Distribution of the Product of Bessel Distribution of. Second Kind and Gamma Distribution - Properties and Characterization

Transcription

1 Inrnaional Journal o Mahaial Analysis Vol. 9 5 no HIKARI Ld hp://d.doi.org/.988/ia.5.53 Disriuion o h Produ o Bssl Disriuion o Sond Kind and Gaa Disriuion - Propris and hararizaion M. Ahsanullah M. Shail B. M. Gola Kiria 3 and. Gorg 4 Ridr Univrsiy Lawrnvill NJ 8648 USA Miai Dad ollg Hialah L 33 USA 34 lorida Inrnaional Univrsiy Miai lorida 3399 USA opyrigh 5 M. Ahsanullah M. Shail B. M. Gola Kiria and. Gorg. This is an opn ass aril disriud undr h raiv oons Ariuion Lins whih pris unrsrid us disriuion and rproduion in any diu providd h original wor is proprly id. Asra In his papr a nw disriuion is inrodud y aing h produ o h proailiy dnsiy unions o Bssl disriuion o sond ind and gaa disriuion. So disriuional propris and a hararizaion o h proposd produ disriuion ar providd. Th prnil poins or so sld valus o parars ar also prsnd. I is hopd ha h proposd produ disriuion will usul in any applid rsarh. Mahais Su lassiiaion: 33E 6E5 6E 6E5 Kywords: Bssl disriuion o sond ind hararizaion Gaa disriuion Modiid Bssl unions o irs and sond inds. Inroduion In rn yars any rsarhrs hav ondud signiian rsarh on h disriuions o h produ o rando varials and hir appliaions. Ths disriuions ar syri and swd and hav appliaions in any ilds o applid sins suh as

2 3 M. Ahsanullah al. Dography Eonois Enginring and Environnal Sins o nion a w s or apl Johnson al. [ ] rish and Sorn [3] Sorn [ 3] Sshadri [] Galaos and Sionlli [5] and Koz and Viari [3] aong ohrs. In his papr a nw swd disriuion wih h proailiy dnsiy union pd o h or whr is h noralizing onsan and and dno h proailiy dnsiy unions o Bssl disriuion o sond ind and gaa disriuion or a oninuous rando varial. So disriuional propris and a hararizaion o h proposd produ disriuion ar providd. Our wor has n oivad y h rn wor o Srivasava and Nadaraah [4] Nadaraah [5 7] Nadaraah al. [6] and Shail and Kiria []. In Srivasava and Nadaraah [4] and Nadaraah [5 7] so ailis o swd Bssl disriuions hav n inrodud y aing h produ o h pd s o Bssl disriuions o h irs and sond inds. In Nadaraah al. [6] a swd Bssl union disriuion wih h pd g. G is inrodud whr g and G ar an rspivly o h proailiy dnsiy union pd and h uulaiv disriuion union d o h Bssl union disriuion du o MKay [4]. In Shail and Kiria [] a nw syri sudn s -yp disriuion and is propris ar inrodud y aing h produ o h pd s o noral and sudn s disriuions. Th organizaion o h papr is as ollows. Th drivaion o disriuion o h produ proailiy dnsiy union and so o is propris ar givn in sion. Sion 3 onains a hararizaion o h proposd produ disriuion. Prnag poins o h produ disriuion ar givn in sion 4. inally so onluding rars ar oulind in sion 5. In ordr o ind h prssion or h pd orrsponding o h disriuion o h produ o h proailiy dnsiy unions o Bssl disriuion o sond ind wih h proailiy dnsiy union o gaa disriuion w will us so spial unions and las ha ar availal in Araowiz and Sgun [] Gradshyn and Ryzhi [9] Prudniov al. [9] and Oldha al. [8] aong ohrs.. Disriuion o h Produ o h Proailiy Dnsiy unions o Bssl Disriuion o Sond Kind and Gaa Disriuion In his sion w will driv h proailiy dnsiy union pd o h produ o h proailiy dnsiy unions o Bssl disriuion o sond ind and gaa disriuion as providd low.. Eprssions or h Noralizing onsan and h Proailiy Dnsiy union onsidr h produ union and whr dnos h noralizing onsan o drind and dno h pd s o Bssl disriuion o sond ind and gaa disriuion rspivly or a oninuous rando varial givn y

3 Disriuion o h produ o Bssl disriuion 33 K. whr I I K sin is h odiid Bssl union o sond ind I is h odiid Bssl union o irs ind and. whr is h shap parar is h sal parar Γ and ar rspivly h gaa and inopl gaa unions. Th uulaiv disriuion union d o h rando varial ~ Gaa is givn y..3 Thor. Th produ union whr as sad aov dnos h pd o h Bssl disriuion o sond ind and dnos h pd o gaa disriuion dins a proailiy dnsiy union pd o our proposd produ disriuion i is noralizing onsan and proailiy dnsiy union pd ar rspivly givn y I.4

4 34 M. Ahsanullah al. and K.5 whr and I.6 dnos h assoiad odiid Bssl union o h irs ind. Proo: Sin and ar pd s and w hav. Hn or o a pd w us hav d ha is d K in whih on using Prudniov al. [9] Volu Equaion pag 349 or h ingral h prssions or h noralizing onsan and h pd o h proposd produ disriuion ar sipliiaion ar asily oaind as ollows.7

5 Disriuion o h produ o Bssl disriuion 35 and K.8 whr. This opls h proo o Thor.. Thor.: Th uulaiv disriuion union d o h proposd produ disriuion is givn y sin.9 whr. Proo: or h uulaiv disriuion union d o h proposd produ disriuion on using h Eq..8 w hav h ollowing quaion

6 36 M. Ahsanullah al. d d K.. Now in h Eq.. aov using h ollowing prssion or h odiid Bssl union o h sond ind I I K sin along wih h ollowing sris rprsnaion o h odiid Bssl union o irs ind I aing u and noing h ollowing ingral u du u d w asily oain h uulaiv disriuion union d o our proposd produ disriuion as givn in.9. This opls h proo o Thor.... Disriuional Propris This sion disusss so hararisis o h proposd disriuion.

7 Disriuion o h produ o Bssl disriuion 37.. Shaps Th shaps o h pd o our proposd produ disriuion Eq..8 o Thor. ar providd in igurs a and d low or so sld valus o h parars. Th s o h parars ar ovious ro hs igurs also h proposd disriuion appars o uniodal and righ swd. igur a: or. igur : or igur : or.. igur d: or

8 38 M. Ahsanullah al. igur : or Mons Th rh on o h pd is providd in Thor.3 low. Thor.3 or so ingr r h rh on o a rando varial having h pd o our proposd produ disriuion Eq..8 o Thor. wih h noralizing dind y.4 or.7 is givn y onsan E r I r. I whr r is ingr and I and I Bssl union o h irs ind as dind in Eq..6. r dno h assoiad odiid Proo: W hav E r r d r K d.

9 Disriuion o h produ o Bssl disriuion 39 whr is h noralizing onsan givn y h quaions.7. Thus in viw o h Eq..4 and using Prudniov al. [9] Vol. Eq pag 349 or h ingral in h aov quaion. and sipliying h proo o Thor.3 asily ollows. 3. hararizaion This sion disusss a hararizaion o our proposd produ disriuion. hararizaion o a proailiy disriuion plays an iporan rol in proailiy and saisis. Bor a pariular proailiy disriuion odl is applid o i h ral world daa i is nssary o onir whhr h givn proailiy disriuion saisis h undrlying rquirns y is hararizaion. A proailiy disriuion an hararizd hrough various hods s or apl Ahsanullah al. [] aong ohrs. In rn yars hr has n a gra inrs in h hararizaions o proailiy disriuions y runad ons. or apl h dvlopn o h gnral hory o h hararizaions o proailiy disriuions y runad on gan wih h wor o Galaos and Koz [4]. urhr dvlopn on h hararizaions o proailiy disriuions y runad ons oninud wih h onriuions o any auhors and rsarhrs aong h Koz and Shanhag [] Glänzl [6 7] and Glänzl al. [8] ar noal. Howvr os o hs hararizaions ar asd on a sipl rlaionship wn wo dirn ons runad ro h l a h sa poin. As poind ou y Glänzl [6] hs hararizaions ay also srv as a asis or parar siaion. In wha ollows w will salish a hararizaion rsul o our proposd produ disriuion y using h runad on y onsidring a produ o rvrs hazard ra and anohr union o h runad poin. Ths ar providd in La 3. and Thor 3. low. La 3. Suppos ha is an asoluly oninuous wih rsp o Lsgu asur rando varial wih d and pd. W assu ha / g whr iss or all and E. I E g is a dirnial union o and g' d g or all hn w hav whr / d g g and is drind y h ondiion ha d d. Proo o La 3. W hav u u du g. Thus

10 4 M. Ahsanullah al. u u du g. Dirniaing oh sids o h aov quaion wih rsp o w oain g' g '. On sipliiaion w g ' g'. g Ingraing h aov quaion w oain g' d g whr is drind y h ondiion ha d. This opls h proo o La 3.. Thor 3. Suppos ha h rando varial has an asoluly oninuous wih rsp o Lsgu asur uulaiv disriuion union d and proailiy dnsiy. W assu ha / union pd iss or all and E. L. Thn has h proposd produ disriuion wih h pd givn y Eq..8 i and only i E g ha is g E d P whr Q

11 Disriuion o h produ o Bssl disriuion 4 P and. Q Proo: Nssary ondiion onsidr h pd o h proposd produ disriuion as givn in Eq..8 whih is prssd as ollows. 3.

12 4 M. Ahsanullah al. W hav g E d. 3. Now using 3. in h aov Eq. 3. and applying h diniion o h inopl gaa union z ollowing z d w asily oain ar sipliiaion h P g 3.3 Q P and Q ar givn as or. onsqunly E g whr whr and hn h proo o nssary ondiion ha is i par o Thor 3. ollows ro La 3.. Suiin ondiion W will prov now h suiiny only i par o Thor 3.. Suppos ha g E d P Q P and whr Q ar givn as or and P Qd. Now dirniaing g wih rsp o w oain ar sipliiaion and using La 3. h ollowing: / g / g / Q Q. 3.4 Ingraing h Eq. 3.4 wih rsp o w oain

13 Disriuion o h produ o Bssl disriuion 43. Q 3.5 Sin d i ollows ro Equaions 3. and 3.5 ha and hn h proo o suiin ondiion only i par o Thor 3. ollows ro La 3.. This opls h proo o Thor Prnils This sion opus h prnag poins o our proposd disriuion wih h pd as givn in.8 and d as givn in.9. or any p h h p prnil also alld h quanil o ordr p o h nw disriuion wih pd is a nur p suh ha h ara undr o h l o p is p. Tha is p is any roo o h quaion givn y p du u p p. Th prnag poins p assoiad wih h d o ar opud or so sld valus o h parars y using R-sowar. Ths ar providd in h Tal 4. low.

14 44 M. Ahsanullah al. Tal 4.: Prnag Poins o h Proposd Produ Disriuion Parars % % 3% 4% 5% 6% 7% 8% 85% 9% 95% 98% 99% =.= = θ = = =.=.3 = θ = = =.=.5 = θ = = =.= = θ = = =.=. = θ = = =.=. = θ = = =.=. = θ =3 = =.=. = θ =5 = =.=. = θ = =.5 =.=. = θ = =.5 =.=. = θ = =.3 =.=. = θ = =.5 ==. = θ = = =.5=. = θ = = =.3=. = θ= = =.=. = θ = = =.=. = θ = = =.=. = θ = = =.=. =3 θ = = =.=. =5 θ = = onluding Rars In his papr a nw disriuion is inrodud y aing h produ o h proailiy dnsiy unions o Bssl disriuion o sond ind and gaa disriuion. So disriuional propris o h proposd disriuion ar oulind. Th prnil poins or so sld valus o parars ar providd. I is osrvd ha h nw disriuion is swd o h righ and arris os o h propris o swd disriuions. hararizaion o a proailiy disriuion plays an iporan rol in proailiy and saisis. W hav salishd a hararizaion rsul o h proposd disriuion y using h runad on y onsidring a produ o rvrs hazard ra and anohr

15 Disriuion o h produ o Bssl disriuion 45 union o h runad poin. I is hopd ha h proposd disriuions will usul in any applid rsarh.. To ailia our rsarhrs and radrs w hav providd R- ods in h appndi. Anowldgns. Th auhors would li o han h rrs and h dior or hlpul suggsions whih iprovd h qualiy and prsnaion o h papr. Also h hird auhor B. M. Gola Kiria would li o ddia his papr o his os inlund and rspd ah ahr Shr hia Ranan Bhowi o Baish Rashi Shi Sundary Aady whih has lrad yars o is glory in Dr 4. Rrns [] Araowiz M. and Sgun I. A. 97. Handoo o Mahaial unions wih orulas Graphs and Mahaial Tals. Dovr Nw Yor. [] Ahsanullah M. Kiria B. M. G. and Shail M. 4. Noral and Sudn s Disriuions and Thir Appliaions. Alanis Prss Paris ran 7-5. hp://d.doi.org/.99/ _ [3] rish U. and Sorn D Er dviaions and appliaions. J. Phys. I ran hp://d.doi.org/.5/p:9974 [4] Galaos J. and Koz S hararizaions o proailiy disriuions. A uniid approah wih an phasis on ponnial and rlad odls Lur Nos in Mahais 675 Springr Brlin. hp://d.doi.org/.7/6953 [5] Galaos J. and Sionlli I. 5. Produs o Rando Varials Appliaions o Prols o Physis and o Arihial unions R Prss. Boa Raon / Alana. [6] Glänzl W A hararizaion hor asd on runad ons and is appliaion o so disriuion ailis Mahaial Saisis and Proailiy Thory Bad Tazannsdor 986 Vol. B Ridl Dordrh hp://d.doi.org/.7/ _8 [7] Glänzl W. 99. So onsquns o a hararizaion hor asd on runad ons Saisis hp://d.doi.org/.8/ [8] Glänzl W. Tls A. and Shur A hararizaion y runad ons and is appliaion o Parson-yp disriuions Z. Wahrsh. Vrw. Gi hp://d.doi.org/.7/5357

16 46 M. Ahsanullah al. [9] Gradshyn I. S. and Ryzhi I. M. 98. Tal o Ingrals Sris and Produs 6h Prining. Aadi Prss San Digo. [] Johnson N. L. Koz S. and Balarishnan N oninuous Univaria Disriuions Volu sond diion John Wily and Sons Nw Yor 994. [] Johnson N. L. Koz S. and Balarishnan N oninuous Univaria Disriuions: Volu. Wily Nw Yor. [] Koz S. and Shanhag D.N. 98. So nw approahs o proailiy disriuions. Advans in Applid Proailiy hp://d.doi.org/.37/46748 [3] Koz S. and Viari D. 5. Survy o dvlopns in h hory o oninuous swd disriuions. METRON-Inrnaional Journal o Saisis Vol. LIII No. pp [4] MKay A. T A Bssl union disriuion. Bioria [5] Nadaraah S. 7. Produ Bssl Disriuions o h irs and Sond Kinds. Inrnaional Journal o Mahais and Mahaial Sins Volu 7 Aril ID pags. hp://d.doi.org/.55/7/57956 [6] Nadaraah S. Srivasava H. M. & Gupa A. K. 7. Swd Bssl union disriuions wih appliaion o rainall daa. Saisis hp://d.doi.org/.8/ [7] Nadaraah S. 8. So produ Bssl dnsiy disriuions. Taiwans Journal o Mahais. Vol. No. pp. 9-. [8] Oldha K. B. Myland J. and Spanir J. 9. An Alas o unions wih Equaor h Alas union alulaor. Springr Nw Yor USA. hp://d.doi.org/.7/ [9] Prudniov A. P. Bryhov Y. A. and Marihv O. I Ingrals and Sris Volus and 3 Gordon and Brah Sin Pulishrs. Asrda. [] Sshadri V Th Invrs Gaussian Disriuion: Saisial Thory and Appliaions. Springr Nw Yor. [] Shail M. and Kiria B. M. G. 7. A Nw Sudn s T-Typ Disriuion and Is Propris. Journal o Applid Saisial Sin Volu 6 Issu 3 pp [] Sorn D Mulipliaiv prosss and powr laws. Physial Rviw E hp://d.doi.org/.3/physrv.57.48

17 Disriuion o h produ o Bssl disriuion 47 [3] Sorn D. 4. riial phnona in naural sins haos raals slorganizaion and disordr: onps and ools nd diion Springr Sris in Synrgis. Hidlrg. [4] Srivasava H. M. and Nadaraah S. 6. So ailis o Bssl disriuions and hir appliaions. Ingral Transors and Spial unions. Vol. 7 No hp://d.doi.org/.8/ Appndi R ods or h Produ Bssl II-Gaa Disriuion lirarysas4 liraryl lirarysas #<-rhisq55 lirarygsl Gauss <- uniona{ i>= & <{ hyprg_a }ls{ hyprg_-a-/-/-^ } } #ssli nu pon.sald = ALSE #sslk nu pon.sald = ALSE ################### pd nou<-union { NR=^*+*h^*+*gaa+*^+-*p-/h+/**sslI/ DR=^+*h^*+*gaa*+*Gauss*+/*++/+h^/ +*h^ nou<-nr/dr } #ingranou 5.7$valu oo = sq y=. pros=nuri or i in :lnghoo { pros[i]=ingranou oo[i]$valu }

18 48 M. Ahsanullah al. ploooprosyp="l" prs=sq quans=nuri or i in :lnghprs { quans[i]=inoo[whihroundpros==prs[i]] } quans Rivd: January 5 5 Pulishd: Marh 6 5