ON A GENERALIZED PROBABILITY DISTRIBUTION IN ASSOCIATION WITH ALEPH ( ) - FUNCTION

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1 Intnational Jounal of Engining, Scinc and athmatic Vol. 8, Iu, Januay 8, ISSN: 3-94 Impact Facto: Jounal Hompag: Doubl-Blind P Riwd Rfd Opn Acc Intnational Jounal - Includd in th Intnational Sial Dictoi Indd & Litd at: Ulich' Piodical Dictoy, U.S.A., Opn J-Gag a wll a in Cabll Dictoi of Publihing Oppotuniti, U.S.A ON A GENERALIZED PROBABILITY DISTRIBUTION IN ASSOCIATION WITH ALEPH ( ) - FUNCTION Yahwant Singh Dpatmnt of athmatic, Gonmnt Collg, Kalada, Jaipu, Rajathan, India ABSTRACT In thi pap, a pobability function Pha ( ) bn intoducd in tm of th Alph ( ) - function and it popti a tudid. It ha hown that th claical non-cntal ditibution uch a, non-cntal chi-qua, non-cntal Studnt- t, non-cntal F and almot all claical cntal continuou ditibution can b obtaind a pcial ca of thi gnal dnity function. Thi gnal dnity function Pi ( ) intoducd with th hop that any dnity function, which can b pntd in tm of any nown pcial function a wll a th dnity of th atio of any two indpndnt tochatic aiabl who dnity function can b pntd in tm of any nown pcial function, i containd in Pa ( ) a pcial ca. Th aiou popti of P, ( ) dicud in thi pap, includ th chaactitic function, momnt, cunc lationhip among momnt and th ditibution function. KER WORDS: Non-cntal Ditibution, -function, Gnal Pobability Function, Ratio Ditibution. ( athmatic ubjct claification: 33C99). INTRODUCTION Th - function intoducd by Suland t.al. [] dfind and pntd in th following fom: mn z,, [] : : z pi qi i mn, pi, qi; i: ( a, ),[ ( a, )] z ( b, ),[ ( b, )] L j j, n i n, pi j j, m i m, qi t() z d (.) Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

2 ISSN: 3-94 Impact Facto: Wh ; t () m ( b ) ( a ) j j j j j j qi p i ( b ) ( a i i jm jn n W hall u th following notation: A ( a, ),[ ( a, )] ; B ( b, ),[ ( b, )] * * j j, n i n, pi j j, m i m, qi (.). SOE DEFINITIONS AND PRELIINARY RESULTS Rult. m, n a A* pi, qi : i B* ( b ) ( b ) d,, A* mn, a pi, qi : i: B*,(, ) b b (.) b Wh R i ; i,,..., fo j,,..., m;r( ),, b and R(.) man th al pat of (.). Thi ult follow aily fom th fact that, b (.) ( ) ( b ) d Wh b,, R,R Rult on mploying (.). ( b ) ( b ) d m, n a A* pi, qi : i: B* d ( ) d b! ( ),, A* mn, a pi, qi : i: B*,(, ) b (.3) Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

3 ISSN: 3-94 Impact Facto: b R( d),r R,R i, b. Th ult follow by Wh panding Rult 3 d and intgating tm by tm by applying ult. m, n A* pi, qi : i: B* m, n y ay i i i a d y b Wh R i ; i,,..., 3. A GENERAL PROBABILITY FUNCTION (.4),, A * p, q : : B*,(, ) fo j,,..., m;r( ). H, w intoduc a gnal pobability dnity function Pby ( ) uing th mot gnalizd function, namly th -function. Such a gnalizd fom i not ncay to obtain all th claical cntal and non-cntal ditibution a pcial ca fom thi gnal ditibution. Spcial ca which can b pd in mo compact fom a gin lat. Without any lo of gnality th function Pi ( ) aumd to b non-ngati inc th paamt can alway b chon in uch a way that Pi ( ) alway non-ngati and till al paamt will b lft to ou choic o that al cla of non-ngati function can b obtaind a pcial ca and th gnal natu of Pi ( ) not lot ith. d m, n a A* pi, qi : i: B* ( b ) ( b ) P ( ) (3.) Cd ( ) Fo and P ( ) lwh, wh ( ) ( ) d,, A* mn, a C( d) b p i, qi : i: B*,(, )! b (3.) It hould b pointd out th facto ( b ) can b abobd inid th -function but it i wittn outid fo conninc of manipulation lat and whn d, C( d) can b wittn in a impl compact fom a, C,, A* mn, a pi, qi : i: B*,(, ) b () b Thn th pobability function Pduc ( ) to (3.3) 3 Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

4 ISSN: 3-94 Impact Facto: m, n a A* pi, qi : i: B* ( b ) ( b ) q ( ) (3.4) C() Almot all th claical cntal and non-cntal ditibution can b obtaind fom qwhich ( ) will b n lat. In od to obtain all th uful claical cntal and non-cntal ditibution a pcial ca it i not ncay to ta gnal dnity function in th fom of P. ( ) In th light of th ult and 3 of ction it i aily n that P( ) d 4. SPECIAL CASES If w put, i in (.), (.3) and (.4), w gt th ult gin by athai and Sana [9] with a littl implification a: mn, a ( a j, j ), p ( ) H p, q ( ) ( bj, j ), q b b d H b,,( aj, j ), p mn, a p, q ( bj, j ), q,(, ) b With th condition gin in (.). (4.) H d d m, n a ( a j, j ), p p, q ( bj, j ), q ( b ) ( b ) ( ) d b! ( ) H With th condition gin in (.3).,,( aj, j ), p mn, a p, q ( bj, j ), q,(, ) b (4.) m, n ( a j, j ), p p, q ( bj, j ), q m, n,,( a, ) y H ay, j j, q H a d y (4.3) j j p p, q ( b, ),(, ) With th condition gin in (.4). Non-cntal Chi-qua Ditibution: Th dnity function fo th non-cntal chi-qua i gin by fo!, lwh m ( ) (4.4) 4 Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

5 ISSN: 3-94 Impact Facto: Put,,,,,,,,,, i d b b b a in 4 P.Uing ( ) th fomula,,,, G F ; 4 4 (4.5) And ltting b, P( ) duc to maft ( ) a littl implification. In od to obtain th noncntal F, non-cntal Bta, Studnt-t and a numb of claical cntal ditibution w nd conid only qo ( ) Pwhn ( ) d and it may b noticd that qi ( ) in a compact fom. Non-cntal F Ditibution : Th dnity function fo non-cntal F i gin by m! m m m ( ) ; lwh fo g ( ) (4.6) By putting m m d,,,, b, b, b,, a,,, m n p, q,, th -function duc to th G -function of th did fom h. Thn by uing th gnal popti that, G ( a) F ( a; z; ) () c, ( a ), (, c), and, i G (4.7) PRduc ( ) to g. ( ) By a impl chang of aiabl w gt th non-cntal Bta ditibution, with th dnity function, m m ( ) m F ; ; ; fo m ( ) ; lwh g (4.8) It may b notd that th conditional ditibution of th multipl colation cofficint und th condition of gin alu of th obation on th aiabl in a multiaiabl nomal ca, ([],p.59), i a non-cntal Bta ditibution. Non-cntal Studnt-t Ditibution: Th dnity function fo th non-cntal Studnt-t ditibution i gin by:! h( ) ; (4.9) 5 Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

6 ISSN: 3-94 Impact Facto: Wh i th non-cntality paamt and i th dg of fdom. Fo conninc w will ta th ditibution in th foldd fom, that i h ( ) ( ) h fo (4.), lwh 3 d,,, b, b,, a, Put,, m n p q,, i, plac b by and a by. Thn Pduc ( ) to h ( ). Th gnalizd hypgomtic function: Th autho in [7] intoducd a gnal pobability ditibution fom wh th following ditibution w obtaind a pcial ca: th gnal hypgomtic ditibution, th gnalizd gamma, gamma, gnalizd F, F ', Studnt-t, Bta, Eponntial, Cauchy, Wibull, Raligh, Waiting tim and logitic. Th dnity function mployd wa, c c c a ( ) ( ) c c F, ; ; a ; fo, c,, c c c ( ) ; lwh f( ) (4.) Thi can b obtaind a a pcial ca fom Pby ( ) maing th following ubtitution. Put c, d, b,, a, a, b, b, c,, i.uing th fomula, ( a,),( b,) ( a) ( b) H, (,),( c,) F ( a, b; c; ) () c (4.) W gt f ( ) fom Paft ( ) a littl implification. Th Ratio Ditibution: Th atio ditibution can b obtaind a: f mn, mn ( bq, q;), A* ( ) ( ) p q, p q : : B*,( a, ;) fo i i i i i p p ( ), lwh Wh (4.3) 6 Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

7 ISSN: 3-94 Impact Facto: ( ) j q p j j b a m n pi qi i i a i b i i n n (4.4) Fom th tuctu of f ( ) itlf it i idnt that f ( ) can b obtaind fom Pby ( ) maing uitabl chang in th paamt. Thu Palo ( ) contain th dnity function of th atio of two indpndnt tochatic aiabl who dnity function can b pd in tm of any nown pcial function. 5. THE CHARACTERISTIC FUNCTION AND OENTS Sinc th chaactitic function i dfind a it it ( t) E P( ) d (5.) Wh i ( ), it can b aily obtaind by placing th paamt d by d it and hnc C( d it) () t (5.) Cd ( ) Wh Cd ( ) i gin in (3.). Hnc th momnt and cumulat can b aluatd without much difficulty. omnt: Th th momnt about th oigin, i obtaind by placing by and thn taing th atio of th nomalizing facto in P. ( ) That i Wh in (3.) C( d, ) (5.3) Cd (, ) ( )! (, ) d b C d And if d, thi duc to C (, ) b C(, ), wh C(, ),, A* mn, a pi, qi : i: B*,(, ) b,, A * mn, a p i, qi : i: B*,(, ) b (5.4) (5.5) 7 Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

8 ISSN: 3-94 Impact Facto: A Rcunc Rlationhip: A cunc lationhip among, and can b obtaind by uing th cunc lationhip fo th -function., ( ) d b C( d)!,, A * mn, a p i, qi : i: B*,(, ) b Wh Cd ( ) i gin in (3.). On applying th cunc fomula fo th -function. (5.6) mn, ( a ; ),,[ ( a ; )], a b = j j n i n p q pi, qi : i: ( bj, j ), m,[ i ( b, )] m, q,( bq, ) To m, n ( a, ),( a j ; j ), n,[ i ( a ; )] n, p m, n ( a j ; j ), n,[ i ( a ; )] n, p pi, qi : i: ( bj, j ), m,( b, ) m, q,( bq, ) p i, qi : i: ( bj, j ), m,[ i ( b, )] m, q,( bq, ) of (5.3), w that i qual to (5.7) ( ) d b C( d)!,, A * mn, a p i, qi : i: B*,(, ) b = C( d) ( ) d b b b,, A*,, A* m, n a m, n a pi, qi : i: B*,(3, ) p i, qi : i: B*,(, ) Hnc, w obtain b (5.8),,, 6. THE DISTRIBUTION FUNCTION Th ditibution function o th mulati dnity function y F( y) P( ) d Can b obtaind fo om pcial fom of P. ( ) By putting and taing th limit d and b, P( ) duc to th fom m, n A* ( ) p, : : * i qi i B ( ) a fo, lwh P (6.) By uing ult (.), w gt y m, n (, ), A* ( ) ( ) pi, qi : i: B*,(, ) P d y ay (6.) 8 Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com

9 ISSN: 3-94 Impact Facto: b Wh R i ; i,,..., ; j,,..., m and i ( ) dfind in (4.3). By uing F( y) w can obtain th ditibution of od tatitic and oth latd tatitic which w will not dicu h. REFERENCES. Boma, J.; On a function which i a pcial ca of ij G -function, Compoitio athmatica, 5(96), Baama, B.L.J.; Aymptotic panion and analytic continuation fo Ban-Intgal, Compoitio athmatica,5(964), Edlyi, A. t. al.; High Tancndntal Function, ol. I, cgaw-hill, Nw Yo (953). 4. Edlyi, A. t.al.; Tabl of Intgal Tanfom, ol. I, cgaw-hill, Nw Yo (954). 5. Fo, C. 96.Th G and H -function a Foui nl. Tan. Am. ath. Soc.98: Gupta, K.C.; On H -functiona, Annal d la Socit Scintifiqu d Bull, T. 79,II(965), athai, A.. and Sana, R.K.; On a gnalizd hypgomtic ditibution, atia, (966), athai, A.. and Sana, R.K.; Tabl of th gnalizd hypgomtic ditibution, atia, (966), athai, A.. and Sana, R.K.; A gnalizd pobability Ditibution,Spatata d la ita mathmatica Y Fiica Toica, ol. (97),93-.. Rao, C.R.; Lina Statitical Infnc and It Application, Wily, NwYo (965).. Sudland, N., Baumann, B. and Nonnnmach, T.F.; Opn poblm: who now about th Alph()-function? Fac. Calc. Appl. Annl. (4),(998), Intnational Jounal of Engining, Scinc and athmatic ijmj@gmail.com