❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛▼♦❞❡❧♦s ❊♥✈♦❧✈❡♥❞♦ ❉✉❛s ❉✐str✐❜✉✐çõ❡s
❲❡✐❜✉❧❧ ■♥✈❡rs❛ ❡ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❲❡✐❜✉❧❧
■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛
❱❛❧❞✐❡❣♦ ❙✐q✉❡✐r❛ ▼❡❧♦
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛▼♦❞❡❧♦s ❊♥✈♦❧✈❡♥❞♦ ❉✉❛s ❉✐str✐❜✉✐çõ❡s
❲❡✐❜✉❧❧ ■♥✈❡rs❛ ❡ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❲❡✐❜✉❧❧
■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛
❱❛❧❞✐❡❣♦ ❙✐q✉❡✐r❛ ▼❡❧♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r❛
Pr♦❢✳❛ ❉❛♥✐❡❧❡ ❞❛ ❙✐❧✈❛ ❇❛r❛t❡❧❛ ▼❛rt✐♥s ◆❡t♦
Ficha catalográfica elaborada pela Biblioteca Central da Universidade de Brasília. Acervo 1016196.
Me l o , Va l d i ego S i que i r a .
M528m Mode l os envo l vendo duas d i s t r i bu i ções We i bu l l i nve r sa e ap r esen t ação da We i bu l l i nve r sa gene r a l i zada / Va l d i ego S i que i r a Me l o . - - 2014 .
v i i i , 70 f . : i l . ; 30 cm.
Di sse r t ação (mes t r ado ) - Un i ve r s i dade de Br as í l i a , I ns t i t u t o de Ci ênc i as Exa t as e da Te r r a , Depa r t amen t o de Ma t emá t i ca , 2014 .
I nc l u i b i b l i og r a f i a .
Or i en t ação : Dan i e l e da S i l va Ba r a t e l a Ma r t i ns Ne t o . 1 . Di s t r i bu i ção de We i bu l l . I . Ma r t i ns Ne t o , Dan i e l e da S i l va Ba r a t e l a . I I . Tí t u l o .
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛▼♦❞❡❧♦s ❊♥✈♦❧✈❡♥❞♦ ❉✉❛s ❉✐str✐❜✉✐çõ❡s
❲❡✐❜✉❧❧ ■♥✈❡rs❛ ❡ ❆♣r❡s❡♥t❛çã♦ ❞❛ ❲❡✐❜✉❧❧
■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛
❱❛❧❞✐❡❣♦ ❙✐q✉❡✐r❛ ▼❡❧♦✶
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❈♦♠✐ssã♦ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳❛ ❉❛♥✐❡❧❡ ❙✳ ❇❛r❛t❡❧❛ ▼❛rt✐♥s ◆❡t♦ ✲ ❖r✐❡♥t❛❞♦r❛ ✭▼❆❚✴❯♥❇✮
Pr♦❢✳❛ ❈át✐❛ ❘❡❣✐♥❛ ●♦♥ç❛❧✈❡s ✲ ▼❡♠❜r♦ ✭▼❆❚✴❯♥❇✮
Pr♦❢✳ ❆r② ❱❛s❝♦♥❝❡❧♦s ▼❡❞✐♥♦ ✲ ▼❡♠❜r♦ ✭▼❆❚✴❯♥❇✮
❇r❛sí❧✐❛✱ ✵✶ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✹
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦s ♠❡✉s ♣❛✐s✱ ▲♦✉r❞❡s ❡ ❱❛❧❞♦✱ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r❡♠ ❡♠ ♠✐♠ ❡ ♠❡ ❛♣♦✐❛r❡♠ ♥❡st❛ ✐♠♣♦rt❛♥t❡ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ➚ ♠✐♥❤❛ ✐r♠ã ❡ ❛♠✐❣❛✱ ❱❛♥❡ss❛✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣❛❧❛✈r❛s ❞❡ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♠❛✐s ♣r❡❝✐s❡✐✳
➚ ♣r♦❢❡ss♦r❛ ❉❛♥✐❡❧❡✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ❞❡❞✐❝❛çã♦ ❡ ✐♥❝❡♥t✐✈♦✳ ➚s ♣r♦❢❡ss♦r❛s ❈át✐❛ ❡ ❉é❜♦r❛✱ ❛❣r❛❞❡ç♦ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s✳ ❆♦ ♣r♦❢❡ss♦r ❆r② ♣♦r ❛❝❡✐t❛r ❝♦♠♣♦r ❛ ❜❛♥❝❛ ❞❡ ú❧t✐♠❛ ❤♦r❛ ♥♦ ❧✉❣❛r ❞❛ ♣r♦❢❛ ❉é❜♦r❛✳
❆♦ ♠❡✉ ❛♠✐❣♦✴✐r♠ã♦ ❱✐♥í❝✐✉s ♣❡❧❛ ♣❛r❝❡r✐❛ ❡ ♠♦♠❡♥t♦s ❞❡ ❞❡s❝♦♥tr❛çã♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❊❧✐❛♥❛✱ ❏❤♦♥❛tt❛♥✱ ▼❛①✱ ▲❛ís ❡ ➱❞❡r✳ ❆ ❛♠✐③❛❞❡ ❡ ❝♦♠♣❛♥❤❡✐✲ r✐s♠♦ ❞❡ ✈♦❝ês ❢♦✐ ❡ss❡♥❝✐❛❧✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ♣♦r s✉❛s ❝♦♥tr✐✲ ❜✉✐çõ❡s à ♠✐♥❤❛ ❢♦r♠❛çã♦✳
➚ ❈❛♣❡s ❡ ❛♦ ❈◆P◗ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❡♠ ❢♦r♠❛ ❞❡ ❜♦❧s❛ ❞❡ ❡st✉❞♦s✳
❘❡s✉♠♦
▼♦❞❡❧♦s ❲❡✐❜✉❧❧ tê♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❞✐s❝✉t✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛✳ ❊ss❡ ❣r❛♥❞❡ ✐♥t❡✲ r❡ss❡ t❡♠ r❡✢❡t✐❞♦ ♥♦ s✉r❣✐♠❡♥t♦ ❞❡ ♠♦❞✐✜❝❛çõ❡s ❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ♠♦❞❡❧♦s ❲❡✐❜✉❧❧✳ Pr✐♠❡✐r❛♠❡♥t❡ ♥❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s três ♠♦❞❡❧♦s ✭r✐s❝♦ ❝♦♠♣❡t✐t✐✈♦✱ ♠✉❧t✐♣❧✐✲ ❝❛t✐✈♦ ❡ ♠✐st✉r❛✮ q✉❡ ❡♥✈♦❧✈❡♠ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❜❛s❡❛❞♦s ♥♦ ❛rt✐❣♦ ❞❡ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪✳ ❆s ❢♦r♠❛s ❞❛s ❢✉♥çõ❡s ❞❡♥s✐❞❛❞❡ ❡ t❛①❛ ❞❡ ❢❛❧❤❛ sã♦ ❛♣r❡s❡♥t❛✲ ❞❛s ❡ ♠ét♦❞♦s ❣rá✜❝♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r s❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♦❜s❡r✈❛çõ❡s ♣♦❞❡ s❡r ❛❥✉st❛❞♦ ♣♦r ✉♠ ❞❡ss❡s ♠♦❞❡❧♦s sã♦ ❞✐s❝✉t✐❞♦s✳ ◆✉♠ s❡❣✉♥❞♦ ♠♦♠❡♥t♦✱ ♥ós ❛♣r❡s❡♥✲ t❛♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ ●✉s♠ã♦ ❡t ❛❧✳ ❬✽❪ ❡ ❞✐s❝✉t✐♠♦s ❛ ❡st✐♠❛çã♦ ❞❡ ♠á①✐♠❛ ✈❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠ ❞❛❞♦s ❝❡♥s✉r❛❞♦s✳ ❖ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ❞❡ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ é ✐♥✈❡st✐❣❛❞♦ ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ é ❞❡♠♦♥str❛❞❛✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❲❡✐❜✉❧❧ ■♥✈❡rs❛✱ ▼♦❞❡❧♦ ❞❡ ❘✐s❝♦✱ ▼♦❞❡❧♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✱ ▼♦❞❡❧♦ ❞❡ ▼✐st✉r❛✱ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡✱ ●rá✜❝♦✲■❲PP✳
❆❜str❛❝t
❲❡✐❜✉❧❧ ♠♦❞❡❧s ❤❛✈❡ ❜❡❡♥ ❡①t❡♥s✐✈❡❧② ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❚❤✐s ❣r❡❛t ✐♥✲ t❡r❡st ✐s r❡✢❡❝t❡❞ ✐♥ t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ♠♦❞✐✜❝❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❲❡✐❜✉❧❧ ♠♦❞❡❧s✳ ❋✐rst❧② ✐♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡s❡♥t t❤r❡❡ ♠♦❞❡❧s ✭❝♦♠♣❡t✐♥❣ r✐s❦✱ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛♥❞ ♠✐①t✉r❡✮ ✐♥✈♦❧✈✐♥❣ t✇♦ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥s✱ ❜❛s❡❞ ♦♥ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪ ♣❛♣❡r✳ ❚❤❡ s❤❛♣❡s ♦❢ t❤❡ ❞❡♥s✐t② ❛♥❞ ❢❛✐❧✉r❡ r❛t❡ ❢✉♥❝t✐♦♥s ❛r❡ s❤♦✇♥ ❛♥❞ ❣r❛♣❤✐❝❛❧ ♠❡t❤♦❞s t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❛ ❣✐✈❡♥ ❞❛t❛ s❡t ❝❛♥ ❜❡ ❛❞❥✉st❡❞ ❜② ♦♥❡ ♦❢ t❤❡s❡ ♠♦❞❡❧s ❛r❡ ❞✐s❝✉ss❡❞✳ ❙❡❝♦♥❞❧②✱ ✇❡ ♣r❡s❡♥t ❛ ❣❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥ ❜② ●✉s♠ã♦ ❡t ❛❧✳ ❬✽❪ ❛♥❞ ❞✐s❝✉ss t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ✇✐t❤ ❝❡♥s♦r❡❞ ❞❛t❛✳ ❚❤❡ ♠✐①t✉r❡ ♠♦❞❡❧ ♦❢ t✇♦ ❣❡♥❡r❛❧✐③❡❞ ✐♥✈❡rs❡ ❲❡✐❜✉❧❧ ❞✐str✐❜✉t✐♦♥s ✐s ✐♥✈❡st✐❣❛t❡❞✳ ❚❤❡ ✐❞❡♥t✐✜❛❜✐❧✐t② ♣r♦♣❡rt② ♦❢ t❤❡ ♠✐①❡❞ ♠♦❞❡❧ ✐s ❞❡♠♦♥str❛t❡❞✳
❑❡②✇♦r❞s✿ ■♥✈❡rs❡ ❲❡✐❜✉❧❧✱ ❈♦♠♣❡t✐♥❣ ❘✐s❦ ▼♦❞❡❧✱ ▼✉❧t✐♣❧✐❝❛t✐✈❡ ▼♦❞❡❧✱ ▼✐①t✉r❡ ▼♦❞❡❧✱ ■❞❡♥t✐✜❛❜✐❧✐t②✱ ■❲PP✲♣❧♦t✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐✐✐
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✺
✶✳✶ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸✳✸ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✹ ●rá✜❝♦s ▼♦❞✐✜❝❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✹✳✶ ❖ ●rá✜❝♦✲❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✹✳✷ ❖ ●rá✜❝♦✲■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸✳✹✳✸ ❈♦♠♣❛r❛çã♦ ❊♥tr❡ ♦s ●rá✜❝♦s ❲PP ❡ ■❲PP ❡ ❊st✐✲
♠❛çã♦ ❞❡ P❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹ ▼✐st✉r❛ ❞❡ ❉✐str✐❜✉✐çõ❡s ❡ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷ ❆♥á❧✐s❡ ❞❡ ❉✉❛s ❉✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✷✾
✷✳✶ ▼♦❞❡❧♦ ❞❡ ❘✐s❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✶✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✳✷ ●rá✜❝♦ ▼♦❞✐✜❝❛❞♦ ✲ ■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷ ▼♦❞❡❧♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷✳✷ ●rá✜❝♦ ▼♦❞✐✜❝❛❞♦ ✲ ■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✸ ▼♦❞❡❧♦ ❞❡ ▼✐st✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✷✳✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✸✳✸ ●rá✜❝♦ ▼♦❞✐✜❝❛❞♦ ✲ ■❲PP ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸✳✹ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✸ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛ ✺✻
✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✸ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✹ ❊st✐♠❛çã♦ ❞❡ ▼á①✐♠❛ ❱❡r♦ss✐♠✐❧❤❛♥ç❛ ❝♦♠ ❉❛❞♦s ❈❡♥s✉r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✺ ▼✐st✉r❛ ❞❡ ❉✉❛s ❉✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ●❡♥❡r❛❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✺✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡ ❡ ▼♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✺✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✺✳✸ ■❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❆❥✉st❡s q✉❡ ❝♦♥s✐❞❡r❛♠ ♦ ♠í♥✐♠♦ ❞❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞♦s ❡rr♦s✳ ✻ ✶✳✷ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡β ❡
♣❛r❛ λ= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✶✳✸ ●rá✜❝♦s ❲PP ♣❛r❛ ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ♣❛r❛ α= 5 ❡β = 3,5✳ ✳ ✳ ✶✽
✶✳✹ ●rá✜❝♦s ■❲PP ♣❛r❛ ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ♣❛r❛α = 5 ❡ β= 3,5✳ ✳ ✷✵
✷✳✶ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✺ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✻ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✼ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✽ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✾ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ✲ ❝❛s♦ ✭❛✮ ❡α1 < α2✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✷✳✶✵ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ✲ ❝❛s♦ ✭❛✮ ❡α1 > α2✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✷✳✶✶ ●rá✜❝♦✲■❲PP ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ✲ ❝❛s♦ ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸✳✶ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❛❧✲ ❣✉♥s ✈❛❧♦r❡s ❞❡ α ❝♦♠ β = 5 ❡ γ = 0,5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✸✳✷ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ α ❝♦♠ β = 5 ❡ γ = 0,5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✸✳✸ ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸✳✹ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
■♥tr♦❞✉çã♦
❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ t❡♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞❛ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✳ ❆ ♣♦♣✉❧❛r✐❞❛❞❡ ❞❡st❛ ❞✐str✐❜✉✐çã♦ é ❡✈✐❞❡♥❝✐❛❞❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐✲ ❧✐❞❛❞❡ ❡ ♥❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♥♦ ❝♦♥t❡①t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❢✉♥çõ❡s ❞❡ r✐s❝♦✳ ❙✉❛ ❤❛❜✐❧✐❞❛❞❡ ❞❡ ♠♦❞❡❧❛r t❛①❛s ❞❡ ❢❛❧❤❛s r❡s✐❞❡ ♥❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✳
◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s q✉❡ t❡♠ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❛❞❛ ♣♦r
F(t) = 1−e−(αt)β
, t ≥0, ✭✶✮
♦♥❞❡α✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛✱ ❡β✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✱ sã♦ ♣♦s✐t✐✈♦s✳ ❯♠❛ ✈❛r✐❡❞❛❞❡
❞❡ ♠♦❞❡❧♦s t❡♠ s✐❞♦ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡ ✭✶✮ ♥❛ t❡♥t❛t✐✈❛ ❞❡ ♠♦❞❡❧❛r ♦✉tr❛s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✳ ❯♠ ❞❡ss❡s ♠♦❞❡❧♦s ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪ ❛ ♣❛rt✐r ❞❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦✿ ❝♦♥s✐❞❡r❡Y ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞❡✜♥✐❞❛ ♣♦r
Y = α
2
X,
♦♥❞❡X é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ ❲❡✐❜✉❧❧✳ ❆ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦
❞❡Y é ❞❛❞❛ ♣♦r
F(t) =P(Y ≤t) = e−(α t)
β
, t >0, ✭✷✮
♦♥❞❡ α, β > 0✳ ◆❡st❡ ❝❛s♦✱ s❡ Y é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐✲
çã♦ ❞❛❞❛ ♣♦r ✭✷✮✱ ❡♥tã♦ é ❞✐t❛ t❡r ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♠♣♦rt❛♥t❡ ❞❡st❛ ❞✐str✐❜✉✐çã♦ é q✉❡ ❛ s✉❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ✉♥✐♠♦❞❛❧✳
◆❛ ❧✐t❡r❛t✉r❛ ❛❧❣✉♥s ♠♦❞❡❧♦s ❡♥✈♦❧✈❡♥❞♦ n ❞✐str✐❜✉✐çõ❡s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s sã♦ ♦❜✲
s❡r✈❛❞♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✱ ♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡ ♦ ❞❡ ♠✐st✉r❛✳ P❛r❛ ❡♥t❡♥❞ê✲❧♦s✱ ❛♣r❡s❡♥t❛♠♦s✿
• ❙❡❥❛♠T1, . . . , Tn ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s✱ s❡♥❞♦F1(t), . . . , Fn(t)❛s r❡s✲
■♥tr♦❞✉çã♦ ✷
❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ é ❞❛❞❛ ♣♦r
F(t) = 1−
n Y
i=1
[1−Fi(t)].
❊st❡ ♠♦❞❡❧♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✳ ❖✉tr❛s t❡r♠✐♥♦❧♦❣✐❛s t❛♠❜é♠ sã♦ ✉t✐❧✐③❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♠♦❞❡❧♦ ❝♦♠♣♦st♦✱ ♠♦❞❡❧♦ ❞❡ s✐st❡♠❛ ❡♠ sér✐❡ ❡ ♠♦❞❡❧♦ ♠✉❧t✐rr✐s❝♦✳ ❊❧❡ t❡♠ r❡❝❡❜✐❞♦ ❝♦♥s✐❞❡rá✈❡❧ ❛t❡♥çã♦ ♥❛ ❧✐t❡r❛t✉r❛ ❡ t❡♠ s✐❞♦ ❡st✉❞❛❞♦ ❤á ❜❛st❛♥t❡ t❡♠♣♦✳ ◆♦ ❝♦♥t❡①t♦ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ ♦ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ s✐st❡♠❛ ❞❡n❝♦♠♣♦♥❡♥t❡s
❡♠ sér✐❡✱ s❡♥❞♦ Ti ♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞❛ ❝♦♠♣♦♥❡♥t❡i❡Z ♦ ❞♦ s✐st❡♠❛✳ ❖ s✐st❡♠❛
❢❛❧❤❛ ♥♦ ♣r✐♠❡✐r♦ ✐♥st❛♥t❡ q✉❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❢❛❧❤❛✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ◆❡❧s♦♥ ❬✶✺❪✳
• ❙❡❥❛♠T1, . . . , Tn ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s✱ s❡♥❞♦F1(t), . . . , Fn(t)❛s r❡s✲
♣❡❝t✐✈❛s ❢✉♥çõ❡s ❞❡ ❞✐str✐❜✉✐çã♦ ❛ss♦❝✐❛❞❛s✳ ❙❡ W = max{T1, . . . , Tn}✱ ❡♥tã♦ ❛
s✉❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ é ❞❛❞❛ ♣♦r
F(t) = P(W ≤t) =P
n \
i=1
(Ti ≤t) !
=
n Y
i=1 Fi(t).
❊st❡ ♠♦❞❡❧♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡✱ ❛♣❡s❛r ❞❡❧❡ ♥ã♦ t❡r s❡ ❞❡st❛❝❛❞♦ t❛♥t♦ q✉❛♥t♦ ♦ ♠♦❞❡❧♦ ❞❡ r✐s❝♦✱ ❝♦♠♦ ♦❜s❡r✈❛❞♦ ♣♦r ▼✉rt❤② ❡t ❛❧✳ ❬✶✹❪✱ t❡♠ s✐❞♦ ❛♣❧✐❝❛❞♦ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ♥♦ ❝♦♥t❡①t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ ♣❛r❛❧❡❧♦ ❝♦♠ ❝♦♠♣♦♥❡♥t❡s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❯♠ s✐st❡♠❛ ♣♦❞❡ s❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠ ♦r❣❛♥✐s♠♦ ♦✉ ✉♠ ❞✐s♣♦s✐t✐✈♦ ♠❡❝â♥✐❝♦✱ ❡ ♦ t❡r♠♦ ♣❛r❛❧❡❧♦ s✐❣♥✐✜❝❛ q✉❡ ❛ s✉❛ ❢❛❧❤❛ ❡stá ❝♦♥❞✐❝✐♦♥❛❞❛ à ❢❛❧❤❛ ❞❡ t♦❞❛s ❛s s✉❛s ♣❛rt❡s✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ✈❡❥❛ ❊❧❛♥❞t✲❏♦❤♥s♦♥ ❡ ❏♦❤♥s♦♥ ❬✼❪✳
• ▼♦❞❡❧♦s ❞❡ ♠✐st✉r❛s ✜♥✐t❛s tê♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❞✐s❝✉t✐❞♦s ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✳
❊ss❡ ✐♥t❡r❡ss❡ é ❞❡❝♦rr❡♥t❡ ❞❡ s✉❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ár❡❛s ❞✐✈❡rs❛s ❝♦♠♦ ❇✐♦❧♦✲ ❣✐❛✱ ❋ís✐❝❛✱ ❊❝♦♥♦♠✐❛✱ ▼❡❞✐❝✐♥❛ ❡ ❊♥❣❡♥❤❛r✐❛✳ ❊♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✱ ♦s ❞❛❞♦s ❞✐s♣♦♥í✈❡✐s ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ❝♦♠♦ s❡♥❞♦ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡ ♠✐st✉r❛ ❞❡ ❞✉❛s ♦✉ ♠❛✐s ❞✐str✐❜✉✐çõ❡s✳ ❚✐tt❡r✐♥❣t♦♥ ❡t ❛❧✳ ❬✶✽❪ ❡ ▼❝▲❛❝❤❧❛♥ ❡ P❡❡❧ ❬✶✻❪ sã♦ ❛❧❣✉♠❛s ❞❛s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ♠✐st✉r❛s ✜♥✐t❛s✳
■♥tr♦❞✉çã♦ ✸
❱ár✐♦s ❝❛s♦s ❡s♣❡❝✐❛✐s ❞♦s ♠♦❞❡❧♦s ♠❡♥❝✐♦♥❛❞♦s ❛❝✐♠❛ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s✱ ❞❡♣❡♥✲ ❞❡♥❞♦ ❞♦ ♥ú♠❡r♦ ❞❡ s✉❜♣♦♣✉❧❛çõ❡s ❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❛ss♦❝✐❛❞❛ ❛ ❝❛❞❛ ✉♠❛ ❞❡ss❛s s✉❜♣♦♣✉❧❛çõ❡s✳ ❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❞❡st❡ ❡st✉❞♦ é ❛♥❛❧✐s❛r ❡ss❡s ♠♦❞❡❧♦s ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡n= 2 ❡ ❛♠❜❛s ❛s s✉❜♣♦♣✉❧❛çõ❡s s❡❣✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✳ ▼✉rt❤②
❡t ❛❧✳ ❬✶✹❪ r❡ú♥❡♠ ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❡st✉❞♦s r❡❧❛❝✐♦♥❛❞♦s às ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ♣✉❜❧✐❝❛❞♦s ❛té ♦ ✐♥í❝✐♦ ❞♦s ❛♥♦s ✷✵✵✵ ❡♥✈♦❧✈❡♥❞♦ ❡ss❡s ♠♦❞❡❧♦s✳ ❈♦♠♦ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s ❝✐t❛r ❏✐❛♥❣ ❡ ▼✉rt❤② ❬✾❪ ❡ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪✱ ❞✉❛s ♣r✐♥❝✐♣❛✐s r❡❢❡✲ rê♥❝✐❛s ❞♦ ♥♦ss♦ ❡st✉❞♦✳ ❖ ♣r✐♠❡✐r♦ ❛♣r❡s❡♥t❛ ✉♠ ♠ét♦❞♦ ❣rá✜❝♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ♠♦❞❡❧♦ ❞❡ ♠✐st✉r❛ ❞❡ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧❀ ♦ s❡❣✉♥❞♦ ❛♣r❡s❡♥t❛ ❛ ❢♦r♠❛ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❡ ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ ♠✐st✉r❛✱ ❞❡ r✐s❝♦ ❡ ♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ❞✉❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❛❧é♠ ❞❡ ✉♠❛ té❝♥✐❝❛ ❣rá✜❝❛ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ss❡s ♠♦❞❡❧♦s✳
❚❛♠❜é♠ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ♠❛✐s r❡❝❡♥t❡✱ ♦♥❞❡ ●✉s♠ã♦ ❡t ❛❧✳ ❬✽❪ ♣r♦♣õ❡♠ ✉♠❛ ♥♦✈❛ ❞✐str✐❜✉✐çã♦ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✳ ❆ s✉❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦✱
G(t)✱ ♣r♦♣♦st❛ ♣❡❧♦s ❛✉t♦r❡s ❡♠ ❬✽❪✱ é ❞❛❞❛ ♣♦r
G(t) = [F(t)]γ =e−γ(αt)β
, t >0. ✭✸✮
❆ ❞✐str✐❜✉✐çã♦ ❛ss♦❝✐❛❞❛ à ✭✸✮ é ❝❤❛♠❛❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❣❡♥❡r❛❧✐③❛❞❛✳ ❊st❛ ❞✐str✐❜✉✐çã♦ t❡♠ ✉♠❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❛✐♥❞❛ ♠❛✐♦r✱ ❞❡✈✐❞♦ ❛ ✐♥❝❧✉sã♦ ❞♦ ♣❛râ♠❡tr♦γ✱
q✉❡ ♣❡r♠✐t❡ ♦❜t❡r ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❛s ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ❡①♣♦♥❡♥❝✐❛❧ ✐♥✈❡rs❛ ❡ ❘❛②❧❡✐❣❤ ✐♥✈❡rs❛✳ ❖✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡st❛ ❞✐str✐❜✉✐çã♦ é q✉❡ ❛ s✉❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ t❡♠ ❢♦r♠❛ ✉♥✐♠♦❞❛❧✳
❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ♦s ♣r✐♥❝✐♣❛✐s tó♣✐❝♦s ❛❜♦r❞❛❞♦s ❡♠ ❝❛❞❛ ❝❛♣ít✉❧♦✳
❖ ❈❛♣ít✉❧♦ ✶ é ❞❡❞✐❝❛❞♦ à ❛♣r❡s❡♥t❛çã♦ ❞❡ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ r❡st❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ ▼♦str❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ é ✉♥✐♠♦❞❛❧✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♠ét♦❞♦ ❣rá✜❝♦✱ ❝❤❛♠❛❞♦ ❣rá✜❝♦✲■❲PP✱ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❡ ❞✉❛s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s✱ ❞❡ ❈❤❛♥❞r❛ ❬✹❪ ❡ ❆t✐❡♥③❛ ❡t ❛❧✳ ❬✶❪✱ ♣❛r❛ ❣❛r❛♥t✐r ❛ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠✐st✉r❛s ✜♥✐t❛s ❞❡ ❞✐str✐❜✉✐çõ❡s✳
■♥tr♦❞✉çã♦ ✹
Cap´ıtulo
1
Pr❡❧✐♠✐♥❛r❡s
❊st❡ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ à ❛♣r❡s❡♥t❛çã♦ ❞❡ r❡s✉❧t❛❞♦s✱ ❝♦♥❝❡✐t♦s ❡ ❞❡✜♥✐çõ❡s ✐♠✲ ♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ ■♥✐❝✐❛♠♦s ♥❛ ❙❡çã♦ ✶✳✶ ❝♦♠ ♦ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♣❛r❛ ❛❥✉st❡ ❞❡ r❡t❛s✳ ◆❛ ❙❡çã♦ ✶✳✷✱ ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❞❡ s♦✲ ❜r❡✈✐✈ê♥❝✐❛ ❡ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✱ ✐♠♣♦rt❛♥t❡s ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❡ ❡①❡♠♣❧✐✜❝❛♠♦✲❛s ♣♦r ♠❡✐♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡①♣♦♥❡♥❝✐❛❧ ❡ ❲❡✐❜✉❧❧✳ ❆s r❡❢❡rê♥❝✐❛s ❜ás✐❝❛s ♣❛r❛ ❡ss❛ s❡çã♦ sã♦✿ ❇♦❧❢❛r✐♥❡ ❡t ❛❧✳ ❬✷❪✱ ❈♦❧♦s✐♠♦ ❡ ●✐♦❧♦ ❬✺❪ ❡ ▲❛✇❧❡ss ❬✶✷❪✳ ◆❛ ❙❡çã♦ ✶✳✸✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❝♦♠♦ ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧✱ s❡❣✉✐♥❞♦ ♦ tr❛❜❛❧❤♦ ❞❡ ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪✳ ❆❧é♠ ❞✐ss♦✱ ❡st✉❞❛♠♦s ❛s ❢♦r♠❛s ❞❛s ❢✉♥çõ❡s ❞❡♥s✐❞❛❞❡ ❡ t❛①❛ ❞❡ ❢❛❧❤❛✱ ❝❛❧❝✉❧❛♠♦s ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❡ ❞✐s❝✉t✐♠♦s ❞✉❛s té❝♥✐❝❛s ❣rá✜❝❛s✱ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❜❛s❡❛❞♦s ♥❡ss❡s ♠ét♦❞♦s✳ ◆❛ ❙❡çã♦ ✶✳✹ ✜♥❛❧✐③❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠ ❡st✉❞♦ s♦❜r❡ ♠✐st✉r❛s ✜♥✐t❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❡ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡✱ ♦♥❞❡ ❞❡♠♦♥str❛♠♦s ❞♦✐s r❡s✉❧t❛❞♦s✱ ❞❡ ❈❤❛♥❞r❛ ❬✹❪ ❡ ❆t✐❡♥③❛ ❡t ❛❧✳ ❬✶❪✱ q✉❡ ❣❛r❛♥t❡♠ ❛ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠✐st✉r❛s ✜♥✐t❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ✉♠❛ ♠❡s♠❛ ❢❛♠í❧✐❛ ♣❛r❛♠étr✐❝❛✳ ❆ ✐❞❡♥t✐✜❝❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠✐st✉r❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❲❡✐❜✉❧❧ é t❛♠❜é♠ ❞❡♠♦♥str❛❞❛✳
✶✳✶ ✲ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s
✶✳✶✳ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✻
✉♠ ❜♦♠ ❛❥✉st❡✳ ■st♦ ♦❝♦rr❡ ♣♦rq✉❡ ♦ ❡rr♦ ♣♦❞❡ s❡r t❛♥t♦ ♣♦s✐t✐✈♦ ✭q✉❛♥❞♦ ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❡st✐✈❡r ❛❝✐♠❛ ❞❛ r❡t❛✮ q✉❛♥t♦ ♥❡❣❛t✐✈♦ ✭q✉❛♥❞♦ ♦ ✈❛❧♦r ♦❜s❡r✈❛❞♦ ❡st✐✈❡r ❛❜❛✐①♦ ❞❛ r❡t❛✮✱ ♦ q✉❡ ♣♦❞❡ ❧❡✈❛r ❛ ✉♠❛ s♦♠❛ ♥✉❧❛ ❞♦s ❡rr♦s ♠❡s♠♦ ❝♦♠ ♦s ✈❛❧♦r❡s ♠✉✐t♦ ❞✐st❛♥t❡s ❞❛ r❡t❛✳
❯♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ s✉♣❡r❛r ❡st❡ ♣r♦❜❧❡♠❛ é ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛❜s♦✲ ❧✉t♦s ❞♦s ❡rr♦s✱ ✉♠❛ ✈❡③ q✉❡ ✐st♦ ❡✈✐t❛ q✉❡ ❣r❛♥❞❡s ❡rr♦s ♣♦s✐t✐✈♦s ❡ ♥❡❣❛t✐✈♦s s❡❥❛♠ ❝♦♠♣❡♥s❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ t❛❧ ♠ét♦❞♦ ❛✐♥❞❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❞❡s✈❛♥t❛❣❡♠✳ P❡❧❛ ❋✐❣✉r❛ ✶✳✶✱ ♥♦t❡ q✉❡ ♦ ❛❥✉st❡ ❡♠ (b) s❛t✐s❢❛③ ❡st❡ ❝r✐tér✐♦ ♠❡❧❤♦r q✉❡ ♦ ❛❥✉st❡ ❡♠ (a)✳ ▼❛s
❡st❛ ♥ã♦ é ✉♠❛ ❜♦❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✱ ♣♦✐s ♥❡♥❤✉♠❛ ❛t❡♥çã♦ é ❞❛❞❛ ❛♦ ♣♦♥t♦ ❞♦ ♠❡✐♦✳ ❖ ❛❥✉st❡ ❡♠ (a)é ♣r❡❢❡rí✈❡❧ ♣♦rq✉❡ ❧❡✈❛ ❡♠ ❝♦♥s✐❞❡r❛çã♦ t♦❞♦s ♦s ♣♦♥t♦s✳
❯♠❛ ♦✉tr❛ ♦♣çã♦ ♣❛r❛ t❡♥t❛r♠♦s s✉♣❡r❛r ♦ ♥♦ss♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧ é ♠✐♥✐♠✐③❛r ❛ s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞♦s ❡rr♦s✳ ❊st❡ ❝r✐tér✐♦ é ❝❤❛♠❛❞♦ ❞❡ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s✳ ❊❧❡ é ❜❡♠ út✐❧ ❡ ♣rát✐❝♦✱ ✉♠❛ ✈❡③ q✉❡ ♦s ❝á❧❝✉❧♦s ❡♥✈♦❧✈✐❞♦s sã♦ ❜❡♠ s✐♠♣❧❡s✳ ❆♦ ❝♦♥s✐❞❡r❛r♠♦s ♦ q✉❛❞r❛❞♦ ❞♦s ❡rr♦s✱ ❡st❡s s❡ t♦r♥❛♠ ♣♦s✐t✐✈♦s✱ ♦ q✉❡ r❡s♦❧✈❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ s✐♥❛❧✱ ❡✱ ❛❧é♠ ❞✐ss♦✱ ♦s ❣r❛♥❞❡s ❡rr♦s sã♦ ❡✈✐t❛❞♦s✳
❋✐❣✉r❛ ✶✳✶✿ ❆❥✉st❡s q✉❡ ❝♦♥s✐❞❡r❛♠ ♦ ♠í♥✐♠♦ ❞❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞♦s ❡rr♦s✳
P❛r❛ ❞❡✜♥✐r♠♦s ♦ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♣❛r❛ ❛❥✉st❡ ❞❡ r❡t❛s✱ ❝♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ X ❡Y s❡❥❛♠ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s r❡❧❛❝✐♦♥❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
Y =a0+bX, ✭✶✳✶✮
♦♥❞❡ a0 ❡ b sã♦ ❝♦♥st❛♥t❡s ❞❡s❝♦♥❤❡❝✐❞❛s✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡st✐♠❛r a0 ❡ b s❡♠ ♦s
♣r♦❜❧❡♠❛s ♠❡♥❝✐♦♥❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♠❡❝❡♠♦s t♦♠❛♥❞♦ ✉♠❛ ❛♠♦str❛ ♦❜s❡r✈❛❞❛
(x1, y1),(x2, y2), . . . ,(xn, yn)
❞♦ ✈❡t♦r ❛❧❡❛tór✐♦(X, Y)❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦✿
✶✳✶✳ ❆❥✉st❡ ❞❡ ❘❡t❛ ♣♦r ▼í♥✐♠♦s ◗✉❛❞r❛❞♦s ✼
♣❛r❛ i= 1,2, . . . , n❡ ♦♥❞❡ x¯= 1
n
n X
i=1 xi✳
❆ tr❛♥s❢♦r♠❛çã♦ ♣r♦♣♦st❛ ❝♦♥❞✉③ ❛ ❞♦✐s ❢❛t♦s ✐♠♣♦rt❛♥t❡s✳ ❙ã♦ ❡❧❡s✿
✶✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♦ ❡✐①♦y é tr❛♥s❧❛❞❛❞♦ ❞❡ 0 ♣❛r❛ x¯❡ ❛ ❡q✉❛çã♦ ✭✶✳✶✮ ♣♦❞❡ s❡r
r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛✿
Y =a+bZ,
♦♥❞❡ Z = X −X¯ ❝♦♠ X¯ = 1
n
n X
i=1
Xi ❡ X1, . . . , Xn ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❛r✐á✈❡✐s
❛❧❡❛tór✐❛s ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s ❡ ❝♦♠ ❛ ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ X✳
✷✳ ❆❧❣✉♥s ❝á❧❝✉❧♦s sã♦ s✐♠♣❧✐✜❝❛❞♦s✳ ◆♦t❡ q✉❡✿
n X
i=1 zi =
n X
i=1
(xi−x¯) = n X
i=1
xi−nx¯= n X
i=1
xi −n·
1
n
n X
i=1
xi = 0. ✭✶✳✸✮
P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ ❞❛q✉✐ ❛té ♦ ✜♥❛❧ ❞❡st❛ s❡çã♦✱ ♦♠✐t✐♠♦s ♦s í♥❞✐❝❡s ❞♦ s♦♠❛tór✐♦✱ ✉t✐❧✐③❛♥❞♦ ♦ sí♠❜♦❧♦P ❛♦ ✐♥✈és ❞❡
n X
i=1
✳
❖ ♦❜❥❡t✐✈♦ ❞♦ ♠ét♦❞♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s é ❡♥❝♦♥tr❛r ˆa ❡ ˆb✱ ❡st✐♠❛❞♦r❡s ❞❡
♠í♥✐♠♦s q✉❛❞r❛❞♦s ❞❡ a ❡ b✱ ♣❡❧♦s ✈❛❧♦r❡s ❞❡ a ❡ b q✉❡ ♠✐♥✐♠✐③❛♠ ❛ ❢✉♥çã♦ S(a, b)✱
❞❛❞❛ ♣♦r
S(a, b) =X(yi−a−bzi)2, ✭✶✳✹✮
❊ss❡s ❡st✐♠❛❞♦r❡s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳
❚❡♦r❡♠❛ ✶✳✶✳ ❙❡S(a, b) é ❛ ❢✉♥çã♦ ❞❛❞❛ ❡♠ ✭✶✳✹✮✱ ❡♥tã♦ ♦s ❡st✐♠❛❞♦r❡s ❞❡ ♠í♥✐♠♦s
q✉❛❞r❛❞♦s ♣❛r❛ a ❡ b sã♦ ❞❛❞♦s ♣♦r
ˆ
a = ¯y= 1
n
X
yi ❡ ˆb = P
yizi P
z2
i
.
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦✭s✮ ♣♦♥t♦✭s✮ ❝rít✐❝♦✭s✮ ❞❡ ❙✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ✭✶✳✹✮ ❡♠ r❡❧❛çã♦ ❛a é
S1(a, b) = ∂
∂aS(a, b) = −2
X
(yi−a−bzi). ✭✶✳✺✮
■❣✉❛❧❛♥❞♦ ❛ ③❡r♦ ❡ ✉t✐❧✐③❛♥❞♦ ✭✶✳✸✮✱ s❡❣✉❡ q✉❡
ˆ
a=
P
yi
n = ¯y. ✭✶✳✻✮
❆ ❞❡r✐✈❛❞❛ ❞❡ ✭✶✳✹✮ ❡♠ r❡❧❛çã♦ ❛b é ❞❛❞❛ ♣♦r
S2(a, b) = ∂
∂bS(a, b) = −2
X
✶✳✷✳ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✽
■❣✉❛❧❛♥❞♦ ❛ ③❡r♦ ❡ ✉s❛♥❞♦ ♥♦✈❛♠❡♥t❡ ✭✶✳✸✮✱ ♦❜t❡♠♦s
ˆb= Pyizi P
z2
i
. ✭✶✳✽✮
▲♦❣♦✱ ♦ ♣♦♥t♦(a, b)✱ ❞❡t❡r♠✐♥❛❞♦ ♣♦r ✭✶✳✻✮ ❡ ✭✶✳✽✮✱ é ♦ ú♥✐❝♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡S✳ ❘❡st❛
♠♦str❛r q✉❡ ❡❧❡ é ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞❡ S✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❡ s✉❛s ❞❡r✐✈❛❞❛s
♣❛r❝✐❛✐s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❉❡ ✭✶✳✺✮ ❡ ✭✶✳✼✮✱ ✉s❛♥❞♦ ✭✶✳✸✮✱ s❡❣✉❡ q✉❡
S11(a, b) = 2n, S12(a, b) = 0 =S21(a, b) ❡ S22(a, b) = 2Xz2i.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❤❡ss✐❛♥❛ é ❞❛❞♦ ♣♦r
H(a, b) =
2n 0 0 2Pz2
i = 4n
X
zi2 >0,
❡ ❝♦♠♦ ❛✐♥❞❛ S11(a, b) = 2n >0✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
✶✳✷ ✲ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛
◆❛s ❞❡✜♥✐çõ❡s s❡❣✉✐♥t❡s ❞❡st❛ s❡çã♦✱ T ❞❡♥♦t❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ♥ã♦ ♥❡❣❛t✐✈❛
❡ ❝♦♥tí♥✉❛ q✉❡ r❡♣r❡s❡♥t❛ ♦ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ❞❡ ✉♠ ❡❧❡♠❡♥t♦✱ ♦✉ s❡❥❛✱ ♦ t❡♠♣♦ ❛té ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ ❞❡ ✐♥t❡r❡ss❡✳ ❖ t❡♠♣♦ ❞❡ ❢❛❧❤❛ ♣♦❞❡ s❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ t❡♠♣♦ ❞❡ ✈✐❞❛ ❞❡ ✉♠ ♣❛❝✐❡♥t❡ ❞♦ ♠♦♠❡♥t♦ q✉❡ ❢♦✐ ❞✐❛❣♥♦st✐❝❛❞❛ ❛ ❞♦❡♥ç❛ ❛té ❛ s✉❛ ♠♦rt❡ ♦✉ ❛té ❛ ❝✉r❛✱ ♣♦❞❡ ❛✐♥❞❛ s❡r ♦ t❡♠♣♦ út✐❧ ❞❡ ✉♠ ❝❡rt♦ ❡q✉✐♣❛♠❡♥t♦ ❛té ❛ s✉❛ q✉❡✐♠❛✳ ❙❡❥❛♠ f(t) ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ T ❡ F(t) ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦
❝♦rr❡s♣♦♥❞❡♥t❡✳
❉❡✜♥✐çã♦ ✶✳✷✳ ❆ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ T é ❞❡✜♥✐❞❛ ♣♦r
S(t) = P(T ≥t) =
Z ∞
t
f(x)dx.
◆♦t❡ q✉❡ ❛ ❢✉♥çã♦ S(t) r❡♣r❡s❡♥t❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♦❜s❡r✈❛çã♦ ♥ã♦ ❢❛❧❤❛r
❛té ✉♠ ❝❡rt♦ t❡♠♣♦ t✱ ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♦❜s❡r✈❛çã♦ s♦✲
❜r❡✈✐✈❡r ❛♦ t❡♠♣♦t✳ ◗✉❛♥❞♦ tr❛t❛❞❛ ❡♠ ❝♦♥t❡①t♦s q✉❡ ❡♥✈♦❧✈❡♠ s✐st❡♠❛s ♦✉ t❡♠♣♦s
❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ ✐t❡♥s ♠❛♥✉❢❛t✉r❛❞♦s✱ é t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❞❡ ❝♦♥✜❛❜✐✲ ❧✐❞❛❞❡✳ ❉❛ ❉❡✜♥✐çã♦ ✶✳✷✱ ✉t✐❧✐③❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡✱ s❡❣✉❡ q✉❡
S(t) é ✉♠❛ ❢✉♥çã♦ ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡✱ ❝♦♥tí♥✉❛✱ ❝♦♠ S(0) = 1 ❡ lim
t→∞S(t) = 0✳
❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡T ♣♦r
✶✳✷✳ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✾
❉❡r✐✈❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡st❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s✿
f(t) = −S′
(t), ✭✶✳✾✮
♦♥❞❡′ ❞❡♥♦t❛ ❛ ❞❡r✐✈❛❞❛ ❡♠ r❡❧❛çã♦ ❛t✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❆ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ✭♦✉ ❢✉♥çã♦ r✐s❝♦✮ é ❞❡✜♥✐❞❛ ♣♦r
h(t) = lim
∆t→0
P(t ≤T < t+ ∆t|T ≥t)
∆t .
◆♦t❡ q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❡s♣❡❝✐✜❝❛ ❛ t❛①❛ ✐♥st❛♥tâ♥❡❛ ❞❡ ❢❛❧❤❛ ♦✉ ♠♦rt❡ ♥♦ t❡♠♣♦t✱ ❞❛❞♦ q✉❡ ♦ ❡❧❡♠❡♥t♦ s♦❜r❡✈✐✈❡✉ ❛té ♦ t❡♠♣♦t ✭▲❛✇❧❡ss ❬✶✷❪✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱
❞❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧✱ t❡♠♦s✿
h(t) = lim
∆t→0
P(t≤T < t+ ∆t|T ≥t)
∆t = lim∆t→0
P(t ≤T < t+ ∆t, T ≥t) ∆t P(T ≥t)
= lim
∆t→0
P(t≤T < t+ ∆t) ∆t P(T ≥t) ,
♣♦✐s[t, t+ ∆t)⊂[t,+∞)✳ ❆ss✐♠✱
h(t) = lim
∆t→0
F(t+ ∆t)−F(t) ∆t S(t) =
F′(t)
S(t) =
f(t)
S(t).
◆♦s ❡①❡♠♣❧♦s s❡❣✉✐♥t❡s✱ ❝❛❧❝✉❧❛♠♦s ❛s ❢✉♥çõ❡s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❡ t❛①❛ ❞❡ ❢❛❧❤❛ ♣❛r❛ ❛s ❞✐str✐❜✉✐çõ❡s ❡①♣♦♥❡♥❝✐❛❧ ❡ ❲❡✐❜✉❧❧ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s✳ ◆♦ ❝♦♥t❡①t♦ ❞❛ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♦✉ ❞❛ t❡♦r✐❛ ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ ❛♠❜❛s ♦❝✉♣❛♠ ♣♦s✐çã♦ ❞❡ ❞❡st❛q✉❡ ♣♦r s❡ ❛❞❡q✉❛r❡♠ ❛ ♠✉✐t❛s s✐t✉❛çõ❡s ♣rát✐❝❛s✳
❊①❡♠♣❧♦ ✶✳✹✳ ❆ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T
❝♦♠ ❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ é ❞❛❞❛ ♣♦r
f(t) = λe−λt, λ, t >0.
❆ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡T é ♦❜t✐❞❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
S(t) =
Z ∞
t
λe−λxdx=e−λt.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❞❛❞❛ ♣♦r
h(t) = f(t)
S(t) =λ.
✶✳✷✳ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛ ✶✵
❊①❡♠♣❧♦ ✶✳✺✳ ❆ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♣❛r❛ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T
❝♦♠ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ é ❞❛❞❛ ♣♦r
f(t) = λβ(λt)β−1e−(λt)β
, t >0,
♦♥❞❡ β✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✱ ❡ λ✱ ♦ ❞❡ ❡s❝❛❧❛✱ sã♦ ❛♠❜♦s ♣♦s✐t✐✈♦s✳ ❆ ❢✉♥çã♦ ❞❡
s♦❜r❡✈✐✈ê♥❝✐❛ ❞❡ T é ❡①♣r❡ss❛ ♣♦r
S(t) =
Z ∞
t
λβ(λx)β−1e−(λx)β
dx=
Z ∞
(λt)β
e−udu=e−(λt)β
❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❞❛❞❛ ♣♦r
h(t) = f(t)
S(t) =λβ(λt)
β−1, t >0.
❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ é ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❡ ♥❛ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛✱ ❞❡✈✐❞♦ ❛ s✉❛ ✢❡①✐❜✐❧✐❞❛❞❡ ❡♠ ❛❥✉st❛r ❞✐❢❡r❡♥t❡s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✳ P❛r❛β = 1✱ t❡♠♦s ❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r
❞❛ ❲❡✐❧❜✉❧❧ ❡✱ ❝♦♠♦ ✈✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✶✳✹✱ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❝♦♥st❛♥t❡✳ ❙❡β >1✱
❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❝r❡s❝❡♥t❡ ❡✱ ♥♦ ❝❛s♦ ❡♠ q✉❡β <1✱ ❡❧❛ é ❞❡❝r❡s❝❡♥t❡✳ ❉❡ ❢❛t♦✱
❝♦♠♦
h′
(t) = λ2β(β−1)(λt)β−2
❡ β, λ, t > 0✱ s❡❣✉❡ q✉❡ h′(t)> 0 ♣❛r❛ β > 1 ❡✱ ❛ss✐♠✱ h é ❝r❡s❝❡♥t❡ ❡ q✉❛♥❞♦ β < 1✱
t❡♠♦sh′(t)<0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ h é ❞❡❝r❡s❝❡♥t❡✳ ❱❡❥❛ ❋✐❣✉r❛ ✶✳✷✳
❋✐❣✉r❛ ✶✳✷✿ ❋✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ β ❡
✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✶
✶✳✸ ✲ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛
❆ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ❣❛♥❤♦✉ ❞❡st❛q✉❡ ❛ ♣❛rt✐r ❞♦s ❡st✉❞♦s ❞❡ ❲❛❧♦❞❞✐ ❲❡✐❜✉❧❧ q✉❡ ❡✈✐❞❡♥❝✐❛r❛♠ ❛ s✉❛ ❛♠♣❧❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ ❙❡✉ ♣r✐♠❡✐r♦ tr❛❜❛❧❤♦ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ♥♦ ❛♥♦ ❞❡ ✶✾✸✾ ❡♠ ✉♠ ❥♦r♥❛❧ ❡s❝❛♥❞✐♥❛✈♦ ❡ tr❛t♦✉ ❞❛ r❡s✐stê♥❝✐❛ ❞❡ ♠❛t❡r✐❛✐s✳ ❊♠ ✉♠ ❡st✉❞♦ s✉❜s❡q✉❡♥t❡ ❡♠ ✐♥❣❧ês✱ ❞❡ ✶✾✺✶✱ ❡❧❡ ♠♦str♦✉ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❡♠ ❞✐❢❡r❡♥t❡s ár❡❛s✳ ❉❡s❞❡ ❡♥tã♦✱ ♦ ♠♦❞❡❧♦ t❡♠ s✐❞♦ ❡①t❡♥s✐✈❛♠❡♥t❡ ✉t✐❧✐③❛❞♦ ❡♠ ❡st✉❞♦s ❞❡ ❝♦♥✜❛❜✐❧✐❞❛❞❡✱ ♥❛ ❛♥á❧✐s❡ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ❡ ❡♠ ♦✉tr❛s ár❡❛s ❞❡✈✐❞♦ ❛ s✉❛ ✈❡rs❛t✐❧✐❞❛❞❡✱ ❛ ♠❡❞✐❞❛ q✉❡ ♣♦❞❡ ♠♦❞❡❧❛r t❛①❛s ❞❡ ❢❛❧❤❛s ❝♦♠ ❢♦r♠❛s ❝♦♥st❛♥t❡s✱ ❝r❡s❝❡♥t❡s ❡ ❞❡❝r❡s❝❡♥t❡s ✭✈❡❥❛ ❊①❡♠♣❧♦ ✶✳✺✮✳
▼✉✐t❛s ♠♦❞✐✜❝❛çõ❡s ❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❢♦r❛♠ ♣r♦♣♦st❛s ❡♠ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❡♠ ▼✉rt❤② ❡t ❛❧✳ ❬✶✹❪ ❡ ▲❛✐ ❡t ❛❧✳ ❬✶✶❪✱ q✉❡ ♣❡r♠✐t❡♠ ♠♦❞❡❧❛r ♦✉tr❛s ❢♦r♠❛s ❞❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛✳ ❯♠❛ ❞❡ss❛s é ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ✉♥✐♠♦❞❛❧✱ q✉❡ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞❛ ♣♦r ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝❤❛♠❛❞❛ ❞❡ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✱ ♣r♦♣♦st❛ ♣♦r ❏✐❛♥❣ ❡t ❛❧✳ ❬✶✵❪ ❝♦♠♦ ✉♠❛ ✈❛r✐❛çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧✳
❈♦♥s✐❞❡r❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s q✉❡ t❡♠ ❢✉♥çã♦ ❞❡ ❞✐str✐✲ ❜✉✐çã♦ ❞❛❞❛ ♣♦r
F(t) = 1−e(αt)β
, α, β >0; t≥0, ✭✶✳✶✵✮
♦♥❞❡ α é ♦ ♣❛râ♠❡tr♦ ❞❡ ❡s❝❛❧❛ ❡ β é ♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛✳ ❱❛♠♦s ♥♦s r❡❢❡r✐r ❛
✭✶✳✶✵✮ ❝♦♠♦ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦✳ P♦❞❡♠♦s ✉t✐❧✐③❛r ♣❛r❛♠❡tr✐③❛çõ❡s ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ r❡♣r❡s❡♥tá✲❧♦✱ ❝♦♠♦ ❛s ❧✐st❛❞❛s ❛ s❡❣✉✐r✿
F(t) = 1−e−(λt)β
, ❝♦♠ λ= 1
α;
F(t) = 1−e−tβ
α′, ❝♦♠ α′ =αβ;
F(t) = 1−e−λ′
tβ
, ❝♦♠ λ′
=
1
α
β
.
❊♠❜♦r❛ s❡❥❛♠ t♦❞❛s ❡q✉✐✈❛❧❡♥t❡s✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ❝♦♥t❡①t♦ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❛r✲ t✐❝✉❧❛r ♣♦❞❡ s❡r ♠❛✐s ❛♣r♦♣r✐❛❞❛✳ ◆♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ❛ r❡♣r❡s❡♥t❛çã♦ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦ é ❛ ❡q✉❛çã♦ ✭✶✳✶✵✮✱ ❛ ♠❡♥♦s q✉❡ s❡❥❛ ✐♥❞✐❝❛❞♦ ♦ ❝♦♥trár✐♦✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♠♦❞❡❧♦s t❡♠ s✐❞♦ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡st❡ ♠♦❞❡❧♦ ❜ás✐❝♦✳ ❯♠ ❞❡❧❡s é ♦ ❡st✉✲ ❞❛❞♦ ❡♠ ❬✶✵❪✱ ❡♠ q✉❡ é ♣r♦♣♦st❛ ❛ s❡❣✉✐♥t❡ tr❛♥s❢♦r♠❛çã♦✿ ❝♦♥s✐❞❡r❡Y ✉♠❛ ✈❛r✐á✈❡❧
❛❧❡❛tór✐❛ ❞❡✜♥✐❞❛ ♣♦r
Y = α
2
X,
✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✷
❞❡Y é ♦❜t✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
F(t) =P(Y ≤t) = P(αX2 ≤t) = P(X ≥ αt2) = 1−P(X < αt2) = 1−P(X ≤ αt2) = 1−F(αt2)
= e−(α t)
β
,
♦♥❞❡ α, β > 0 ❡ t > 0✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ Y t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧
✐♥✈❡rs❛✱ ❝✉❥❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ é ❞❛❞❛ ♣♦r
F(t) = e−(αt)β
, α, β >0❡ t >0. ✭✶✳✶✶✮
❆ s❡❣✉✐r✱ ✐♥✐❝✐❛♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛❞❛ ♣♦r ✭✶✳✶✶✮✱ ♦❜t❡♥❞♦ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡✱ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❡ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ♦ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❡st❛ ❞✐str✐❜✉✐çã♦✳ ❋✐♥❛❧✐③❛♠♦s ❛ s❡çã♦ ❛♣r❡s❡♥t❛♥❞♦ ❞✉❛s té❝✲ ♥✐❝❛s ❣rá✜❝❛s✱ ❝❤❛♠❛❞❛s ❣rá✜❝♦✲❲PP ❡ ❣rá✜❝♦✲■❲PP✱ ❞❡s❝r❡✈❡♥❞♦ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❜❛s❡❛❞♦s ♥❡ss❡s ♠ét♦❞♦s✳
✶✳✸✳✶ ❋✉♥çã♦ ❉❡♥s✐❞❛❞❡
❉✐❢❡r❡♥❝✐❛♥❞♦F(t) ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛✿
f(t) =αββt−β−1e−(α t)
β
, α, β >0 ❡t >0. ✭✶✳✶✷✮
❱❛♠♦s ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦ ❞❛ ♠♦♥♦✲ t♦♥✐❝✐❞❛❞❡ ❞❡ ✉♠❛ ❢✉♥çã♦✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❡ s✉❛ ❞❡r✐✈❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ q✉❡ é ♦❜t✐❞❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
f′
(t) = αββt−β−1e−(α t)
β
[(−β−1)t−1+t−β−1αββ]
= αββt−β−1e−(αt)β
t−β−1[−(β+ 1)tβ+αββ]
= f(t)t−β−1[
−(β+ 1)tβ+αββ]. ✭✶✳✶✸✮
❈♦♠♦f(t)t−β−1 >0✱ s❡❣✉❡ q✉❡
f′
(t) = 0 ⇐⇒ tβ = α
ββ
β+ 1 ⇐⇒ t =α
β+ 1
β
−1
β
⇐⇒ t=α
1 + 1
β
−1
β
.
❉❡♥♦t❛♥❞♦ tm = α(1 + β1)− 1
β✱ t❡♠♦s q✉❡ f′(t) > 0 ♣❛r❛ t < t
m ❡✱ ❛ss✐♠✱ f(t) é ✉♠❛
❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡ q✉❛♥❞♦t > tm✱ t❡♠♦sf′(t)<0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡f(t)é ❞❡❝r❡s❝❡♥t❡✳
▲♦❣♦✱ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ é ✉♥✐♠♦❞❛❧✳
■st♦ ❞✐❢❡r❡ ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦✱ ❡♠ q✉❡ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ é ❞❡❝r❡s❝❡♥t❡ ✭♣❛r❛
✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✸
❉❡r✐✈❛♥❞♦ F(t) ❡♠ ✭✶✳✶✵✮✱ ♦❜t❡♠♦s ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧
❜ás✐❝♦✱
f(t) =βα−βtβ−1e−(t α)
β
, t >0.
❆ ❞❡r✐✈❛❞❛ ❞❡ f(t) é ❞❛❞❛ ♣♦r
f′
(t) = f(t)tβ−1[(β
−1)t−β
−βα−β].
❈♦♠♦ α, β, t, f(t) > 0✱ s❡❣✉❡ q✉❡ f′(t) < 0 s❡ β
≤ 1 ❡✱ ❛ss✐♠✱ f(t) é ❞❡❝r❡s❝❡♥t❡✳ ◆♦
❝❛s♦ ❡♠ q✉❡ β >1✱ t❡♠♦s q✉❡ f′(t)>0♣❛r❛ t < t
m ❡ f′(t)<0 ♣❛r❛ t > tm✱ ♦♥❞❡
tm =α
1− 1
β
1
β
,
♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ é ✉♥✐♠♦❞❛❧✳
✶✳✸✳✷ ❋✉♥çã♦ ❚❛①❛ ❞❡ ❋❛❧❤❛
❆ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ T ❝♦♠ ❞❡♥s✐❞❛❞❡ ❲❡✐❜✉❧❧
✐♥✈❡rs❛✭α, β✮ é ❞❛❞❛ ♣♦r
S(t) = 1−e−(αt)β
,
♦♥❞❡ t > 0 ❡ α, β > 0✳ ❆ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❝♦rr❡s♣♦♥❞❡♥t❡ é ❞❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡
❢♦r♠❛✿
h(t) = α
ββt−β−1e−(α t)
β
1−e−(α t)β
,
♦♥❞❡t >0 ❡ α, β >0✳
▲❡♠❛ ✶✳✻✳ ❆ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ s❛t✐s❢❛③ ♦s s❡❣✉✐♥t❡s ❧✐♠✐t❡s✿
(i) lim
t→0h(t) = 0 ❡ (ii) limt→∞h(t) = 0.
❉❡♠♦♥str❛çã♦✳
(i) ◆❛ ❡①♣r❡ssã♦ ♣❛r❛ h(t)✱ t❡♠♦s q✉❡ ♦ ❞❡♥♦♠✐♥❛❞♦r t❡♥❞❡ ❛ 1✱ q✉❛♥❞♦ t t❡♥❞❡
❛ 0✳ ❊♥tã♦ ❜❛st❛ ♠♦str❛r q✉❡ ♦ ♥✉♠❡r❛❞♦r t❡♥❞❡ ❛ 0✱ q✉❛♥❞♦ t t❡♥❞❡ ❛ 0✳ ❯s❛♥❞♦
▲✬❍♦s♣✐t❛❧✱ t❡♠♦s q✉❡✿
lim
t→0 βα
βt−β−1e−(α t)
β
= βαβlim t→0
1
tβ+1
e(αt)β
=βαβlim
t→0
−(β+ 1)t−β−2
−βαβt−β−1e(α t)β
= (β+ 1) lim
t→0
1
t e(α t)β
= 0,
♣♦✐s
lim
t→0
t e(αt) β
✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✹
P❛r❛ ♣r♦✈❛r ♦ ❧✐♠✐t❡ ❡♠ ✭✶✳✶✹✮✱ ♥♦t❡ ✭❢❛③❡♥❞♦h= t1β✮ q✉❡
lim
t→0t e
1
tβ = lim
h→∞
1
hβ1
eh = lim
h→∞
eh 1
β h 1
β−1
.
❙❡♥❞♦ ❛ss✐♠✱ ❞♦✐s ❝❛s♦s ♣♦❞❡♠ s❡r ❛♥❛❧✐s❛❞♦s s❡♣❛r❛❞❛♠❡♥t❡✳ ❙ã♦ ❡❧❡s✿
• β >1
❚❡♠♦s q✉❡ 1
β <1 ❡ ❛ss✐♠
lim
t→0t e
1
tβ = lim
h→∞βe hh1−1
β = +∞.
• β <1
◆❡st❡ ❝❛s♦✱ ❛♣❧✐❝❛♥❞♦ ▲✬❍♦s♣✐t❛❧ ✈ár✐❛s ✈❡③❡s ❡ s❡♥❞♦ 1
β >1✱ ♦❜t❡♠♦s✿
lim
t→0t e
1
tβ = lim
h→∞
eh
1
βh 1
β−1
= lim h→∞ eh 1 β 1
β −1
hβ1−2
=· · · = lim h→∞ eh 1 β 1
β −1
· · ·1
β − 1
β
−1h1β−
1
β
= lim
h→∞Ce hh1β
−β1
= +∞,
♦♥❞❡
C = 1
1
β
1
β −1
· · ·1
β − 1
β
−1 ❡ [x] := min{n ∈Z;n≥x}.
(ii) ❯s❛♥❞♦ ▲✬❍♦s♣✐t❛❧ ♥♦✈❛♠❡♥t❡ ❡ ❛s ❡①♣r❡ssõ❡s ✭✶✳✶✸✮ ❡ ✭✶✳✾✮✱ t❡♠♦s q✉❡✿
lim
t→∞h(t) = limt→∞
f(t)
S(t) = limt→∞
f(t)t−β−1[−(β+ 1)tβ +αββ]
−f(t)
= lim
t→∞
β+ 1
t −
αββ
tβ+1
= 0.
✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✺
h′(t) = f
′(t)S(t) + [f(t)]2
[S(t)]2 ♣♦✐s S
′
(t) = −f(t)
= f(t)t
−β−1[−(β+ 1)tβ+αββ]S(t) + [f(t)]2
[S(t)]2
= f(t)
S(t)
t−β−1[−(β+ 1)tβ+αββ]S(t) +f(t)
S(t)
= h(t)t−β−1 −(β+ 1)tβ+αββ+αββ e −(α
t) β
1−e−(α t)β
!
= h(t)t−β−1
−(β+ 1)tβ +α
ββ1−e−(α t)
β
+αββ e−(α t)
β
1−e−(α t)
β
= h(t)t−β−1
−(β+ 1)tβ+ αββ
1−e−(α t)β
.
❚❡♠♦s q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ t❡♠ ❢♦r♠❛ ✉♥✐♠♦❞❛❧ ❡ t❡♠ ♠á①✐♠♦ ❡♠ t = tM
❞❛❞♦ ♣❡❧❛ s♦❧✉çã♦ ❞❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿
(tα
M)
β
1−e(tMα )β
= 1 + 1
β.
■st♦ ❞✐❢❡r❡ ❞♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦✱ ❡♠ q✉❡ ❛ ❢✉♥çã♦ t❛①❛ ❞❡ ❢❛❧❤❛ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ✭♣❛r❛ β > 1✮✱ ❡str✐t❛♠❡♥t❡ ❞❡❝r❡s❝❡♥t❡ ✭♣❛r❛ β < 1✮ ♦✉ ❝♦♥st❛♥t❡ ✭♣❛r❛
β= 1✮✱ ❝♦♥❢♦r♠❡ ❊①❡♠♣❧♦ ✶✳✺ ✭❝♦♥s✐❞❡r❡λ= α1✮✳
✶✳✸✳✸ ▼♦♠❡♥t♦s
❙❡❥❛ Y ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❞❡ ♣❛râ♠❡tr♦sα ❡ β✱ ❞❛❞❛ ♣♦r ✭✶✳✶✶✮✳ ❖ ❦✲és✐♠♦ ♠♦♠❡♥t♦ ❞❡ Y é ❞❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
E(Yk) =
Z ∞
−∞
tkf(t)dt=
Z ∞
0
αββ t(k−β−1)e−(α t)
β
dt.
❋❛③❡♥❞♦u =αβt−β✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s✿ du=−αββ t−β−1dt; u→ ∞ q✉❛♥❞♦ t→0 ❡ u→0q✉❛♥❞♦ t → ∞; tk= (t−β)−kβ
= αuβ
−kβ
✳ ❉❡ss❡ ♠♦❞♦✱ ♦❜t❡♠♦s q✉❡
E(Yk) =
Z ∞ 0 u αβ −k β
e−udu= (α−β)−kβ Z ∞
0
u[(1−kβ)−1]e−udu
= αkΓ
1− k
β
. ✭✶✳✶✺✮
✶✳✸✳ ❆ ❉✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛ ✶✻
✶✳✸✳✹ ●rá✜❝♦s ▼♦❞✐✜❝❛❞♦s
●rá✜❝♦s ♠♦❞✐✜❝❛❞♦s tê♠ s✐❞♦ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❛❧t❡r♥❛t✐✈♦ ♥❛ r❡♣r❡s❡♥✲ t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ❞❛❞♦s✱ s❡❣✉♥❞♦ ▼✉rt❤② ❡t ❛❧✳ ❬✶✹❪✳ ❆ ✐❞❡✐❛ é tr❛♥s❢♦r♠❛r ♦s ❞❛❞♦s ♥♦ ✐♥t✉✐t♦ ❞❡ ❧✐♥❡❛r✐③❛r ❛ ❢✉♥çã♦ ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ ✭♦✉ ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦✮ ❞♦ ♠♦❞❡❧♦ ♣r♦♣♦st♦✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❛♠♦s ❞✉❛s ♠♦❞✐✜❝❛çõ❡s ❞❡ ◆❡❧s♦♥ ❬✶✺❪ ❡ ❉r❛♣❡❧❧❛ ❬✻❪✱ ❝❤❛♠❛❞❛s ❲❡✐❜✉❧❧ Pr♦❜❛❜✐❧✐t② P❛♣❡r ✭❲PP✮ ❡ ■♥✈❡rs❡ ❲❡✐❜✉❧❧ Pr♦❜❛❜✐❧✐t② P❛♣❡r ✭■❲PP✮✱ ❛s q✉❛✐s✱ ♥❡st❡ ❡st✉❞♦✱ ♥♦s r❡❢❡r✐♠♦s ❝♦♠♦ ❣rá✜❝♦✲❲PP ❡ ❣rá✜❝♦✲■❲PP✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ ♦❜❥❡t✐✈♦ ❛q✉✐ é ❞✐s❝✉t✐r t❛✐s té❝♥✐❝❛s ❣rá✜❝❛s ❡ ❛♣r❡s❡♥t❛r ✉♠ ♣r♦✲ ❝❡❞✐♠❡♥t♦ ♣❛r❛ ❡st✐♠❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❢♦r♠❛ ❡ ❡s❝❛❧❛ ❞❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ✭♦✉ ❲❡✐❜✉❧❧✮ ❜❛s❡❛❞♦ ♥❡ss❡s ♠ét♦❞♦s ❣rá✜❝♦s✳
✶✳✸✳✹✳✶ ❖ ●rá✜❝♦✲❲PP
●rá✜❝♦s✲❲PP ♣♦❞❡♠ s❡r ❝♦♥str✉í❞♦s ❞❡ ✈ár✐❛s ♠❛♥❡✐r❛s✳ ❯♠❛ ❞❡❧❛s é ❛ ❛♣r❡s❡♥✲ t❛❞❛ ♣♦r ◆❡❧s♦♥ ❬✶✺❪✱ q✉❡ ❜❛s❡✐❛✲s❡ ♥❛ ♠♦❞✐✜❝❛çã♦ ❞❛❞❛ ♣♦r
y= ln[−ln(1−F(t))] ❡ x= lnt. ✭✶✳✶✻✮
❈♦♥s✐❞❡r❛♥❞♦ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦ ✭✶✳✶✵✮✱ ♦❜t❡♠♦s✿
y= ln[−ln(1−F(t))] = ln[−lne−(αt)β
] = ln
t α
β
=βlnt−βlnα.
❆ss✐♠✱ s❡♥❞♦ x= lnt✱ ♦ ♠♦❞❡❧♦ ❲❡✐❜✉❧❧ ❜ás✐❝♦ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ r❡❧❛çã♦ ❧✐♥❡❛r
y =βx−βlnα.
❖ ❣rá✜❝♦ ❞❡ y ♣♦r x é ❝❤❛♠❛❞♦ ❞❡ ❣rá✜❝♦✲❲PP✳ ◆♦t❡ q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠é✲
tr✐❝❛ ❞♦ ❣rá✜❝♦✲❲PP é ✉♠❛ r❡t❛✱ ❝♦♠ ✐♥❝❧✐♥❛çã♦ ❞❛❞❛ ♣❡❧♦ ♣❛râ♠❡tr♦ ❞❡ ❢♦r♠❛ β ❡
q✉❡ ✐♥t❡r❝❡♣t❛ ♦ ❡✐①♦ y ❡♠ −βlnα✳
❆❣♦r❛ s❡ ❛♣❧✐❝❛r♠♦s ❛ ♠♦❞✐✜❝❛çã♦ ✭✶✳✶✻✮ à ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ✐♥✈❡rs❛ ❞❛❞❛ ♣♦r ✭✶✳✶✶✮✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿
y = ln[−ln(1−e−z)], ✭✶✳✶✼✮
♦♥❞❡
z =α
t
β
=e[βlnα−βx].
◆♦t❡ q✉❡y é ✉♠❛ ❢✉♥çã♦ ♥ã♦✲❧✐♥❡❛r ❞❡x✳ ❖s t❡♦r❡♠❛s ❛ s❡❣✉✐r ❞ã♦ ❞♦✐s r❡s✉❧t❛❞♦s
❞♦ ❣rá✜❝♦✲❲PP ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❲❡✐❜✉❧❧ ■♥✈❡rs❛✳