Relations for Moments of Kumaraswami Power Function Distribution Based on Ordered Random Variables and a Characterization

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1 Relato fo Moet of Kuaawa Powe Fucto Dtbuto Baed o Odeed Rado Vaable ad a Chaactezato M. J. S. Kha, Sude Kua, Aay Kua Depatet of Stattc ad Opeato Reeach, Algah Mul Uvety, Alagh, Ida. Depatet of Appled Stattc, Babaaheb Bhao Abedka Uvety, Luckow, Ida. Depatet of Appled Stattc, Babaaheb Bhao Abedka Uvety, Luckow, Ida. ABSTRACT: I th pape, ecuece elato fo gle ad poduct oet of geealzed ode tattc fo Kuaawa powe fucto dtbuto have bee obtaed. Thee elato ae copoated a a patcula cae of ecuece elato fo ode tattc ad k th ecod. Futhe, chaactezato eult fo th dtbuto ha bee obtaed fo ecuece elato of gle oet of geealzed ode tattc. KEYWORDS: Geealzed ode tattc, ode tattc, uppe ecod, ecuece elato, Kuaawa Powe fucto dtbuto, chaactezato. 2 Matheatc Subect Clafcato. 62G3, 62E. I. INTRODUCTION The cocept of geealzed ode tattc (go) wa gve by Kap (995), whch gve a below: Let X, X 2,, X be a equece of depedet ad detcally dtbuted (d) ado vaable (v) wth abolutely cotuou dtbuto fucto (df ) F (x) ad pobablty dety fucto ( pdf ) f (x), x (, ). Let, k, ~ (, 2,..., ), M,, be the paaete uch that k M, fo all {,2,., }. The X (,,, k), X (2,,, k),, X (,,, k ),, 2,, ae called go f the ot pdf gve by f o the coe ( x, x2,, x ) X (,, ~, k),, X (,, ~, k) k [ F ( x )] f ( x ) [ F ( x)] f ( x) (.) k F ( ) x x F () of whee F( x) F( x) deote the uvval fucto Choog the paaete appopately, odel uch a oday ode tattc, k,. e.,k th ecod value, k,. e. k, equetal ode tattc ;, 2,,, ode tattc wth o-tegal aple ze ;, Pfefe ecod value ;,,, ad pogeve type Copyght to IJARSET 525

2 II ceoed ode tattc, k ca be obtaed a patcula cae of go. Fo plcty we have aued that 2. The pdf of th go gve by Kap (995) C f,,, x F x g F x f x, x X k! Ad the ot pdf of X,,, k ad,,, X k, gve by (.2) whee ad,,,,,,,, f x y F x g F x X k X k C h C!! h F y h F x F y f x f y, x, k ( )( ), ( x) ( x) l( x),,, g x h x h x, Recuece elato ae qute ueful coputg the oet.the eult gve peet pape ca be ued to copute the oet of odeed ado vaable f the paet populato follow Kuaawa powe fucto dtbuto. Seveal autho deved the ecuece elato fo go fo dffeet dtbuto. See, Kap ad Gathe (997), Ahaullah (2), Ahad ad Fawzy (23), Al-Hua et al.(25), Ahad (27), Kha et al. (27), Kua (2), Kua ad Kha (23), Kha et al.(25a, 25b), Kha ad Kha (26). A ado vaable X ad to be Kuaawa-Powe fucto dtbuto (Kw-PFD) Abdul Moe (27) wth the followg pobablty dety fucto pdf (.3) a b a ab x x f x ; a, b,, x (.4) ad t dtbuto fucto df gve by a b x F x ; a, b,, x the coepodg uvval fucto SF ae (.5)` Copyght to IJARSET 525

3 I vew of (.4) ad (.6), we get a b x F x ; a, b,, x (.6) a a F x x x f x ab Relato (.7) ued to obta the ecuece elato fo oet of geealzed ode tattc. The Kuaawa Powe fucto dtbuto wa toduced by Abdul-Moe (27), addg the ew hape ad cale paaete (Abdul-Moe (27)).Whch the geealzato of Kuaawa dtbuto. The Kuaawa powe fucto dtbuto educe to Kuaawa dtbuto fo ad The Kuaawa dtbuto wa appecated fo t ue hydologcal pheoea ad fo othe pupoe. Fo oe detal ad t applcato ee Kuaawa (98), Suda ad Subbah (989), Fletche ad Poabala (996), Sefet al. (2), Gaet al.(26), Couad-Hau (27) ad Sachez et al.(27). Th pape ogazed fou ecto. I ecto 2, we have poduced ecuece elato fo gle oet of go fo Kw-PFD. I ecto 3, we have deduced the ecuece elato fo poduct oet of go fo Kw-PFD. I ecto 4, the chaactezato eult baed o the ecuece elato fo gle oet of go fo Kw-PFD deduced. II. Recuece elato fo gle oet fo Kw-PFD fo go I th ecto, ecuece elato fo gle oet of Kw-PFD fo go ha bee obtaed. Futhe, the patcula cae of ecuece elato fo gle oet of ode tattc ad ecod value ae deduced fo go. Theoe 2.: Let X be a o-egatve cotuou ado vaable ad follow Kuaawa powe fucto dtbuto (.4). Suppoe that ad,,,,,, E X k E X k (.7) Poof: Fo (.2), we have a a E X,,, k X,,, k ab ab (2.) C,,, E X k x F x g F x f xdx! Itegatg above tegal teatg F x f x a tegad ad et pat fo dffeetato, we have C C 2,,,! 2! E X k x F x g F x dx x F x g F x f x dx whch ple that C,,,,,,! E X k E X k x F x g F x dx Now vew of (.7), we get,,,,,, E X k E X k Copyght to IJARSET

4 a C a x F x g F x f xdx! ab ad hece the eult. C! ab x F x g F x f x dx Reak 2.: Settg ad k Theoe (2.), we get the ecuece elato fo gle oet of ode tattc of Kw-PFD a a : : : : E X E X E X X ab ab Reak 2.2: Settg ad k Theoe (2.), we get the ecuece elato fo gle oet of uppe th k ecod of Kw-PFD a,,,,,, E X k E X k a a E X,,, k E X,,, k abk abk Reak 2.3: Settg ad (2.) we get ecuece elato fo gle oet of go fo Kuaawa dtbuto. III. Recuece elato fo poduct oet of go fo Kw-PFD I th ecto, the ecuece elato fo poduct oet of go fo Kw-PFD ha bee obtaed. Futhe, the ecuece elato fo poduct oet of ode tattc ad ecod value ae obtaed a patcula cae of go. Theoe 3.: Let X be a o-egatve cotuou ado vaable ad follow Kw-PFD (.4). Suppoe that, ad the followg ecuece elato atfed.,,,,,,,,,,,, E X k X k E X k X k Poof: Fo (.3), we have a ab,,,,,, E X k X k a E X,,, k X,,, k E X,,, k X,,, k ab (3.) whee C x F x f x g F xi xdx!! (3.2) I x h F y h F x F y f y dy (3.3) x Solvg the tegal I(x) by pat ad ubttutg the eultg expeo (3.2), we get,,,,,,,,,,,, E X k X k E X k X k Copyght to IJARSET

5 I vew of (.7), we obta c!! x x y F x f x g F x F y h F y h F x dydx c!! x x y F x f x g F x F y ad hece the theoe. a a h F y h F x x x f xdydx ab Reak 3.: Settg ad k Theoe (3.), we get the ecuece elato fo poduct oet of ode tattc of Kw-PFD a E X : X: E X : X : a a E X : X : E X : X : ab ab Reak 3.2: Settg ad k Theoe (3.), we get the ecuece elato fo poduct oet of uppe th k ecod of Kw-PFD a,,,,,,,,,,,, E X k X k E X k X k a a E X,,, k X,,, k E X,,, k X,,, k abk abk Reak 3.3: Settg ad (3.) we get ecuece elato fo poduct oet of go fo Kuaawa dtbuto. IV. Chaactezato of Kw-PFD I th ecto we have dcued the chaactezato of Kw-PFD. Chaactezato of pobablty dtbuto play a potat ole pobablty ad tattcal cece. Befoe appled a patcula odel o eal wold data t eceay to check that gve cotuou pobablty dtbuto atfed the udelyg aupto. I th ecto, we have obtaed a chaactezato eult of Kw-PFD baed o ecuece elato of gle oet fo go. Theoe 4.: The eceay ad uffcet codto fo a ado vaable X to be atfed wth pdf gve (.4) fo that a ab If ad oly f a E X,,, k E X,,, k ab,,,,,, E X k E X k (4.) Copyght to IJARSET

6 a a F x x x f x ab Poof: The eceay pat follow edately fo (4.) o the othe had f ecuece elato (4.) atfed, the Let,,,,,, a a a E X k E X k,,,,,, ab E X k ab E X k c c 2 ab x F x g F x f xdx x F x g F x f xdx!! c g 2 F x F x ab x F x g F x f x dx! F x h x F x g F x (4.2) Dffeetatg both de of (4.2), we get Now ' g 2 h x F x g F x f x F x F x F x a a,,, a c,,, E X k E X k ab x h x dx! (4.3) Itegatg RHS (4.3) by pat ad ug the value of whch educe to h x fo (4.2), we have a a F x x F x g F x f x x x ab dx (4.4) c! f x Now applyg geealzato of Mütz-Száz Theoe Hwag ad L (984) to (4.4), whch tate that o a pace La, b of uable fucto defed o ab,, a equece of fucto f x coplete o ab, f fo ay g L a, b the equalte b a,,2,3, f x g x dx ple that gx o, ab, the we get ' Copyght to IJARSET

7 x a a F x x x ab f x F x a a ab x x f a a ab F x x x f x Th pove that a a F x x x f x ab f x ha the fo a (.4). REFERENCES [] Abdul Moe, I.B., The Kuaawa powe fucto dtbuto, J. Stat. Appl. Po. vol. 6, o., 27, pp [2] Ahad, A.A., Relato fo gle ad poduct oet of geealzed ode tattc fo doubly tucated Bu type XII dtbuto, J. Egypt. Math. Soc., vol. 5, 27, pp [3] Ahad, A.A. ad Fawzy, M., Recuece elato fo gle oet of geealzed ode tattc fo doubly tucated dtbuto, J. Statt. Pla. Ife., vol. 7, 23, o. 2, pp [4] Ahaullah, M., Geealzed ode tattc fo expoetal dtbuto, Statt. Pla. Ife, vol. 85, 2, pp [5] Al-Hua, E.K., Ahad, A.A. Ad Al-KAhf, M.A., Recuece elato fo oet ad codtoal oet geeatg fucto of geealzed ode tattc, Metka, vol. 6, 25, pp [6] Couad-Hau, D., Ug otecalo aaly to vetgate the elatohp betwee ovecoupto ad uceta acce to oe peoal utlty fucto, Ecologcal Ecooc, vol. 64, 27, pp [7] Fletche, S.C. ad Poabala, K., Etato of eevo yeld ad toage dtbuto ug oet aaly, J. Hydology, vol. 82, 996, pp [8] Ga, A. Poabala, K. Khall, D. ad Kaaouz, M., Ga yeld elablty aaly wth cop wate dead ucetaty, Stochatc Evoetal Reeach ad Rk Aeet, vol. 2, 26, pp [9] Hwag, J.S. ad L, G.D., O a geealzed oet poble II, Poc. Ae. Math. Soc., vol. 9, 984, pp [] Kap, U., A Cocept of Geealzed Ode Stattc, B. G. Teube Stuttgat., 995. [] Kap, U. Gathe, U., Chaactetc popety of geealzed ode tattc fo expoetal dtbuto, Appl. Math. (Waaw), vol. 24, 997, pp [2] Kha, M.I. Kha, M.A.R. ad Kha, M.A., Geealzed ode tattc fo q-expoetal type I dtbuto ad t chaactezato, It. J. Coput. Theo. Stat., vol. 2, 25a, o., pp. -6. [3] Kha, M.I. Kha, M.A.R. ad Kha, M.A., Geealzed ode tattc fo q-expoetal type II dtbuto ad t chaactezato, It. J. Coput. Sc., vol. 5, 25b, o. 4, pp [4] Kha. R.U., Awa, Z. ad Atha, H., Recuece elato fo gle ad poduct oet of geealzed ode fo doubly tucated Webull dtbuto, Algah, J. Statt., vol. 27, 27 pp [5] Kha, R.U. ad Kha, M.A., Moet popete of geealzed ode tattc fo expoetal-webull lfete dtbuto, Joual of Advaced Stattc, vol., 26, pp [6] Kua, D., Geealzed ode tattc fo Kuaawa dtbuto ad t chaactezato, Tau Oxfod Joual of Ifoato ad Matheatcal Scece, vol. 27, 2, o. 4, pp [7] Kua, D. ad Kha, M.I., Relato fo geealzed ode tattc fo doubly tucated geealzed expoetal dtbuto ad t chaactezato, Math. Sc. Lett., vol. 2, 23, o., pp [8] Kuaaway, P., Geealzed pobablty dety fucto fo double- bouded ado-pocee, Joual of Hydology, vol. 46, 98, pp [9] Sachez, S. Acheyta, J. ad McCaffey, W. C., Copao of pobablty dtbuto fucto fo fttg dtllato cuve of petoleu, Eegy ad Fuel, vol., 27, pp [2] Sef, A., Poabala, K. ad Vlach, J., Maxzato of aufactug yeld of yte wth abtay dtbuto of copoet value, Aal of Opeato Reeach, vol. 99, 2, pp [2] Suda, V. ad Subbah, K., Applcato of double bouded pobablty dety fucto fo aaly of ocea wave, Ocea Egeeg, vol. 6, 989, pp Copyght to IJARSET

8 AUTHOR S BIOGRAPHY Mohd. Jahag Sabb Kha wokg a Atat Pofeo at Depatet of Stattc ad Opeato eeach Algah Mul Uvety, Algah. He ha e yea of teachg expeece. He ha alo woked a a Atat Pofeo at Depatet of Appled Stattc, Babaaheb Bhao Abedka Uvety (A Cetal Uvety), Luckow ad at Depatet of Matheatc ad Stattc, Baathal Uvety, Baathal, Raatha. D. Mohd. Jahag Sabb Kha ha publhed eveal pape eputed oual of Natoal ad Iteatoal level. Sude Kua wokg a Pofeo at Depatet of Appled Stattc, School fo Phycal Scece,Babaaheb Bhao Abedka Uvety (A Cetal Uvety), Luckow. He ha eghtee yea of teachg expeece. He ha alo woked a a Atat Pofeo at Depatet of Stattc, Kuau Uvety, Uttaakhad. D. Sude Kua ha publhed eveal pape eputed oual of Natoal ad Iteatoal level. Aay Kua a eeach chola Depatet of Appled Stattc Babaaheb Bhao Abedka Uvety (A Cetal Uvety), Luckow. He ha publhed oe pape eputed oual of Iteatoal level. Copyght to IJARSET