The Existence of the Moments of the Cauchy Distribution under a Simple Transformation of Dividing with a Constant

Transcription

1 Theoreticl Mthemtics & Applictions, vol., no., 0, 7-5 ISSN: (print), (online) Interntionl Scientific Press, 0 The Eistence of the Moments of the Cuch Distriution under Simple Trnsformtion of Dividing with Constnt Johnson Ohkwe nd Bright Osu Astrct In this pper, we estlish the eistence of the moments of Cuch distriution with prmeters, nd denoted Cuch (,, ) vi simple trnsformtion of dividing with suitle constnt. As result of this trnsformtion ever Cuch (, ) would e distriuted on the intervl [-, ]. The results for the first four crude moments were given in view of their importnce in otining the ver importnt sttisticl mesures nmel vrince, skewness nd kurtosis. Mthemtics Suject Clssifiction: 6E0 Kewords: Cuch distriution, Trnsformtion, Moments Deprtment of Mthemtics nd Sttistics, Fcult of Sciences, Anmr Stte Universit, P.M.B. 0, Uli, Anmr Stte, Nigeri, e-mil: ohkwe.johnson@hoo.com Deprtment of Mthemtics, Fcult of Biologicl nd Phsicl Sciences, Ai Stte Universit, P.M.B. 000, Uturu, Ai Stte, Nigeri, e-mil: megorit@hoo.com Article Info: Revised : Jul 9, 0. Pulished online : Octoer, 0

2 8 The Eistence of the Moments Introduction The proilit densit function of Cuch distriution (cd) with the loction prmeter, nd scle prmeter is given f,,, 0 ( ( ) ). () Eqution () is referred to s Cuch (., ) The stndrd form of () cn e otined replcing with 0 nd with nd is given f 0,, ( ) () The Cuch distriution represents n etreme cse nd serves s counter emples for some well ccepted results nd concepts in sttistics. For emple the centrl limit theorem does not hold for the limiting distriution of the men of rndom smple from Cuch distriution []. Becuse of this specil nture, some uthors consider the Cuch distriution s pthologicl cse. However, it cn e postulted s model for descriing dt tht rise s n reliztions of the rtio of two norml rndom vriles. Other pplictions given in literture: [4] found tht Cuch distriution descries the distriution of velocit differences induced different vorte elements. An ppliction of the Cuch distriution to stud the polr nd non-polr liquids in porous glsses is given in [5]. In [] pointed out tht the Cuch distriution descries the distriution of hpocenters on focl spheres of erthqukes. It is shown in the pper [7] tht the source of fluctutions in contct window dimensions is vrition in contct resistivit, nd the contct resistivit is distriuted s Cuch rndom vrile. In spite of the wide pplictions of the Cuch distriution in vrious fields, the non-eistence of its moments is of huge concern insmuch s it hs undefined epecttion vlue nd higher moments diverge. For the epecttion vlue, the integrl E ( ) d ()

3 J. Ohkwe nd B.Osu 9 is not completel convergent, tht is lim d, (4) does not eist. However the principl vlues lim d (5) does eist nd is equl to zero [6]. In n cse the convention is to regrd the epecttion vlue of the Cuch distriution s undefined. Attempts hd previousl een mde to solve the prolem of non-definition of the Cuch distriution through trunction. For smmetric trunction, the renormlized proilit densit function given f( ). rctn (6) which hs epecttion vlue, E ( ) 0, nd vrince, Vr( ), rctn third centrl moment = 0, nd fourth centrl moment 4 ( ). rctn The frction of the originl Cuch distriution within the smmetric intervl is f ( ) rctn. Now the crucil question is wh would someone truncte? simpl ecuse of the need to otin distriution whose moments re mthemticll trctle. Therefore the purpose of this pper is to emplo simple trnsformtion on finite distriution of Cuch rndom vrile in order to otin distriution whose moments re mthemticll trctle. For this purpose we would consider the Cuch rndom vrile X, whose distriution is given in eqution () nd whose reliztions re from the non-compct rel spce ( ). Tht is, rel spce ( ) without the two points, nd []. Tht is X : r q, with rq, or

4 0 The Eistence of the Moments X,...,,,,..., (7) r 0 q If we let M,...,,,,..., (8) m r 0 q nd dividing (8) m, we otin the distriution of Y, whose vlues will lie in the intervl, where m X r,...,, 0,,..., q Y (9) Under the trnsformtion in (9), the proilit densit function (pdf) of Y is given f ( ), ( ) (0) where is the pdf stiliztion term which cn e otined evluting the integrl in (0), equting to unit nd solving for. If ( ) ( ) rctn( ) d d ( ) ( ) 0 m 0 which implies tht. rctn( ) Therefore the pdf of Y is given f ( ), rctn( ) ( ) () Hving otined the pdf of Y, the net tsk is to otin the first four moments of k Y, E ( Y ), k,,,4. Interest is limited to these moments ecuse the sic sttisticl mesures nmel; men, vrince, skewness nd kurtosis re otined from them. For the purpose of estlishing the moments, we would dopt the nottion

5 J. Ohkwe nd B.Osu K rctn( ) () Some Moments of the Cuch Distriution Under Trnsformtion. First Moment E(Y) B definition thus, EY ( ) f( ) dk d ( ) ln ( ) rctn( ) K {ln( ) ln( )} EY ( ) rctn( ) {rctn( ) rctn( )}. Second Moment E(Y ) ( ) ( ) ( ) ( ) 0 E Y K f d K d K d K ln( ) rctn( ) This implies tht {ln ( ) ln( )} EY ( ) rctn ( ) {rctn ( ) rctn( )} 0

6 The Eistence of the Moments. Third Moment E(Y ) B definition, ( ) ( ) ( ) ( ) 0 E Y K f d K d K d K ln( ) rctn( ) therefore EY ( ) rctn( ) {rctn( ) rctn( )} 4 {ln( ) ln( )}.4 Fourth Moment E(Y 4 ) ( ) ( ) ( ) ( ) 0 E Y K f d K d K d therefore ( )ln( ) K ( 6) rctn( ) ( )ln( ) 4 EY ( ) ( 6)rctn( ) ( ) ln( ) rctn( ) ( 6)rctn( ) k Hving otined the moments, E ( Y ) using the trnsformed distriution, one cn otin those of X using k k EY ( ) m 0

7 J. Ohkwe nd B.Osu Lemm. The following epression for the men, vrince, skewness for (, ) nd kurtosis for (, ) given s follows: EY ( ) K( Y) (, ) (, ) VrY K Y ( ) ( ) (, ) (, ) ( ) (, ) (, ) (, ) K ( Y) (, ) (, ) of rndom vrile Y defined in (9) re K Y 4( ) (, ) (, ) (, ) K ( Y) (, ) (, ) K Y Proof. The first four cummulnts Ki ( Y ), i,,,4 re given respectivel where K( Y) (, ) (, ), tn ( ) ( ) (, ) ln( ) nd (, ) {tn ( ) tn( )}. where K ( Y) (, ) (, ), (, ) ln( ) nd (, ) {tn ( ) tn( )}

8 4 The Eistence of the Moments where K( Y) (, ) (, ), ( ) (, ) 4ln( ) nd (, ) {tn ( ) tn( )}. Finll, ( ) (, ) (, ) K4 Y, where ( ) (, ) ln nd (, ) ( 6){tn ( ) tn( )}. From the ove four epressions for the moments of Y, we hve tht lim (, ) 0 nd lim (, ) 0 0, 0 0, Hence the limiting vlues of the kurtosis s tends to zero nd tends to infinit re given lim (, ) nd lim (, ). 0, 0, 0 Conclusion Considering the non-eistence of the moments of the Cuch distriution with prmeters nd denoted X such tht X ~ Cuch (, ), r q, the fundmentl findings of this stud is tht, we cn otin the moments of Cuch distriution through simple trnsformtion of dividing

9 J. Ohkwe nd B.Osu 5 with suitle constnt which converts X to e distriuted on the intervl [-, ] denoted Y such tht Y ~ Cuch (, ),. The eut of this stud is tht hving otined the moments of Y ~ Cuch (,, ), we cn esil generte those of those of X ~ Cuch (,, ) r q. ACKNOWLEDGEMENTS. The uthors wish to thnk the referees for the comments nd suggestions tht helped to improve the qulit of the pper. References [] Y.Y. Kgn, Correltions of erthquke focl mechnisms, Geophsicl Journl Interntionl, 0, (99), [] J.F.C. Kingmn nd S.J. Tlor, Introduction to Mesure nd Proilit, Cmridge Universit Press, 97. [] K. Krishnmoorth, Hndook of Sttisticl Distriutions with Applictions, Chpmn nd Hll,/CRC Press, Boc Rton, 006. [4] I.A. Min, I. Mezic nd A. Leonrd, Lev stle distriutions for velocit nd velocit difference in sstems of vorte elements, Phsics of fluids, 8, (996), [5] S. Stpf, R. Kimmich, R.O. Seitter, A.I. Mklkov nd V. D. Skid, Proton nd deuteron field-ccling NMR relometr of liquids confined in porous glsses, Colloids nd Surfces: A Phsicochemicl nd Engineering Aspects, 5, (996),07 4. [6] C. Wlck, Hnd-ook on Sttisticl Distriutions for Eperimentlists, Prticle phsics Group, Fsikum, Universit of Stockholm, 000. [7] S.S. Winterton, T.J. Sm nd N. G.Trr, On the source of sctter in contct residence dt, Journl of Electronic Mterils,, (99), 97 9.