Measures of Entropy based upon Statistical Constants

Transcription

1 Proceedgs of the World Cogress o Egeerg 07 Vol I WCE 07, July 5-7, 07, Lodo, UK Measures of Etroy based uo Statstcal Costats GSButtar, Member, IAENG Abstract---The reset artcle deals wth mortat vestgatos ad develomets of some ew robablstc measures of formato based uo certa stadard statstcal costats The fdgs of our vestgatos have bee llustrated grahcally as well as aalytcally Ide Terms--- Etroy, Easblty, Cocavty, Measures of cetral tedecy, Degeerate dstrbutos, Symmetrc fucto INTRODUCTION To quatfy the ucertaty cotet of a radom eermet ruled by the robablty dstrbuto P,, t was Shao [5],, who for the frst tme remarked that ucertaty s always assocated wth every robablty dstrbuto ad hece t must be a fucto of robabltes Wth ths assumto ad also wth certa desrable ostulates, Shao [5] vestgated ad develoed a measure of ucertaty, today kow as etroy fucto gve by HP ( ) = I () Later o Shao roved that the etroy fucto () s etremely useful may dscles of Mathematcs ad other Sceces After the veto of Shao s [5] measure of etroy, Rey [4] troduced etroy of order, gve by the followg eresso: H ( P) I,, > 0 Mauscrt receved: March 04, 07 GSButtar s wth Deartmet of Mathematcs,Khalsa College Amrtsar (Ida), emal: gurcharabuttar@gmalcom Cotact No ISBN: ISSN: (Prt); ISSN: (Ole) () Recetly, Parkash ad Kakkar [0, ] vestgated ad develoed the followg arametrc ad o-arametrc measures of etroy, the alcatos of whch have bee rovded to codg theory M( P) (3) M ( P), 0, (4) Moreover, keeg vew the alcato areas of dfferet measures of etroy, Parkash ad Mukesh [, 3] troduced the followg geeralzed measures of etroy H P, 0, 0 (5) log, H P, 0 (6) Ad rovded ther alcatos the felds of Statstcs ad Oeratos Research Some alcatos of etroy measures the feld of queueg theory have bee rovded by Buttar [4] The other establshmets regardg the robablstc etroc models have bee rovded by Kaur [6, 7], Herremoes [5], Nada ad Paul [9], Sharma ad Taeja [6], Cohe ad Merlo [3], Chakrabart [], Laveda [8] etc It s worth metog here that most of the Bologcal Sceces, researchers usually measure dversty ad equtablty of dfferet commutes Some of the oeers ethusastcally volved the study are Shao [5], Rey [4], Smso [7], Weer [8] etc It has bee realzed that Shao s measure s most wdely alcable ad ossesses may terestg ad desrable roertes but stll there s a eed for develog ew measures to eted the scoe of ther alcatos I secto II, we have WCE 07

2 Proceedgs of the World Cogress o Egeerg 07 Vol I WCE 07, July 5-7, 07, Lodo, UK vestgated ad troduced some ew robablstc measures of formato based uo certa statstcal costats II SOME NEW MEASURES OF ENTROPY BASED UPON STATISTICAL CONSTANTS A Iformato Measure terms of Measures of Cetral Tedecy Let a radom varable X takes the values,,, The, geometrc mea G, arthmetc mea M ad harmoc mea H of these observatos are gve by: G 3, 0 (7) M H (8) / (9) Equatos (7), (8) ad (9) ca be rewrtte as G (0) where Also M H or M H 3, () Aga equato (0) ca be wrtte as G 3 M () G log log M (3) From equatos () ad (3), we get G M log = log (4) M H Net, we demostrate the roosal of a formato theoretc measure volvg geometrc mea G, arthmetc mea M ad harmoc mea H Ths measure s gve by ( P) log,, 0 (5) To rove that (5) reresets a formato measure, we study ts essetal roertes as follows () P s ermutatoally symmetrc ()Sce s cotuous fucto for 0 < s also cotuous everywhere the same terval () Sce 0 < <, P 0 <, P Ths roerty of egatvty s due to the fact that the etroy fucto (5) volves Burg s[] etroy whch always gves egatve value (v) Determato of Mamum Value Let us cosder the followg fucto: L log L Smlarly, L L Thus L L L L 0, gves 3 ISBN: ISSN: (Prt); ISSN: (Ole) WCE 07

3 Proceedgs of the World Cogress o Egeerg 07 Vol I WCE 07, July 5-7, 07, Lodo, UK = whch further gves mlyg that Also usg, we get = = showg that the mamum value occurs at the uform dstrbuto Further, the mamum value s gve by ma ( P ) log f ( ) (say) Also f ( ) 0 ' Hece, the mamum value s a decreasg fucto of ad t resembles wth Burg s [] measure of etroy ' (v)we have Thus ad 0 " 3 s a cocave fucto of Thus, we otce that P roertes rovg that P formato,,, satsfes all the essetal s a ew measure of we have reseted ( P ) grahcally Fg- whch obvously show that ( P ) s a cocave fucto ( P ) Fg-: Cocavty of ( P ) B Iformato Measure terms of Measures of Dserso ad Cetral Tedecy The varace of a dscrete dstrbuto of observatos,,, s defed as (6) The above equato (6) ca be rewrtte as where M Thus, we have M M (7) ISBN: ISSN: (Prt); ISSN: (Ole) WCE 07

4 Proceedgs of the World Cogress o Egeerg 07 Vol I WCE 07, July 5-7, 07, Lodo, UK Addg equatos () ad (7), we get whch gves M M H M The above relato mles that theoretc measure deedg uo arthmetc mea M, harmoc mea H ad varace Ths measure s gve by (8) Now, we tate the develomet of a formato ( P),, 0 Also usg, we (9) get Thus, the mamum value arses whe the dstrbuto s uform (v) Cocavty: To study ts cocavty, we have ' ( P) Also, '' ( P) 3 0 We shall rove that the RHS of equato (9) s a whch shows that ( P ) s cocave ad wth these essetal formato measure roertes, we coclude that ( P ) s aother ew measure To rove ths, we study ts followg roertes: () Obvously ( P ) () ( P ) s of formato 0 Net, we have reseted the values of ( P ) ad cotuous fucto of 0 () obtaed the Fg- whch shows that the measure troduced equato (9) s a cocave fucto ( P ) s ermutatoally symmetrc fucto of 0 (v) ( P) Mamum Value: To obta, the mamum value of the etroy measure (9), we cosder the followg Lagrage s fucto: L Thus, we have ( P) Fg- : Cocavty of ( P ) Smlarly,,, For mamum value, we ut 0, 3 ISBN: ISSN: (Prt); ISSN: (Ole) III CONCLUSION The kow fact that formato theory deals wth a varety of measures of etroy mles that we have to develo those measures whch ca be successfully aled to a dversty of mathematcal dscles Moreover, f we have a varety of formato measures, we shall be more bedable alyg a stadard measure deedg uo the stuato of ts alcatos The dea has eforced us for the develoed of some ew measures ad cosequetly, for the kow values of some statstcal WCE 07

5 Proceedgs of the World Cogress o Egeerg 07 Vol I WCE 07, July 5-7, 07, Lodo, UK costats, we have vestgated ad rojected may ew robablstc formato theoretc measures of etroy REFERENCES [] JP Burg, The relatosh betwee mamum etroy sectra ad mamum lkelhood sectra, I: Chldrers, DG(ed) Moder Sectral Aalyss, 30-3, 97 [] C G Chakrabart, ad I Chakrabarty, Shao etroy: Aomatc characterzato ad alcato, Iteratoal Joural of Mathematcs ad Mathematcal Sceces, vol 7, , 005 [3] E G D Cohe, ad R L Merlo Clausus, etroy revsted, Moder Physcs Letters B, vol 8, o9, (-5), 04 [4] GS Buttar, Theoretc Models Iformato Theory ad ther Alcato PhD Thess, Deartmet of Statstcs, Pujab Uversty,Patala,Ida,07 [5] P Herremoes, ad C Vgat, A etroy ower equalty for the bomal famly Joural of Iequaltes Pure ad Aled Mathematcs, vol 4, o 6, 003 [6] J N Kaur, Geeralzed etroy of order ad tye, Mathematcs Semar, vol 4, 79-84, 967 [7] J N Kaur, Some ew measures of etroy ad dvergece, Joural of Mathematcal ad Physcal Sceces, vol 9, , 985 [8] B H Laveda, Mea etroes, Oe Systems & Iformato Dyamcs, vol, 89-30, 005 [9] A K Nada, ad P Paul, Some results o geeralzed resdual etroy, Iformato Sceces, vol 76, 7-47, 006 [0] O Parkash, ad P Kakkar, New measures of formato ad ther alcatos codg theory, Caada Joural of Pure ad Aled Sceces, vol 8, o, 905-9, 04 [] O Parkash, ad P Kakkar New formato theoretc models, ther detaled roertes ad ew equaltes Caada Joural of Pure ad Aled Sceces, vol 8, o 3, 35-33, 04 [] O Parkash, ad Mukesh New geeralzed arametrc measure of etroy ad cross etroy Amerca Joural of Mathematcs ad Sceces, vol, o, 9-96, 0 [3] Parkash, O ad Mukesh Cotrbuto of mamum etroy rcle the feld of queueg theory Commucatos Statstcs-Theory ad Methods, vol 45, o, [4] A Rey, O measures of etroy ad formato, Proceedgs 4th Berkeley Symosum o Mathematcal Statstcs ad Probablty, vol, , 96 [5] C E Shao, A mathematcal theory of commucato, Bell System Techcal Joural, vol7, , [6] B D Sharma, ad I J Taeja, Etroes of tye (,) ad other geeralzed measures of formato theory, Metrca, vol, 0-5, 975 [7] E H Smso, Measuremet of dversty, Nature, vol 63, , 949 ISBN: ISSN: (Prt); ISSN: (Ole) WCE 07