On Probability Density Function of the Quotient of Generalized Order Statistics from the Weibull Distribution
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2 8 Aee whee k x y k ad a ea ube. g x x og x x I h pape we ue he Me afo echque o fd he dbuo of he quoe of ay wo geeaed ode ac fo he Webu dbuo. We have ao gve pecfc eu ad appcao.. The Me Tafo Epe 948 wa he f o ugge a yeac appoach o he udy of poduc ad quoe of depede ado vaabe by ug a Me afo echque. Fox 957 exeded h ega afo o he wodeoa cae ode o evauae poduc ad ao of ado vaabe X f x y whch o-egave he f quada ad eo Y wh pdf eewhee. The Me afo of f x y defed a: M x y f x y dxdy. whee ad ae copex vaabe. Ude uabe codo dcued by Fox 957 poee he vee h k f x y M x y dd h k wh he pah of egao beg wo e paae o he agay ax ad o he gh of he og he Agad pae. Exeo whe f x y pove a fou quada ae aed Fox 957 bu ae o equed h pape. Howeve we w be geay eeed wo pacua cae of he above: M u M. M M.3
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4 Aee whee x y k a ea ube. ad k The Me afo. of 3.3 gve by: M S S M S S!! y x y y x y e e dxdy.!! e y. I. dy whee x ad y w w I e d w wh ad ax copee gaa fuco.
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7 O Pobaby Dey Fuco of he Quoe of Geeaed Ode Sac fo he Webu Dbuo 3 4. Specfc Reu a. If we pu 3. we ge he pdf of he ao of he exee geeaed Ode Sac fo Webu dbuo a: g k! k 4. whee k ad a ea ube. b. If we pu 3. we ge he pdf of he ao of he coecuve geeaed Ode Sac a: g k! 4. whee k ad a ea ube. c. If we ake o be odd ad pug p p 3. we ge he pdf of he ao of peak o eda geeaed Ode Sac of a ado ape of e p fo he Webu dbuo a: p p p p p g p p k p! p! p p p 4.3 whee p k ad a ea ube. Reak a. By pug ad k. 3. educe o he dey of he quoe of ay wo oday ode ac Mak 98. b. By pug ad k wh R. e 3. educe o he
8 4 Aee dey of he quoe of ay wo ode ac wh o-ega ape e. a pove ea ube fo. e ad k 3. educe o he dey of he quoe of ay wo equea ode ac. c. By pug 5. Appcao The coepodg eu fo geeaed ode ac wh ad k fo he Webu dbuo 3. ca eay be deduced a peca cae of he eu 3.. The h oe ca be deduced fo 3. gve a: c / k =!! 4.4 / whee k E Y k he h oe of Y k. I pacua: a. Wh 9 9 k ad fo 4.4 educe o he h oe of he ao of he exee geeaed ode ac fo Webu dbuo a: 7 7 / 9 = b. Wh 9 k ad fo 4.4 educe o he h oe of he ao of he coecuve geeaed ode ac fo Webu dbuo a:
9 O Pobaby Dey Fuco of he Quoe of Geeaed Ode Sac fo he Webu Dbuo 5 / 9 = c. Wh 5 k ad fo 4.4 educe o he h oe of he ao of peak o eda geeaed ode ac fo Webu dbuo a: / 59 = R S Ackowedge The auho hak he efeee fo he coe ad fufu uggeo. Refeece. Abaow M. ad Segu I. M Hadbook of Maheaca Fuco wh Foua Gaph ad Maheaca Tabe. oue Dove Pubcao. Ic. New Yok.. Epe B Soe appcao of he Me afo Sac. Aa of Maheaca Sac Edey A. Magu W. Obehege F. ad Tco F. G Hghe Tacedea Fuco. Vo.. McGaw- H Book opay New Yok. 4. Fox Soe Appcao of Me Tafo o he heoy of Bvaae Saca Dbuo. Poceedg of he abdge Phoophca Socey Kap U A ocep of Geeaed Ode Sac. Joua of Saca Pag ad Ifeece Mak H. J. 98. Pobaby dey fuco of quoe of ode ac fo Paeo Powe ad Webu dbuo. oucao Sac- Theoy ad Mehod
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