as beg he leme o a srume, he Equao (1) saes ha he probably ha he srume lves or a leas or a leas s + hours gve ha has survved hours s he same as he al probably ha lves or a leas s hours. I oher words, I he srume s alve a me, he he dsrbuo o he remag amou o me ha survves s he same as he orgal leme dsrbuo, ha s, he srume does o remember ha has already bee use or a me.the codo Equao (1) s equvale o = {X > s} Or {X > s + }={X > s}{x > } () Sce Equao () s sased whe X s expoeally dsrbued (or ), ollows ha expoeally dsrbued radom varables are memory-less ad orgeul. Followg Feller (1971), We shall reer o hs lack o memory as he Markov propery o he expoeal dsrbuo. Feller (1971) has deed he ollowg as a characersc propery o he expoeal dsrbuo. Le Ω ad ( are oempy regos, also le ad be wo depede radom varables wh deses ad, ad deoe he desy o her sum S = + by g. The pars ( ) ad ( ) are relaed by her lear rasormao = S wh deerma 1 ad Sce he eves ( ) Ω ad ( ) are decal s see ha he jo desy o ( ) s gve by g( ) = ( ). Where = > 0 (3) he jo desy o he par (X 1,S) s gve by Iegrag over all x we oba he margal desy g o S. The codoal desy o gve ha S = s sases =. ) I he specal case o expoeal deses (x) = (x) = (where x > 0) we ge = or 0 < x < s. oher words, gve ha +, he varable s uormly dsrbued over he erval (0,s). uvely speakg, he kowledge ha S = s gves us o clue as o he possble poso o he radom po wh he erval (0, s). Ths resul coorms wh he oo o complee radomess here he expoeal dsrbuo. Also ced rom Feller (1971) o he radomess o he expoeal dsrbuo. Le be depede wh he commo desy or x > 0. u The (, ) s obaed rom ( ) by a lear rasormao o he orm wh deerma = > 0,) wh deerma 1. Deoe by Ω he oca o pos The desy o ( ) s coceraed o Ω ad s gve by I The varables map Ω oo he rego deed by 0 < < ad see (3) wh he desy o s gve by The margal desy o s kow o be he gamma desy /! ad hece he codoal desy o he uple gve ha = s equals! or 0 < (ad zero elsewhere). I oher words, gve ha = s he varables are uormly dsrbued over her possble rage. We may say ha gve = s, he varables represe pos chose depedely ad a radom he erval (0, s) umbered her aural order rom le o rgh. Followg Tms(1988), Le X be a posve radom varable wh he probably dsrbuo uco F() havg e mea E(x) ad e sadard devao. The radom varable X may represe he leme o some em or he me o complee some ask. The coece o varao o he posve radom varable X s deed by. I applcaos oe oe works wh he squared coece o varao raher ha wh. The (squared) coece o varao s a measure o he varably o he radom varable X. For example, he deermsc dsrbuo has he expoeal 31
JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord dsrbuo has ad he Errlag-k dsrbuo has he ermedae value A radom varable wh creasg (decreasg) alure rae has a propery ha s coece o varao s smaller(larger) ha or equal o 1. The alure rae s a cocep ha eables us o dscrmae bewee dsrbuo ucos o physcal cosderaos. We ow dscuss he gamma, logormal ad webull dsrbuos or a posve radom varable T ad dey he releva properes o hese dsrbuos o he egave-expoeal dsrbuo. The gamma dsrbuo :The desy () gve by () =, 0, where he shape parameer α ad he scale parameer λ are boh posve. Here Γ(α) s he complee gamma uco deed by Γ( ) = > 0, Ad havg he he propery or ay > 0. The probably dsrbuo uco F( ) may be wre as F() =. The laer egral s kow as he complee gamma uco. I he shape parameer s a posve eger k, he gamma dsrbuo s he well-kow Erlag-k ( ) dsrbuo or whch () = ad F() = 1, he Erlag-K dsrbuo has a very useul erpreao. A radom varable wh a Erlag-K dsrbuo ca be represeed as he sum o k depede radom varables havg a commo expoeal dsrbuo. The mea ad he squared coece o varao o he gamma dsrbuo are gve by E(X) = ad = Sce E(X) ad ca assume arbrarly posve values or he gamma desy, a uque gamma dsrbuo ca be ed o each posve radom varable wh gve rs wo momes. To characerze he shape ad he alure rae o he gamma desy, We dsgush bewee he cases ( ) ad ( ). The gamma desy s always umodal, ha s he desy has oly oe maxmum. For he case o he desy rs creases o he maxmum a = ( )/ λ > 0 ad ex decreases o zero as, whereas or he case o he desy has s maxmum a = 0 ad hus decreases rom = 0. The alure rae uco s creasg rom zero o λ ad s decreasg rom y o zero. The expoeal dsrbuo ( ) has a cosa alure rae λ ad s a aural boudary bewee he cases ad. The logormal dsrbuo: The desy ()s gve by () = > 0, where he shape parameer α s a posve ad he scale parameer λ may assume each real value. The probably dsrbuo uco F() equals F() = φ, > 0, Where φ(x) = s he sadard ormal probably dsrbuo uco. The mea ad he squared coece o varao o he logormal dsrbuo are gve by E(X) = ad - 1. Thus a uque logormal dsrbuo ca be ed o each posve radom varable wh gve rs wo momes. The logormal desy s always umodal wh a maxmum a = > 0. The alure rae uco rs creases ad ex decreases o zero as ad hus he alure rae s oly decreasg he log-le rage. The Webull dsrbuo: The desy () s gve by () = αλ, > 0, Wh he shape parameer ad scale parameer λ > 0. I s observed ha he Webull dsrbuo reduces o he expoeal dsrbuo whe he shape parameer s uy. The probably dsrbuo uco F( ) equals F() = 1 -, The mea ad he squared coece o varao o he webull desy are gve by E(X) = ad = A uque Webull dsrbuo ca be ed o each posve radom varable wh gve rs wo momes. For ha purpose a o-lear equao α mus be solved umercally. The Webull desy s 3
JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord always umodal wh a maxmum a = (α > 1) ad a = 0 (α ). The alure rae uco s creasg rom 0 o y ad s decreasg rom y > 1. The gamma ad webull deses are smlar shape, ad or he logormal deses akes o shapes smlar o he gamma ad webull deses. For he case o 1 he gamma ad webull deses have her maxmum value a = 0 so ha mos oucomes wll be small ad very large oucomes occur oly occasoally. The logormal desy eds o zero as aser ha ay power o, ad hus he logormal dsrbuo wll ypcally produce ewer small oucomes ha he oher wo dsrbuos. Ths laer ac explas he popular use o he logormal dsrbuo acuares sudes. The dereces bewee he gamma, webull ad logormal deses become mos sgca her al behavor. The deses or large go dow lke, ad. Thus or gve mea ad coece o varao he logormal desy has always he loges al oly, ha s oly s coece o varao s less ha oe. I he gure, we llusrae hese acs by drawg he gamma, webull ad logormal deses or = 0.5, where E(X) = 1 s ake. 1.0 Logormal 1.10 1.00 Gamma 0.90 0.80 Webull 0.70 0.60 0.50 0.40 0.30 0.0 0.10 0.00 0.00 0.05 1.00 1.50.00.50 3.00 Fg The gamma, logormal ad webull deses wh E(x) = 1 ad = 0.5. Source,Tms(1988). We dscuss some useul exesos o Erlaga mxure o expoeally dsrbued compoes, or a (expoeal) dsrbuos. A Erlag-K ( ) combao o boh. A parcularly covee dsrbued radom varable ca be represeed as he sum o k depede expoeally dsrbued dsrbuo arses whe hese compoes have he same meas. radom varables wh he same meas. A geeralzed Erlaga dsrbuo s oe bul ou o sum or 33
JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord Markov process. Followg Korve(1993), we may gve a mahemacal deo o a Markov cha as a sequece X 0, X 1, - - - - o dscree radom varables wh he propery ha he codoal probably dsrbuo o X + 1 gve X 0, X 1, - - - - X deped oly o he value o X bu o urher o X 0, X 1, - - - - X - 1. Tha s or ay se o values, h,, - - - j he dscree sae space, (X + 1 = j / X 0 = h, - - - - -, X = ) = (X + 1 = j / X = ) =., j =1,, The marx whose eres are he s s called he raso probably marx or he process. The above cha s a rs order Markov cha. I hs process, he probably o makg raso o a uure sae does o deped o he prevous sae bu oly deped o he prese sae. I oher words, he probably o makg a raso o a uure sae does o deped o he pas hsory A Markov cha wh dscree sae ad couous me s reerred o as Markov process. Followg Howard(1960) we le a represe he raso rae o he process rom sae o sae j, j. I a shor me erval,, he process currely sae wll make a raso o sae j wh probably, j. X s he a sae o he process a me, he we have (X + = j x = ) = a. Suppose ha he raso rae do o chage wh me ( a s are cosas) saoary. We descrbe he process by a raso rae marx A wh compoes a. Aer some mapulao, we have d j 1 Wh he propery a a jj j j marx orm, we have d a, j 1,,3...,, j 1,,3 A. Ad hs s he geeral represeao o d () = p a k ( ) kj (Chapma Kolmogorov equao). I s a lear, rs-order dereal equao wh cosa coeces he a kj s.ad hese are he elemes o he raso marx A whch are erpreed as he mea o egave expoeal dsrbuo. Tha s, he dsrbuo o me spe sae whe j s he ex sae. Sem-Markov process.the Markov process dscussed above has he propery ha sae chages ca oly occur a he approprae me sas. However, gve he aure o some processes, raso may o acually occur a hese me sas. We hereore cosder a suao where he me bewee rasos may be several o us o me ad where he raso me ca deped o he raso ha s beg made. Ths leads o a geeral orm o Markov process called a sem-markov process Howard( 1971). Suppose he process eers sae. Le Y be he me he spe sae beore movg ou o he sae. The Y s called he wag me sae. We le w ( ) be he probably dsrbuo uco o Y. The w (m) = (Y = m) = m j1 The cumulave probably dsrbuo W ( ) ad he complmeary cumulave probably dsrbuo W ( ) or he wag mes are gve as ollows m1 j1 j1 W (m) = (Y ) = W m m m1 Ad W () = (Y > ) = 1 - W () = w m m1 j1 j1 m m1 (4) 34
JORIND 9() December, 011. ISSN 1596 8308. www.rascampus.org., www.ajol.o/jourals/jord,j=1,, deog saes ad m=1,, represeg me. Ierval raso probables. We dee () o be he probably ha he process wll be sae j year gve ha he eered sae year zero. Ths s called he erval raso probably rom sae o sae j he erval (0, ]. The W m m. 1 j 0 j k 1 k m1, j = 1,, = 1,, 3,.... W () s as deed (4). We observe ha he quay m m1 k has o be descrbed by egave expoeal dsrbuo perhaps gamma or Webul. I some o he applcaos Abubakar( 007) ulzed boh he expoeal ad Webull respecvely o descrbe he wag me he saes o sem- Markov model or leprosy reame. Also Abubakar(010) cosdered as a radom varable he me akes or suda savaah o be rasormed o sahel savaah ad cosequely expressed he me wh Webull dsrbuo uco a sem-markov model or desercao. Cocluso I s observed ha he Markov ad saoary assumpos mply ha he mes bewee eves mus be egave-expoeally dsrbued. The parameers o hese dsrbuos, he λ s may be depede o he sae occuped,, ad he ex sae, j, bu all o he dsrbuos mus be o he egave expoeal orm. No oher dsrbuo amly ca eve be cosdered as a caddae or descrbg he mes bewee eves. I was meoed ha may applcaos, he mes bewee eves are mos aurally coceved o as havg a desy uco o he geeral orm show Fg (perhaps a gamma or webull or logorma l). k kj Tha s, oe eds o hk erms o some omal value, he mea, plus or mus some relavely mor varao. Or, pu aoher ways, he mos lkely values are cosdered o be clusered abou he mea, ad large devaos rom he mea are vewed as creasgly ulke. However, he orm o he egave expoeal desy ucos mples ha he mos lkely mes are close o zero, ad very log mes are creasgly ulkely. I hs characersc o he egave expoeal dsrbuo seems compable wh he applcao you have md, perhaps a Markov model s approprae Ths s a mpora udersadg o be able o dsgush bewee hose processes whch mgh properly be modeled as saoary Markov process ad hose whch should o. Reereces Abubakar,U.Y. (007), A Sochasc Model Descree Saes ad Descree Tme or he Corol o Leprosy Dsease. Leoardo Joural o Sceces, 11, July-December. p 51-60. www.ljs.academcdrec.org Abubakar, U.Y (010), A Sochasc Model wh a Cos Srucure ad he Applcao o he Sudy o Deser Ngera.Joural o Mahemacal Sceces. A joural o Naoal Mahemacal Cere Abuja, Ngera. Vol.1, No.1 pp 49-66. Feller,W.(1971). A Iroduco o robably Theory ad s Applcaos. Vols 1&11,3 rd ed, Wley, New York. HOWARD, R.A. (1960). Dyamc programmg ad Markov processes. The M.I.T. RESS, Massachuses. HOWARD, R. A. (1971) Dyamc probablsc sysems, Vols I& II, Joh Wley, New York. Kohlas,J(198). Sochasc Mehods o Operaos Research. Cambrdge Uvery ress. KORVE K. N. (1993), Models or he Maageme o Ashma h.d. Thess, Faculy o Mahemacal sudes, Uversy o Souhampo Eglad. Ross,S.M.(1989). Iroduco o robably Models. Academc ress,inc New York. Tms H. C. (1988) Sochasc Modellg ad Aalyss: A Compuaoal Approach. Joh Wley & Sos Ld. New York. 35