Universidade do Estado do Rio de Janeiro (UERJ) Instituto de Física Armando Dias Tavares (IFADT) Departamento de Física Nuclear e Altas Energias (DFNAE) Rio de Janeiro, 23 de setembro de 2011

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✐ •Ú❧t✐♠❛ •❙❛✐r

❋ís✐❝❛ ❊st❛tíst✐❝❛

❱✐t♦r ❖❣✉r✐

❯♥✐✈❡rs✐❞❛❞❡ ❞♦ ❊st❛❞♦ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✭❯❊❘❏✮

■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❆r♠❛♥❞♦ ❉✐❛s ❚❛✈❛r❡s ✭■❋❆❉❚✮

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ◆✉❝❧❡❛r ❡ ❆❧t❛s ❊♥❡r❣✐❛s ✭❉❋◆❆❊✮

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✐✐ •Ú❧t✐♠❛ •❙❛✐r

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ▼✐❝r♦❡st❛❞♦s✱ ♠❛❝r♦❡st❛❞♦s ❡ ❧❡✐ ③❡r♦ ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷

✶✳✷ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❡♥❡r❣✐❛ ♥❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✷ ❊❧❡♠❡♥t♦s ❞❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✾

✷✳✶ ●r❛♥❞❡③❛s ❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡s ❛ ♣r✐♦r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✷✳✸ Pr♦❜❛❜✐❧✐❞❛❞❡s ❛ ♣♦st❡r✐♦r✐ ❡ ❞✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✸ ❊❧❡♠❡♥t♦s ❞❡ ❚❡r♠♦❞✐♥â♠✐❝❛ ✷✹

✸✳✶ ❱❛r✐á✈❡✐s ❡ ❡q✉❛çõ❡s ❞❡ ❡st❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸✳✶✳✶ ❆ ❡q✉❛çã♦ ❞❡ ❈❧❛♣❡②r♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✸✳✶✳✷ ❆ ❧❡✐ ❞❡ ❈✉r✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✶✳✸ Pr♦♣r✐❡❞❛❞❡s ❡①t❡♥s✐✈❛s ❡ ✐♥t❡♥s✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸✳✷ ❆ ✶✲❛ _{❡ ❛ ✷✲}❛ _{❧❡✐ ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛ ♣❛r❛ s✐st❡♠❛s ❢❡❝❤❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾} ✸✳✸ ❊♥tr♦♣✐❛ ❡ ✐rr❡✈❡rs✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✸✳✸✳✶ ❊♥tr♦♣✐❛ ❡♠ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸✳✸✳✷ ❊♥tr♦♣✐❛ ❡♠ ✉♠❛ ❡①♣❛♥sã♦ ❧✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸✳✹ ❈❛❧♦r❡s ❡s♣❡❝í✜❝♦s ❞♦s só❧✐❞♦s ❡ ❞♦s ❣❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✸✳✺ P♦t❡♥❝✐❛✐s q✉í♠✐❝♦s ❡ t❡r♠♦❞✐♥â♠✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✐✐✐ •Ú❧t✐♠❛ •❙❛✐r

✸✳✼ ❊♥tr♦♣✐❛ ❡ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❞❡ ✉♠ ❣ás ✐❞❡❛❧ ♠♦❧❡❝✉❧❛r ♠♦♥♦❛tô♠✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✸✳✽ ❊♥tr♦♣✐❛ ❞❛ r❛❞✐❛çã♦ ❞❡ ❝♦r♣♦ ♥❡❣r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

✹ ▲✐♠✐t❡s ❞♦s ❣❛s❡s ✐❞❡❛✐s ❞❡❣❡♥❡r❛❞♦s ✼✸

✹✳✶ ❉❡♥s✐❞❛❞❡ ❞❡ ❡st❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

✹✳✷ ●❛s❡s ♥ã♦✲❞❡❣❡♥❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✹✳✸ ●❛s❡s ❞❡❣❡♥❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

✹✳✸✳✶ ❋ér♠✐♦♥s ❞❡❣❡♥❡r❛❞♦s ♥ã♦✲r❡❧❛t✐✈íst✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

✹✳✸✳✷ ❇ós♦♥s ❞❡❣❡♥❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾

✺ ❖ ♠ét♦❞♦ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❡ ❇♦❧t③♠❛♥♥✲P❧❛♥❝❦ ✾✶

✺✳✶ ❖ ❝♦♥❝❡✐t♦ ❡st❛tíst✐❝♦ ❞❡ ❡♥tr♦♣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

✺✳✷ ❊♥tr♦♣✐❛ ❡ ✷✲❛ _{❧❡✐ ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺} ✺✳✸ P♦♣✉❧❛çõ❡s ❞♦s ❣❛s❡s ✐❞❡❛✐s ♥ã♦✲❞❡❣❡♥❡r❛❞♦s ❡ ❞❡❣❡♥❡r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻

✺✳✹ ❆ ❞✐str✐❜✉✐çã♦ ❝❛♥ô♥✐❝❛ ❞❡ ●✐❜❜s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹

✻ ❖ ♠ét♦❞♦ ❞♦s ❡♥s❡♠❜❧❡s ❞❡ ❇♦❧t③♠❛♥♥✲●✐❜❜s ✶✵✽

✻✳✶ ❆ ❞✐str✐❜✉✐çã♦ ♠✐❝r♦❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺

✻✳✶✳✶ ❖ ❣ás ✐❞❡❛❧ ❝❧áss✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✻

✻✳✶✳✷ ❖ ❣ás ❞❡ s♣✐♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

✻✳✷ ❆♣❧✐❝❛çõ❡s ❞❛ ❞✐str✐❜✉✐çã♦ ❝❛♥ô♥✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

✻✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✐✈ •Ú❧t✐♠❛ •❙❛✐r

✻✳✷✳✸ ❆ ❛♥❛❧♦❣✐❛ ❚❡r♠♦❞✐♥â♠✐❝❛ ❞❡ ●✐❜❜s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✽

✻✳✷✳✹ ❋✉♥çõ❡s t❡r♠♦❞✐♥â♠✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✶

✻✳✷✳✺ ●❛s❡s ✐❞❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✻

✻✳✷✳✻ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ P❧❛♥❝❦ ✭✶✾✵✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✶

✻✳✷✳✼ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ▼❛①✇❡❧❧✲❇♦❧t③♠❛♥♥ ✭✶✽✻✵✲✶✽✼✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✸

✻✳✷✳✽ ❋❧✉t✉❛çõ❡s ❡ ❝♦♥❡①ã♦ ❡♥tr❡ ❛s ❞✐str✐❜✉✐çõ❡s ❝❛♥ô♥✐❝❛ ❡ ♠✐❝r♦❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✺

✻✳✸ ❆ ❞✐str✐❜✉✐çã♦ ❣r❛♥✲❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✵

✻✳✸✳✶ ❆s ❞✐str✐❜✉✐çõ❡s ❞❡ ❋❡r♠✐✲❉✐r❛❝ ❡ ❇♦s❡✲❊✐♥st❡✐♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻

✻✳✸✳✷ ❖ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❞❡ ❣❛s❡s ❞❡❣❡♥❡r❛❞♦s ❞❡ ❢ér♠✐♦♥s ♥ã♦✲r❡❧❛t✐✈íst✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✷

✼ ❖✉tr♦s ♠ét♦❞♦s ✶✻✽

✼✳✶ ❊♥tr♦♣✐❛✱ ❞❡s♦r❞❡♠ ❡ ✐♥❢♦r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✽

✼✳✷ ❖ ♠ét♦❞♦ ❞❡ ❉❛r✇✐♥✲❋♦✇❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✷

✼✳✸ ❆ ❢♦r♠✉❧❛çã♦ ❞❡ ❱♦♥ ◆❡✉♠❛♥♥✲▲❛♥❞❛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✻

✽ ❙✐st❡♠❛s ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ✶✽✹

✽✳✶ ❖ ❝♦♥❝❡✐t♦ ❞❡ q✉❛s❡✲♣❛rtí❝✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✻

✽✳✷ ❖s só❧✐❞♦s ❝r✐st❛❧✐♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✵

✽✳✷✳✶ ❖ ♣r♦❜❧❡♠❛ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✵

✽✳✷✳✷ ❖ át♦♠♦ ❝♦♠♦ ✉♠ ♦s❝✐❧❛❞♦r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✶

✽✳✷✳✸ ❖ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞♦s ♠❡t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✵

✽✳✸ ❆ r❛❞✐❛çã♦ ❞❡ ❝♦r♣♦ ♥❡❣r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✷

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✈ •Ú❧t✐♠❛ •❙❛✐r

✽✳✸✳✷ ❆ q✉❛♥t✐③❛çã♦ ❞❛ r❛❞✐❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✶

✽✳✹ ❖ ❣ás ✐❞❡❛❧ ♠♦❧❡❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✼

❆ ❖ t❡♦r❡♠❛ ❞❡ ❈❛r♥♦t ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❧❛✉s✐✉s ✷✷✶

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶ •Ú❧t✐♠❛ •❙❛✐r

✶✳ ■♥tr♦❞✉çã♦

❚❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❝❧áss✐❝♦ ❝♦♠♦ q✉â♥t✐❝♦✱ ❛ ❞❡s❝r✐çã♦ ♣✉r❛♠❡♥t❡ ♠❡❝â♥✐❝❛ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ❝♦r♣♦s ♠❛❝r♦s❝ó♣✐❝♦s é ♣r❛t✐❝❛♠❡♥t❡ ✐♠♣♦ssí✈❡❧✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠✐❝r♦s❝ó♣✐❝♦✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s s✐st❡♠❛s ♠❛❝r♦s❝ó♣✐❝♦s r❡s✉❧t❛ ❞♦ ❡st❛❞♦ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❡ s✉❛s ♣❛rtí❝✉❧❛s ❝♦♥st✐t✉✐♥t❡s✱ ❝♦♠♦ ❞❛s ♠♦❧é❝✉❧❛s ❞❡ ✉♠ ❣ás ♦✉ ❞❡ ✉♠ ❧íq✉✐❞♦✱ ❞♦s át♦♠♦s ❞❡ ✉♠ só❧✐❞♦✱ ♦✉ ❞♦s ❡❧étr♦♥s ❞❡ ✉♠ ♠❡t❛❧ ♦✉ ❞❡ ✉♠ s❡♠✐❝♦♥❞✉t♦r✳

❍✐st♦r✐❝❛♠❡♥t❡✱ ❜❛s❡❛♥❞♦✲s❡ ❡♠ r❡❧❛çõ❡s ❡ ❧❡✐s ❡♠♣ír✐❝❛s✱ ❛s t❡♥t❛t✐✈❛s ❞❡ ✉♠❛ ❞❡s❝r✐çã♦ s✐st❡♠át✐❝❛ ❡ q✉❛♥✲ t✐t❛t✐✈❛ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ s✐st❡♠❛s ♠❛❝r♦s❝ó♣✐❝♦s✱ ❡♠ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ♠❡✐♦ ❝✐r❝✉♥❞❛♥t❡ ✭✈✐③✐♥❤❛♥ç❛✮ ♦✉ ✉♠ ❝❛♠♣♦ ❡①t❡r♥♦✱ ❝✉❧♠✐♥❛r❛♠ ❝♦♠ ❛ t❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛ ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛ ❬✻✱ ✶✹✱✶✺✱✷✺✱ ✸✶✱✸✸✱✸✼✱

✹✶✱✹✼❪✳ ❆ ♣❛rt✐r✱ ❡♥tã♦✱ ❞❛ ❞❡✜♥✐çã♦ ❡ ❞❛ ♠❡❞✐çã♦ ❞❡ ♣♦✉❝❛s ❣r❛♥❞❡③❛s ❡ ♣❛râ♠❡tr♦s✱ ❛s ❝❤❛♠❛❞❛s ✈❛r✐á✈❡✐s ❞❡ ❡st❛❞♦✱ ❝❤❡❣♦✉✲s❡ à ❢♦r♠✉❧❛çã♦ ❞❡ ❧❡✐s ❣❡r❛✐s ✭❧❡✐s ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛✮ q✉❡ r❡❣❡♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡ ❛ ❡✈♦❧✉çã♦ ❞♦s s✐st❡♠❛s ♠❛❝r♦s❝ó♣✐❝♦s✳ ❊♥tr❡t❛♥t♦✱ ❝♦♠♦ q✉❛❧q✉❡r t❡♦r✐❛ ❢❡♥♦♠❡♥♦❧ó❣✐❝❛✱ ❛ ❚❡r♠♦❞✐♥â♠✐❝❛ ♥ã♦ ♣♦ss✐❜✐❧✐t❛ ❛ ♣r❡✈✐sã♦ ❞♦s ✈❛❧♦r❡s ❞❡ ♣❛râ♠❡tr♦s ♦✉ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠ s✐st❡♠❛✱ t❛✐s ❝♦♠♦ ♦ ❝❛❧♦r ❡s♣❡✲ ❝í✜❝♦ ❡ ❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡✱ ♥❡♠ ❡st❛❜❡❧❡❝❡ ❛s r❡❧❛çõ❡s ❞❡ ✈í♥❝✉❧♦s ✭❡q✉❛çõ❡s ❞❡ ❡st❛❞♦✮ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❞❡ ❡st❛❞♦✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✷ •Ú❧t✐♠❛ •❙❛✐r

♣❛rtí❝✉❧❛s✱ ❝♦♥❞✐❝✐♦♥❛❞❛s às ❧❡✐s ❞❛ ▼❡❝â♥✐❝❛✱ ❞❡♥♦♠✐♥❛❞❛ ❚❡♦r✐❛ ❈✐♥ét✐❝❛ ❞♦s ●❛s❡s✳ ❆ ❚❡♦r✐❛ ❈✐♥ét✐❝❛ ❞♦s ●❛s❡s✱ ❝♦♥str✉í❞❛ s♦❜r❡ ❤✐♣ót❡s❡s ❣❡r❛✐s ❛❝❡r❝❛ ❞❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s ❝♦♥st✐t✉✐♥t❡s ❞❡ ✉♠ s✐st❡♠❛ ❞✉r❛♥t❡ s✉❛s ❝♦❧✐sõ❡s✱ ♣❡r♠✐t❡ ❡st❛❜❡❧❡❝❡r ❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦ ❞❡ ✉♠ ❣ás ♠♦❧❡❝✉❧❛r à ❜❛✐①❛ ♣r❡ssã♦ ❡ ♦ ✈❛❧♦r ❞❡ s❡✉ ❝❛❧♦r ❡s♣❡❝í✜❝♦✳

◗✉❛s❡ q✉❡ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛♦ s✉r❣✐♠❡♥t♦ ❞❛ ❚❡♦r✐❛ ❈✐♥ét✐❝❛✱ ♦ r❡✜♥❛♠❡♥t♦ ❞❡ss❡ ♣r♦❝❡❞✐♠❡♥t♦✱ ❛ ♣❛rt✐r t❛♠❜é♠ ❞❡ ❛r❣✉♠❡♥t♦s ❡st❛tíst✐❝♦✲♣r♦❜❛❜✐❧íst✐❝♦s✱ ♠❛s s❡♠ ❤✐♣ót❡s❡s ♦✉ ♠♦❞❡❧♦s t❡ór✐❝♦s s♦❜r❡ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s ❝♦♥st✐t✉✐♥t❡s ❞❡ ✉♠ s✐st❡♠❛✱ ❣r❛ç❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛♦s tr❛❜❛❧❤♦s ❞❡ ❇♦❧t③♠❛♥♥ ❡ ●✐❜❜s✱ ❡st❛❜❡❧❡❝❡✉✲s❡ ✉♠❛ ♥♦✈❛ t❡♦r✐❛ ✐♥t❡r♣r❡t❛t✐✈❛✱ ♥ã♦ ❛♣❡♥❛s ♣❛r❛ ♦s ❣❛s❡s✱ ♠❛s t❛♠❜é♠ ❛♣♦✐❛❞❛ ♥❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛✱ ❞❡♥♦♠✐♥❛❞❛ ♣♦r ▼❛①✇❡❧❧ ❞❡ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛ ❬✹✱✶✶✱✶✷✱✶✸✱✶✹✱✶✺✱✶✼✱✶✾✱✷✸✱✷✺✱✷✾✱✸✺✱✸✻✱

✸✼✱✹✶✱✹✸✱✹✼❪✱ q✉❡ ❛❧é♠ ❞❡ ♣❡r♠✐t✐r ♦ ❝á❧❝✉❧♦ ❞❡ ✈ár✐♦s ♣❛râ♠❡tr♦s ❞❡ ✉♠ s✐st❡♠❛✱ ♣♦ss✐❜✐❧✐t❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ❡st❛❞♦s ❡ ❛s ♣ró♣r✐❛s ❧❡✐s ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛ ❡♠ ♣r♦❝❡ss♦s ♥♦s q✉❛✐s ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ♦❝♦rr❡ ♣♦r ✉♠❛ s✉❝❡ssã♦ r❡✈❡rsí✈❡❧ ❞❡ ❡st❛❞♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ❞✐t♦s ♣r♦❝❡ss♦s r❡✈❡rsí✈❡✐s✳

✶✳✶✳ ▼✐❝r♦❡st❛❞♦s✱ ♠❛❝r♦❡st❛❞♦s ❡ ❧❡✐ ③❡r♦ ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛

❈♦r♣♦s ♠❛❝r♦s❝ó♣✐❝♦s sã♦ s✐st❡♠❛s ❝♦♠ ✉♠ ♥ú♠❡r♦ ✏♠✉✐t♦ ❣r❛♥❞❡✑ ✶ _{❞❡ ♣❛rtí❝✉❧❛s ❝♦♥st✐t✉✐♥t❡s ❝✉❥❛s}

✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s q✉❡ ♦s ❞❡s❝r❡✈❡♠ ❝❧❛ss✐❝❛♠❡♥t❡✱ ❝♦♠♦ s✉❛s ♣♦s✐çõ❡s ❡ ♠♦♠❡♥t❛✱ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ♣♦r ✉♠ ♣❡q✉❡♥♦ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ✈í♥❝✉❧♦s✳ ❆ss✐♠✱ ❡♠ ❣❡r❛❧✱ é ♥❡❝❡ssár✐♦ t❛♠❜é♠ ✉♠ ♥ú♠❡r♦ ♠✉✐t♦ ❣r❛♥❞❡ ❞❡ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ♠✐❝r♦s❝ó♣✐❝❛ ❞❡ ✉♠ ❝♦r♣♦✳ ❊ss❡ ♥ú♠❡r♦✱ ❞❡♥♦t❛❞♦ ♣♦rη

✶ _{❚✐♣✐❝❛♠❡♥t❡✱ ♠❛✐♦r q✉❡10}19

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✸ •Ú❧t✐♠❛ •❙❛✐r

✱ é ❝❤❛♠❛❞♦ ❞❡ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞♦ s✐st❡♠❛✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❝♦r♣♦s ♠❛❝r♦s❝ó♣✐❝♦s sã♦ ❞✐t♦s s✐st❡♠❛s ❝♦♠ ♠✉✐t♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ ✷ _{❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s q✉❡✱ ❡♠ ✉♠ ❞❛❞♦ ✐♥st❛♥t❡✱}

❝❛r❛❝t❡r✐③❛♠ ✉♠ s✐st❡♠❛ é ❞❡♥♦♠✐♥❛❞♦ ✉♠ ♠✐❝r♦❡st❛❞♦✳

P❛r❛ ❛ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛✱ ♦ ❡st❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ❡♠ ✐♥t❡r❛çã♦ ❝♦♠ ♦ ♠❡✐♦ ❡①t❡r♥♦ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ♣❛râ♠❡tr♦s ♠❛❝r♦s❝ó♣✐❝♦s ❝♦♠♦ ❛ ♣r❡ssã♦ ❡ ❛ t❡♠♣❡r❛t✉r❛ é ❞❡♥♦♠✐♥❛❞♦ ♠❛❝r♦❡st❛❞♦✳ ❖s ♠❛❝r♦❡st❛✲ ❞♦s ❞❡ ✉♠ s✐st❡♠❛ ❝❛r❛❝t❡r✐③❛♠✲s❡ ♣♦r ♣❛râ♠❡tr♦s q✉❡✱ ❛ss♦❝✐❛❞♦s ❛ ✈❛❧♦r❡s ♠é❞✐♦s ❞❡ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s✱ ❛♣r❡s❡♥t❛♠ ♣❡q✉❡♥íss✐♠❛s ✢✉t✉❛çõ❡s r❡❧❛t✐✈❛s ❡✱ sã♦ ❞✐t♦s ❡♠ ❚❡r♠♦❞✐♥â♠✐❝❛✱ ❡st❛❞♦s ❞❡ ❡q✉✐❧í❜r✐♦ t❡r✲ ♠♦❞✐♥â♠✐❝♦✳ ❆ss✐♠✱ ♣r♦❝❡ss♦s t❡r♠♦❞✐♥❛♠✐❝❛♠❡♥t❡ r❡✈❡rsí✈❡✐s s❡r✐❛♠ ❛q✉❡❧❡s ♥♦s q✉❛✐s ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❛r✲s❡✲✐❛ ♣♦r ✉♠❛ s✉❝❡ssã♦ r❡✈❡rsí✈❡❧ ❞❡ ♠❛❝r♦❡st❛❞♦s✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ✈❡r✐✜❝❛✲s❡ q✉❡ ♥ã♦ ❜❛st❛♠ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❡q✉✐❧í❜r✐♦ ✉s✉❛✐s ❞❛s ❣r❛♥❞❡③❛s ♠❡❝â♥✐❝❛s✱ ❡❧étr✐❝❛s ❡ ♠❛❣♥ét✐❝❛s✱ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ♦ ❡q✉✐❧í❜r✐♦ t❡r♠♦❞✐♥â♠✐❝♦✳ ❖ ❡q✉✐❧í❜r✐♦ t❡r♠♦❞✐♥â♠✐❝♦ ♣r❡ss✉♣õ❡ ❛✐♥❞❛ ✉♠ ♦✉tr♦ t✐♣♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s✐st❡♠❛ ❝♦♠ ❛ s✉❛ ✈✐③✐♥❤❛♥ç❛✱ ♦ ❝❤❛♠❛❞♦ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦✱ q✉❡ é ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ♠❛❝r♦s❝ó♣✐❝❛ ❡♠❡r❣❡♥t❡✱ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♥ã♦✲♠❡❝â♥✐❝❛✱ ❛ t❡♠♣❡r❛t✉r❛✱ ❆ ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠ ❛ s✉❛ ✈✐③✐♥❤❛♥ç❛ ❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♥♦✈❛ ❣r❛♥❞❡③❛✱ ❛ t❡♠♣❡r❛t✉r❛✱ ♣❛r❛ ❝❛r❛❝t❡r✐③á✲❧♦✱ ❢♦✐ ❞❡♥♦♠✐♥❛❞♦ ♣♦r ❋♦✇❧❡r ❞❡ ❧❡✐ ③❡r♦ ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛✳

❊♠ ♣r✐♥❝í♣✐♦✱ ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ♣♦r ✉♠❛ s✉❝❡ssã♦ r❡✈❡rsí✈❡❧ ❞❡ ❡st❛❞♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦✉ ✉♠ ♣r♦❝❡ss♦ r❡✈❡rsí✈❡❧✱ ❞❡✈❡r✐❛ ♦❝♦rr❡r ♦ ♠❛✐s ❧❡♥t❛♠❡♥t❡ ♣♦ssí✈❡❧✱ ♣❛r❛ q✉❡ ❛s ✢✉t✉❛çõ❡s r❡❧❛t✐✈❛s ❞❛s ✈❛r✐á✈❡✐s ❞❡ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ♣❡r♠❛♥❡❝❡ss❡♠ ♣r❛t✐❝❛♠❡♥t❡ ❞❡s♣r❡③í✈❡✐s✳ ◆♦ ❡♥t❛♥t♦✱ ❞✉r❛♥t❡ q✉❛❧q✉❡r ❛❧t❡r❛çã♦ ❞❛s ✷_{▼❡s♠♦ ♣❛r❛ s✐st❡♠❛s ❝♦♠ ♣♦✉❝♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✱ s❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠ ♠❡✐♦ ❝✐r❝✉♥❞❛♥t❡ ♥ã♦ é ♣❡r❢❡✐t❛♠❡♥t❡}

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✹ •Ú❧t✐♠❛ •❙❛✐r

❝♦♥❞✐çõ❡s ❡①t❡r♥❛s ❞❡ ✉♠ s✐st❡♠❛✱ ❛s ♣❛rtí❝✉❧❛s ❝♦♥st✐t✉✐♥t❡s ❞✐str✐❜✉❡♠✲s❡ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ♦ ❝❤❛♠❛❞♦ t❡♠♣♦ ❞❡ r❡❧❛①❛çã♦✱ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ❛s ♠é❞✐❛s ❞❛s ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s✱ q✉❡ ❝♦♥st✐t✉❡♠ ♦s ♠✐❝r♦❡st❛❞♦s✱ ❝♦rr❡s♣♦♥❞❛♠ ❛ ✉♠ ♥♦✈♦ ♠❛❝r♦❡st❛❞♦✱ é ♠✉✐t♦ ♣❡q✉❡♥♦✱ q✉❛♥❞♦ ❝♦♠♣❛r❛❞♦ ❛♦s ✐♥t❡r✈❛❧♦s ❞❡ t❡♠♣♦ ❡①♣❡r✐✲ ♠❡♥t❛✐s✱ ♠❡❞✐❞♦s ❡♠ ❡s❝❛❧❛ ♠❛❝r♦s❝ó♣✐❝❛✳ ❆ss✐♠✱ t♦❞♦ ♣r♦❝❡ss♦ ✐♥✜♥✐t❡s✐♠❛❧✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛❝r♦s❝ó♣✐❝♦✱ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠❛ s✉❝❡ssã♦ ❞❡ ❡st❛❞♦s ❞❡ ❡q✉✐❧í❜r✐♦✳

❆♣❡s❛r ❞❡ ♥ã♦ s❡r ❞✐❢í❝✐❧ s❛t✐s❢❛③❡r ❛ ❝♦♥❞✐çã♦ ❞❡ r❡❛❧✐③❛çã♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ♠❛❝r♦s❝♦♣✐❝❛♠❡♥t❡ ❧❡♥t♦✱ ❝❤❛♠❛❞♦ t❛♠❜é♠ ❞❡ ♣r♦❝❡ss♦ q✉❛s❡✲❡stát✐❝♦✱ ❡st❛ é ❛♣❡♥❛s ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ r❡✈❡rs✐❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣r♦❝❡ss♦✳ ❯♠ ♣r♦❝❡ss♦ q✉❛s❡✲❡stát✐❝♦ ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ r❡✈❡rsí✈❡❧✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ✉♠ ❝✐r❝✉✐t♦ RC✱ ❛ ❞❡s❝❛r❣❛ ❞♦ ❝❛♣❛❝✐t♦r ✭C✮✱ ✐♥✐❝✐❛❧♠❡♥t❡ ❝❛rr❡❣❛❞♦✱ s♦❜r❡ ♦ r❡s✐st♦r ✭R✮ ❛ ✉♠❛ ❞❛❞❛ t❡♠♣❡r❛t✉r❛ ✐♥✐❝✐❛❧✱ ♣♦❞❡ s❡r tã♦ ❧♦♥❣❛ q✉❛♥t♦ s❡ q✉❡✐r❛✱ ❛✉♠❡♥t❛♥❞♦✲s❡ ♦ ✈❛❧♦r ❞❛ r❡s✐stê♥❝✐❛✳ ❈♦♠♦✱ ❡♠ ❝❛❞❛ ✐♥st❛♥t❡✱ ❛ ❝❛r❣❛ ❛r♠❛③❡♥❛❞❛ ♥♦ ❝❛♣❛❝✐t♦r✱ ❛ t❡♥sã♦ ❡♠ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❡ ❛ t❡♠♣❡r❛t✉r❛ ❞♦ r❡s✐st♦r tê♠ ✈❛❧♦r❡s ❞❡ ❡q✉✐❧í❜r✐♦ ♠❛❝r♦s❝♦♣✐❝❛♠❡♥t❡ ❞❡✜♥✐❞♦s✱ ♦ ♣r♦❝❡ss♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ q✉❛s❡✲❡stát✐❝♦✳ ◆♦ ❡♥t❛♥t♦✱ ❞✉r❛♥t❡ ♦ ♣r♦❝❡ss♦✱ ❛ ❡♥❡r❣✐❛ ✐♥✐❝✐❛❧♠❡♥t❡ ❛r♠❛③❡♥❛❞❛ ♥♦ ❝❛♣❛❝✐t♦r ✈❛✐ s❡♥❞♦ ❝❡❞✐❞❛ ❛♦ r❡s✐st♦r✱ ❝♦♠♦ ❝❛❧♦r✱ ❛✉♠❡♥t❛♥❞♦ ❛ s✉❛ t❡♠♣❡r❛t✉r❛ ✭❡❢❡✐t♦ ❏♦✉❧❡✮✳ ❆♣ós ❛ ❞❡s❝❛r❣❛ t♦t❛❧ ❞♦ ❝❛♣❛❝✐t♦r✱ q✉❛♥❞♦ ❛ ❝❛r❣❛ ❡ ❛ t❡♥sã♦ ❝❛❡♠ ❛ ③❡r♦✱ s❡ ♦ r❡s✐st♦r ❢♦r r❡s❢r✐❛❞♦ ❞❡ ♠♦❞♦ q✉❡ ❛ s✉❛ t❡♠♣❡r❛t✉r❛ ✈♦❧t❡ ❛♦ ✈❛❧♦r ✐♥✐❝✐❛❧✱ ♦ ❝❛♣❛❝✐t♦r ♣❡r♠❛♥❡❝❡rá ❞❡s❝❛rr❡❣❛❞♦✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ s❡ ♦ ❝❛♣❛❝✐t♦r ❢♦r ❧❡♥t❛♠❡♥t❡ ❝❛rr❡❣❛❞♦✱ r❡❝♦♥❞✉③✐❞♦ ❛♦ s❡✉ ❡st❛❞♦ ✐♥✐❝✐❛❧✱ ❛ t❡♠♣❡r❛t✉r❛ ❞♦ r❡s✐st♦r ♥ã♦ r❡t♦r♥❛rá ❛♦ s❡✉ ✈❛❧♦r ✐♥✐❝✐❛❧✳ ❉❡ss❡ ♠♦❞♦✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❞❡s❝❛r❣❛✱ ❛♣❡s❛r ❞❡ s❡r r❡❛❧✐③❛❞♦ ♣♦r ✉♠❛ s✉❝❡ssã♦ q✉❛s❡✲❡stát✐❝❛ ❞❡ ❡st❛❞♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ ♥ã♦ é r❡✈❡rsí✈❡❧✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✺ •Ú❧t✐♠❛ •❙❛✐r

❛♦s ✈❛❧♦r❡s ✐♥✐❝✐❛✐s✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ❧❡✐s ❞❛ ❚❡r♠♦❞✐♥â♠✐❝❛✱ t♦❞♦s ♦s ♣r♦❝❡ss♦s q✉❡ ♦❝♦rr❡♠ ❡s♣♦♥t❛♥❡❛♠❡♥t❡ ♥❛ ♥❛t✉✲ r❡③❛ ❡♥✈♦❧✈❡♠ ❡❢❡✐t♦s ❞✐ss✐♣❛t✐✈♦s✱ ❝♦♠♦ ❛ tr♦❝❛ ❞❡ ❝❛❧♦r✱ ♦ ❛tr✐t♦✱ ❛ ❤✐st❡r❡s❡ ♦✉ ❛ ❞✐❢✉sã♦ ❞❡ ♣❛rtí❝✉❧❛s ❡✱ ♣♦rt❛♥t♦✱ sã♦ ✐rr❡✈❡rsí✈❡✐s✳ ❉❡ss❡ ♠♦❞♦✱ ✉♠ ♣r♦❝❡ss♦ r❡✈❡rsí✈❡❧ é ✉♠❛ ✐❞❡❛❧✐③❛çã♦ q✉❡ s❡ ❛♣r♦①✐♠❛ ❡①♣❡r✐✲ ♠❡♥t❛❧♠❡♥t❡✱ q✉❛♥❞♦ ✉♠ s✐st❡♠❛ ❡stá ❡♥✈♦❧✈✐❞♦ ❡♠ ♣r♦❝❡ss♦s q✉❛s❡✲❡stát✐❝♦s ♥♦s q✉❛✐s ♣♦❞❡✲s❡ ❞❡s♣r❡③❛r ♦s ❡❢❡✐t♦s ❞✐ss✐♣❛t✐✈♦s✳

◆♦ ❝♦♥t❡①t♦ ❝❧áss✐❝♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❛♥á❧✐s❡ ❞❛ ❡♥❡r❣✐❛ r❡✢❡t❡ ❡ r❡✈❡❧❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ❞❡ ✉♠ s✐st❡♠❛✱ ❛ ❢♦r♠✉❧❛çã♦ ❞❛ ▼❡❝â♥✐❝❛ ❞❡ ❍❛♠✐❧t♦♥✱ ♥♦ ❡s♣❛ç♦ ❞❡ ❢❛s❡s✱ é ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❛ ❛♠♣❧❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛✱ ✐♥✐❝✐❛❞❛ ♣❡❧♦ ❢ís✐❝♦ ❛✉strí❛❝♦ ▲✉❞✇✐❣ ❇♦❧t③♠❛♥♥ ✭✶✽✹✹✲✶✾✵✻✮✱ ❡♠ ✶✽✼✼✱ ❡ s✐♥t❡t✐③❛❞❛ ♣❡❧♦ ❢ís✐❝♦ ♥♦rt❡✲❛♠❡r✐❝❛♥♦ ❏♦s✐❛❤ ❲✐❧❧❛r❞ ●✐❜❜s ✭✶✽✸✾✲✶✾✵✸✮✱ ❬✶✷❪ ❡♠ ✶✾✵✶✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✻ •Ú❧t✐♠❛ •❙❛✐r

✶✳✷✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❡♥❡r❣✐❛ ♥❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✱ ❛ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❝♦♥✜♥❛❞❛ ❡♠ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦✱ s♦❜ ❛ ❛çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❝♦♥s❡r✈❛t✐✈♦✱ é q✉❛♥t✐③❛❞❛✱ ♦✉ s❡❥❛✱ ♦s ✈❛❧♦r❡s ♣❡r♠✐t✐❞♦s ♣❛r❛ ❛ s✉❛ ♠❡❞✐❞❛ ❝♦♥st✐t✉❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❞✐s❝r❡t♦ {ǫi}✱ ❞❡♥♦♠✐♥❛❞♦ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛✱ t❛❧ q✉❡ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ♣♦ssí✈❡❧ ❞❡ ❡♥❡r❣✐❛

ǫi ❝♦rr❡s♣♦♥❞❡ ✉♠❛ ❢✉♥çã♦Ψi(x, y, z, t)✱ ❞❡ s✉❛s ❝♦♦r❞❡♥❛❞❛s ❡s♣❛❝✐❛✐s ✭x, y, z✮ ❡ ❞♦ t❡♠♣♦ ✭t✮✳ ❉✐③✲s❡ ❡♥tã♦

q✉❡ ❝❛❞❛ ❢✉♥çã♦ Ψi(x, y, z, t)❝❛r❛❝t❡r✐③❛ ♦ ❡st❛❞♦ ❞❛ ♣❛rtí❝✉❧❛✱ ❛ss♦❝✐❛❞♦ ❛♦ ❝❤❛♠❛❞♦ ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛ǫi✳ ❉♦

♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦✱ ❛s ❢✉♥çõ❡s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦ ❡st❛❞♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛✱ sã♦ s♦❧✉çõ❡s ❞♦ ❝❤❛♠❛❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❞❡ ❙❝❤rö❞✐♥❣❡r✱

HΨi=ǫiΨi

♦♥❞❡ H é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r✱ ❝❤❛♠❛❞♦ ❤❛♠✐❧t♦♥✐❛♥♦✱ q✉❡ r❡♣r❡s❡♥t❛ ❛s ❡♥❡r❣✐❛s ❝✐♥ét✐❝❛ ❡ ♣♦t❡♥❝✐❛❧✱ ❛ss♦❝✐❛❞❛s à ♣❛rtí❝✉❧❛✳ ✸

❙❡ ❛ ♣❛rtí❝✉❧❛✱ ❡♠ ✉♠ ❞❛❞♦ ✐♥st❛♥t❡✱ ❡♥❝♦♥tr❛✲s❡ ❡♠ ✉♠ ❡st❛❞♦ ♣❛rt✐❝✉❧❛rΨi✱ ❛ss♦❝✐❛❞♦ ❛♦ ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛

ǫi✱ q✉❛❧q✉❡r ♠❡❞✐❞❛ ❞❡ ❡♥❡r❣✐❛ ♣♦st❡r✐♦r r❡s✉❧t❛rá ♥♦ ♠❡s♠♦ ✈❛❧♦rǫi✳ ◆❡ss❡ s❡♥t✐❞♦✱ ♦s ❡st❛❞♦s ❝❛r❛❝t❡r✐③❛❞♦s

♣❡❧❛s ❢✉♥çõ❡s Ψi sã♦ ❞❡♥♦♠✐♥❛❞♦s ❛✉t♦❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛ ♦✉ ❡st❛❞♦s ❡st❛❝✐♦♥ár✐♦s ❞❛ ♣❛rtí❝✉❧❛✳

❯♠❛ ✈❡③ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r ❞❡ ❙❝❤rö❞✐♥❣❡r é ❧✐♥❡❛r ❡ ❤♦♠♦❣ê♥❡♦✱ ❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛ é ✉♠❛ t❡♦r✐❛ ❧✐♥❡❛r ❝♦♠ r❡❧❛çã♦ ❛♦s ❡st❛❞♦s ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ ♦✉ s❡❥❛✱ ♦s ❛✉t♦❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛ ♦❜❡❞❡❝❡♠ ❛♦ ❝❤❛♠❛❞♦ ✸ _{❊♠ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✱ ♦s ❡st❛❞♦s ❡ ♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡♠ ✉♠ ❝❛♠♣♦ ❝♦♥s❡r✈❛t✐✈♦✱ sã♦✱ r❡s♣❡❝t✐✈❛✲}

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✼ •Ú❧t✐♠❛ •❙❛✐r

♣r✐♥❝í♣✐♦ ❞❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ❡st❛❞♦s✱ ♥♦ s❡♥t✐❞♦ q✉❡ q✉❛❧q✉❡r ❢✉♥çã♦Ψ✱ r❡s✉❧t❛♥t❡ ❞❛ s✉♣❡r♣♦s✐çã♦ ❧✐♥❡❛r

❞❡ ❡st❛❞♦s ❡st❛❝✐♦♥ár✐♦s ✭Ψi✮✱

Ψ =X

i

ciΨi (❡st❛❞♦ r❡s✉❧t❛♥t❡ ❞❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ❛✉t♦❡st❛❞♦s)

♦♥❞❡ ♦s ❝♦❡✜❝✐❡♥t❡sci sã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ é ✉♠ ♣♦ssí✈❡❧ ❡st❛❞♦ ❞❛ ♣❛rtí❝✉❧❛✳

❆ss✐♠✱ ♠❡s♠♦ ❡♠ ✉♠ ❝❛♠♣♦ ❝♦♥s❡r✈❛t✐✈♦✱ ❡①✐st❡♠ ❡st❛❞♦s ❞❛ ♣❛rtí❝✉❧❛ q✉❡ ♥ã♦ ♣♦ss✉❡♠ ❡♥❡r❣✐❛ ❞❡✜♥✐❞❛ ♣♦✐s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ✐♥t❡r♣r❡t❛çã♦ ♣r♦❜❛❜✐❧íst✐❝❛ ❞❡ ▼❛① ❇♦r♥✱ ♦s ❝♦❡✜❝✐❡♥t❡s ✭ci✮ ❞❛ ❡①♣❛♥sã♦ ❧✐♥❡❛r

❞❡t❡r♠✐♥❛♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ P(ǫi) ❞❡ q✉❛❧q✉❡r ✈❛❧♦r ♣❛rt✐❝✉❧❛r ǫi ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❞❛ ♣❛rtí❝✉❧❛✱

s❡❣✉♥❞♦

P(ǫi) =|ci|2

❉❡ss❡ ♠♦❞♦✱ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛✱ ♦♥❞❡ q✉❛❧q✉❡r ❡st❛❞♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡♠ ✉♠ ❝❛♠♣♦ ❝♦♥s❡r✈❛t✐✈♦ ❡stá ❛ss♦❝✐❛❞♦ ❛ ✉♠ ú♥✐❝♦ ✈❛❧♦r ❞❡ ❡♥❡r❣✐❛✱ ♦✉ s❡❥❛✱ ❛ ❡♥❡r❣✐❛ é ✉♠❛ ❝♦♥st❛♥t❡ ❞♦ ♠♦✈✐♠❡♥t♦✱ ♥❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✱ ✉♠ ❡st❛❞♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡♠ ✉♠ ❝❛♠♣♦ ❝♦♥s❡r✈❛t✐✈♦ ♥ã♦ ❡stá ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ú♥✐❝♦ ✈❛❧♦r ❞❡ ❡♥❡r❣✐❛✳ ❆♣❡♥❛s ❡st❛❞♦s ❡st❛❝✐♦♥ár✐♦s ❡stã♦ ✉♥✐✈♦❝❛♠❡♥t❡ ❛ss♦❝✐❛❞♦s ❛ ✉♠ ♣♦ssí✈❡❧ ✈❛❧♦r ❞❡ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✳

◆❡ss❡ s❡♥t✐❞♦✱ ❛ ❡♥❡r❣✐❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡♠ ✉♠ ❝❛♠♣♦ ❝♦♥s❡r✈❛t✐✈♦ ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠❛ ❝♦♥st❛♥t❡ ❞♦ ♠♦✈✐♠❡♥t♦✳ ❖ ♣❛râ♠❡tr♦ q✉❡ ♦❜❡❞❡❝❡ ❛ ✉♠❛ ❧❡✐ ❞❡ ❝♦♥s❡r✈❛çã♦ é ♦ ✈❛❧♦r ♠é❞✐♦ ❞❛ ❡♥❡r❣✐❛✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✽ •Ú❧t✐♠❛ •❙❛✐r

♣❛r❛ ❛❧❣✉♥s s✐st❡♠❛s✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r ♦ ♦♣❡r❛❞♦r ❤❛♠✐❧t♦♥✐❛♥♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ❝♦♠♦

H(t) =H_♦+V(t)

♦♥❞❡ H_♦ é ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞♦ s✐st❡♠❛ ♥❛ ❛✉sê♥❝✐❛ ❞❡ q✉❛❧q✉❡r ✐♥t❡r❛çã♦ ❝♦♠ ♦ ♠❡✐♦ ❡①t❡r♥♦✱ ❡ V(t) é ✉♠

t❡r♠♦✱ ✉s✉❛❧♠❡♥t❡✱ ❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦ ✭t✮✱ q✉❡ ❝❛r❛❝t❡r✐③❛ ❛s ✐♥t❡r❛çõ❡s ❞♦ s✐st❡♠❛ ❝♦♠ ♦ ♠❡✐♦ ❡①t❡r♥♦✳

◆❡ss❡s ❝❛s♦s✱ ♣♦❞❡✲s❡ ❛❞♦t❛r ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❉✐r❛❝ ❬✶✵❪✱ ♥♦ q✉❛❧ ♦ t❡r♠♦ ❝♦♥st❛♥t❡ H_♦ ❞❡t❡r♠✐♥❛ ♦s ♣♦ssí✈❡✐s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛✱ ❡ ♦ t❡r♠♦ ❞❡ ✐♥t❡r❛çã♦ V(t) é ♦ r❡s♣♦♥sá✈❡❧ ♣♦r tr❛♥s✐çõ❡s ❞♦

s✐st❡♠❛ ❡♥tr❡ ♦s ♣♦ssí✈❡✐s ❛✉t♦❡st❛❞♦s ❞❡ ❡♥❡r❣✐❛✳ ❖ t❡r♠♦ ❞❡ ✐♥t❡r❛çã♦✱ ♣♦rt❛♥t♦✱ ❛❝❛rr❡t❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ s❡ ❡♥❝♦♥tr❛r ❡♠ q✉❛✐sq✉❡r ❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡st❛❞♦s ❞❡✜♥✐❞♦s ❞❡ ❡♥❡r❣✐❛✱ ♦✉ s❡❥❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ t♦❞♦s ♦s ❡st❛❞♦s ❝♦♠♣❛tí✈❡✐s ❝♦♠ ♦s ✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ ♣❡r♠✐t✐❞♦ ♣❡❧❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✳

◆♦ ❝❛s♦ ❞❡ s✐st❡♠❛s ♠❛❝r♦s❝ó♣✐❝♦s ❡♠ ❡q✉✐❧í❜r✐♦ ❝♦♠ s✉❛ ✈✐③✐♥❤❛♥ç❛✱ ♦ t❡r♠♦ ❞❡ ✐♥t❡r❛çã♦ é tã♦ ♠❡♥♦r ❞♦ q✉❡ q✉❛❧q✉❡r ♥í✈❡❧ ❞❡ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ q✉❛❧q✉❡r ❡st❛❞♦ ❝♦♠ ✉♠❛ ❞❛❞❛ ❡♥❡r❣✐❛✱ ♣r❛t✐❝❛♠❡♥t❡✱ ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ t❡♠♣♦✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❛s ✐♥❝❡rt❡③❛s r❡❧❛t✐✈❛s ❛ss♦❝✐❛❞❛s ❛♦s ❞✐✈❡rs♦s ♣❛râ♠❡tr♦s ♠❛❝r♦s❝ó♣✐❝♦s ❞❡ ✉♠ s✐st❡♠❛ sã♦ tã♦ ♣❡q✉❡♥❛s q✉❡ ❛s ♣r❡✈✐sõ❡s ❞❛ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛ ❞♦ ❡q✉✐❧í❜r✐♦ sã♦ ✏♣r❛t✐❝❛♠❡♥t❡ ❡①❛t❛s✑✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✾ •Ú❧t✐♠❛ •❙❛✐r

✷✳ ❊❧❡♠❡♥t♦s ❞❛ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s

❖s ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞♦s ❝♦♠♦ ♣r♦❝❡ss♦s ❞❡t❡r♠✐♥íst✐❝♦s ♦✉ ❛❧❡❛tór✐♦s✳ ❙❡ ♦s ❡❢❡✐t♦s ❛ss♦❝✐❛❞♦s ❛ ✉♠ ❢❡♥ô♠❡♥♦ ❞❡✈✐❞♦ ❛ ❞❡t❡r♠✐♥❛❞❛s ✐♥✢✉ê♥❝✐❛s ♦✉ ❝❛✉s❛s sã♦ ✐♥❡q✉✐✈♦❝❛♠❡♥t❡ ♣r❡✈✐sí✈❡✐s✱ ❞✐③✲ s❡ q✉❡ ♦s ♣r♦❝❡ss♦s ❡♥✈♦❧✈✐❞♦s sã♦ ❞❡t❡r♠✐♥íst✐❝♦s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ♦s ❡❢❡✐t♦s ❛ss♦❝✐❛❞♦s ❛ ✉♠ ❢❡♥ô♠❡♥♦ ♥ã♦ sã♦ ❡①❛t❛♠❡♥t❡ ♣r❡✈✐sí✈❡✐s✱ ♠❛s ♣♦❞❡♠ s❡r ❛ss♦❝✐❛❞♦s ❛ ❝❡rt❛s ❡①♣❡❝t❛t✐✈❛s r❡❧❛t✐✈❛s ❞❡ ♦❝♦rrê♥❝✐❛✱ ♦s ♣r♦❝❡ss♦s ❡♥✈♦❧✈✐❞♦s sã♦ ❞✐t♦s ❛❧❡❛tór✐♦s✳

❊♠ ❣❡r❛❧✱ ❛ ♥ã♦ ♣r❡✈✐s✐❜✐❧✐❞❛❞❡ ❞♦s ❡❢❡✐t♦s ❞❡ ✉♠ ❢❡♥ô♠❡♥♦ ❡stá ❛ss♦❝✐❛❞❛ ❛ ♣r♦❝❡ss♦s ❝♦♠♣❧❡①♦s q✉❡ ❡♥✈♦❧✈❡♠ ❛ ✐♥t❡r❛çã♦ ❞❡ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ s✐st❡♠❛s s✐♠♣❧❡s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❛té ♦ s✉r❣✐♠❡♥t♦ ❞❛ ▼❡❝â♥✐❝❛ ◗✉â♥t✐❝❛✱ ❡♠ ✶✾✷✻✱ ❛s t❡♦r✐❛s ❢ís✐❝❛s ♣r♦❜❛❜✐❧íst✐❝❛s ❞❡s❝r❡✈✐❛♠ ❢❡♥ô♠❡♥♦s ♦✉ s✐st❡♠❛s ❢ís✐❝♦s q✉❡✱ ♣♦r ❡♥✈♦❧✲ ✈❡r❡♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s✱ ❡r❛♠ t❛♠❜é♠ t❡♦r✐❛s ❡st❛tíst✐❝❛s✳ ❆ss✐♠✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡st❛✈❛ ✈✐♥❝✉❧❛❞♦ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❡st❛tíst✐❝♦ ❞❛s ♠♦❧é❝✉❧❛s ❞❡ ✉♠ ❣ás✱ ♦✉ ❞❛ ❡♥♦r♠❡ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú❝❧❡♦s q✉❡ ♣❛rt✐❝✐♣❛♠ ❞♦ ❢❡♥ô♠❡♥♦ ❞❛ r❛❞✐♦❛t✐✈✐❞❛❞❡✳

❈♦♠♦ ❛ t❡♦r✐❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❋ís✐❝❛ ❈❧áss✐❝❛ ✕ ❛ ▼❡❝â♥✐❝❛ ❞❡ ◆❡✇t♦♥✲●❛❧✐❧❡✉ ✕ ♣r❡ss✉♣✉♥❤❛ q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠ ✉♠ ♣❡q✉❡♥♦ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s s❡r✐❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦ ♣♦r s✉❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱✹ _{❛ ❛❧❡❛t♦r✐❡❞❛❞❡ ❡ ♦ ❛❝❛s♦ ❡♠ ✉♠ ❢❡♥ô♠❡♥♦ ♦✉ ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❡r❛♠ ❛tr✐❜✉í❞♦s ❛}

✐♥❝❛♣❛❝✐❞❛❞❡ ❞♦ ♦❜s❡r✈❛❞♦r ❞❡t❡r♠✐♥❛r ❡ss❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♦✉ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦s ❛rr❛♥❥♦s ❡①♣❡r✐♠❡♥t❛✐s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ♦❜s❡r✈❛çã♦ ❞♦s ❢❡♥ô♠❡♥♦s✳ ❊♠ ♣r✐♥❝í♣✐♦✱ ❛ ♣❛rt✐r ❞❡ ❞❛❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s✱ ✉♠❛ t❡♦r✐❛ ❞❡t❡r♠✐♥íst✐❝❛ ♣❡r♠✐t✐r✐❛ ❛ ♣r❡❞✐çã♦ ❞❡ s♦♠❡♥t❡ ✉♠ ú♥✐❝♦ r❡s✉❧t❛❞♦ ♣❛r❛ ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦✱

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✵ •Ú❧t✐♠❛ •❙❛✐r

❡♥q✉❛♥t♦ t❡♦r✐❛s ♣r♦❜❛❜✐❧íst✐❝❛s ♣♦❞❡r✐❛♠ ❛❞♠✐t✐r ✈ár✐♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ♣❛r❛ ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛✱ ❛♦ ❛ss♦❝✐❛r ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛ ❝❛❞❛ ✉♠ ❞❡ss❡s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s✳ ✺

✷✳✶✳ ●r❛♥❞❡③❛s ❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s

❙❡ ❛s ♠❡❞✐❞❛s ❛ss♦❝✐❛❞❛s ❛ ✉♠❛ ❣r❛♥❞❡③❛ ♥ã♦ sã♦ ✐♥t❡✐r❛♠❡♥t❡ ♣r❡✈✐sí✈❡✐s q✉❛♥❞♦ ❡❢❡t✉❛❞❛s s♦❜ ❛s ♠❡s✲ ♠❛s ❝♦♥❞✐çõ❡s ❡①♣❡r✐♠❡♥t❛✐s ♣ré✲❞❡t❡r♠✐♥❛❞❛s✱ ❞✐③✲s❡ q✉❡ ❛ ❣r❛♥❞❡③❛ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳ ◆♦ ❡♥t❛♥t♦✱ ❣r❛♥❞❡③❛s ❛❧❡❛tór✐❛s ♥ã♦ s❡ ♠❛♥✐❢❡st❛♠ ❞❡ ♠♦❞♦ t♦t❛❧♠❡♥t❡ ✐♠♣r❡✈✐sí✈❡❧ ❥á q✉❡✱ ❡♠ ❣❡r❛❧✱ s❡✉s ✈❛❧♦r❡s ♦✉ ♠❡❞✐❞❛s✱ ❛❧é♠ ❞❡ ❧✐♠✐t❛❞♦s ❛ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡✜♥✐❞♦✱ ❡stã♦ ❛ss♦❝✐❛❞♦s ❛ ❝❡rt❛s ❡①♣❡❝t❛t✐✈❛s ❞❡ ♦❝♦rrê♥❝✐❛✳ ❆ ❛❧❡❛t♦r✐❡❞❛❞❡✱ ♦✉ ❛ ✐♠♣r❡❞✐❝❛❜✐❧✐❞❛❞❡ ♣❛r❝✐❛❧✱ s✐❣♥✐✜❝❛ q✉❡ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ q✉❛❧q✉❡r ✈❛❧♦r ♥ã♦ é ♣r❡✈✐sí✈❡❧ ❞❡t❡r♠✐♥✐st✐❝❛♠❡♥t❡ ♠❛s✱ s✐♠✱ ♣r♦❜❛❜✐❧✐st✐❝❛♠❡♥t❡✱ ♦✉ s❡❥❛✱ ❛ ❝❛❞❛ ✈❛❧♦r ❞❡♥tr♦ ❞❡ss❡ ✐♥t❡r✈❛❧♦ ❛ss♦❝✐❛✲s❡ ✉♠❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ♦❝♦rrê♥❝✐❛ ♦✉ ♣r♦❜❛❜✐❧✐❞❛❞❡✳

✷✳✷✳ Pr♦❜❛❜✐❧✐❞❛❞❡s ❛ ♣r✐♦r✐

❆ t❡♦r✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s t❡✈❡ s✉❛ ♦r✐❣❡♠ ♥❛ ❛♥á❧✐s❡ ❞♦s ❥♦❣♦s ❞❡ ❛③❛r✱ ❛♦ s❡ q✉❛♥t✐✜❝❛r ❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ❢❡♥ô♠❡♥♦ ♥ã♦ ♣r❡✈✐sí✈❡❧ ❡ ❛❧❡❛tór✐♦✱ ❝♦♠♦ ✉♠ ❥♦❣♦ ❞❡ ❝❛rt❛s ✺ _{❙❛❜❡✲s❡ ❤♦❥❡ q✉❡✱ ♠❡s♠♦ ♣❛r❛ s✐st❡♠❛s ❝♦♠ ♣♦✉❝♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡s❝r✐t♦s ♣♦r t❡♦r✐❛s✱ ❡♠ ♣r✐♥❝í♣✐♦✱ ❞❡t❡r♠✐♥íst✐❝❛s✱}

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✶ •Ú❧t✐♠❛ •❙❛✐r

♦✉ ♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ❞❛❞♦s✳ ❊ss❛ q✉❛♥t✐✜❝❛çã♦ ❢♦✐ r❡❛❧✐③❛❞❛ ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r P❛s❝❛❧ ❡ ❋❡r♠❛t✱ ❡♠ ✶✻✺✹✱ ❡ s✐♥t❡t✐③❛❞❛ ♣♦r ▲❛♣❧❛❝❡✱ ❡♠ ✶✽✶✹ ❬✹✵❪✳

❖s ú♥✐❝♦s r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ♣❛r❛ ♦s ❧❛♥ç❛♠❡♥t♦s ❞❡ ✉♠ ❞❛❞♦ sã♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s{1,2,3,4,5,6}✳ P❛r❛ ✉♠ ❞❛❞♦ ♥ã♦ ✈✐❝✐❛❞♦✱ ❞❡✈✐❞♦ à s✐♠❡tr✐❛ ❞♦ ♣r♦❜❧❡♠❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣r✐♦r✐ ❛tr✐❜✉í❞❛ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♥ú♠❡r♦ ♣♦ssí✈❡❧ é ✐❣✉❛❧ ❛ ✶✴✻✳ ❆ss✐♠✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ s❡❣✉♥❞♦ ▲❛♣❧❛❝❡✱ é ❞❛❞❛ ♣❡❧❛ r❛③ã♦ ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❢❛✈♦rá✈❡✐s ♣♦ssí✈❡✐s ♣❛r❛ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ ❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❛❧t❡r♥❛t✐✈❛s ✐❣✉❛❧♠❡♥t❡ ♣♦ssí✈❡✐s✱ ♦✉ ❡qü✐♣r♦✈á✈❡✐s✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✷ •Ú❧t✐♠❛ •❙❛✐r

P❛r❛ ❡✈❡♥t♦s ❝♦♠♣♦st♦s✱ ♠❡s♠♦ q✉❡ ❡♠ ♥ú♠❡r♦ ✜♥✐t♦✱ ❛ s✐t✉❛çã♦ ♣♦❞❡ s❡r ♠❛✐s ❝♦♠♣❧✐❝❛❞❛✳ ●❛❧✐❧❡✉✱ ❡♠ ✉♠ ❛rt✐❣♦ ❡s❝r✐t♦ ♣♦r ✈♦❧t❛ ❞❡ ✶✻✶✸✱ ✐❞❡♥t✐✜❝♦✉ ❝♦rr❡t❛♠❡♥t❡ ♦s ✷✶✻ r❡s✉❧t❛❞♦s ❡qü✐♣r♦✈á✈❡✐s ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ três ❞❛❞♦s✱ s❡♥❞♦ ❝❛♣❛③ ❞❡ ♣r❡✈❡r ❛ ❧✐❣❡✐r❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s três ❞❛❞♦s s♦♠❛r❡♠ ✾ ❡ ✶✵✳ ❆♣❡s❛r ❞❡ ❝❛❞❛ r❡s✉❧t❛❞♦ ❡st❛r ❛ss♦❝✐❛❞♦ ❛ s❡✐s ♣❛rt✐çõ❡s ❞✐st✐♥t❛s✱ ❝❛❞❛ ♣❛rt✐çã♦ ❝♦rr❡s♣♦♥❞❡ ❛ ❞✐❢❡r❡♥t❡s ♠✉❧t✐♣❧✐❝✐❞❛❞❡s✳

s♦♠❛ ✭S✮ ❞❡ ♣❛rt✐çõ❡s ♣❛rt✐çõ❡s ♠✉❧t✐♣❧✐❝✐❞❛❞❡

três ❞❛❞♦s ❡q✉✐✈❛❧❡♥t❡s

9 (126) 126−162−216−261−612−621 6 (135) 135−153−315−351−513−531 6 (144) 144₋414₋441 3 (225) 225−252−522 3 (234) 234−243−324−342−423−432 6

(333) 333 1

10 (136) 136₋163₋316₋361₋613₋631 6 (145) 145₋154₋415₋451₋514₋541 6 (226) 226−262−622 3 (235) 235₋253₋325₋352₋523₋532 6 (244) 244₋424₋442 3 (334) 334−343−433 3

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✸ •Ú❧t✐♠❛ •❙❛✐r

✾✱ ❡ ❞❡ ✷✼ ♣❛r❛ ❛ s♦♠❛ ✶✵✱ ❛s r❡s♣❡❝t✐✈❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s sã♦ ❞❛❞❛s ♣♦r

P(S= 9) = 25/216 ❡ P(S = 10) = 27/216

❖ ❢❛t♦ ❞❡ ❡ss❡ r❡s✉❧t❛❞♦✱ ❛♣❡s❛r ❞❡ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦s ❥♦❣❛❞♦r❡s ♠❛✐s ❡①♣❡r✐❡♥t❡s ❞❛ é♣♦❝❛✱ ♥ã♦ s❡r ♣♦r ❡❧❡s ❡①♣❧✐❝❛❞♦✱ ♠♦str❛ q✉❡ ❡✈❡♥t♦s ❝♦♠♣♦st♦s ♣♦❞❡♠ s❡r ❝♦♠♣❧❡①♦s ❞❡♠❛✐s ♣❛r❛ q✉❡ s✉❛s ❛❧t❡r♥❛t✐✈❛s ♦✉ r❡s✉❧t❛❞♦s ❡qü✐♣r♦✈á✈❡✐s s❡❥❛♠ ❞✐r❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s ✳

❆ ❛①✐♦♠❛t✐③❛çã♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❢♦✐ r❡❛❧✐③❛❞❛ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ❬✽✱✷✽❪✱ ❡♠ ✶✾✸✸✱ ♣❡❧♦ ♠❛t❡♠át✐❝♦ r✉ss♦ ❆♥❞r❡✐ ◆✐❦♦❧❛❡✈✐❝❤ ❑♦❧♠♦❣♦r♦✈ ✭✶✾✵✸✲✶✾✽✼✮✳ ❙❡❣✉♥❞♦ ❑♦❧♠♦❣♦r♦✈✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ S =_{x1, x2, x3, ...} ❞♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦✱ ✻ ❝♦♠♦ ❛s ♠❡❞✐❞❛s ❞❡ ✉♠❛ ❣r❛♥❞❡③❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡P(xi) ❛ss♦❝✐❛❞❛ ❛ ✉♠ ❞❛❞♦ r❡s✉❧t❛❞♦xi é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ t❛❧ q✉❡

X

i

P(xi) = 1

❖✉ s❡❥❛✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦✱ ♥ã♦ é ♥❡❝❡ssár✐♦ ❛tr✐❜✉✐r ♥❡♥❤✉♠ s✐❣♥✐✜❝❛❞♦ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❜❛✲ ❜✐❧✐❞❛❞❡❀ ❜❛st❛ q✉❡ s❡ ❛ss♦❝✐❡ ❛ ❝❛❞❛ ❡✈❡♥t♦ ✭♦✉ r❡s✉❧t❛❞♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ✉♠ ❢❡♥ô♠❡♥♦✮✱ ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ♠❡♥♦r q✉❡ ❛ ✉♥✐❞❛❞❡✱ ❝✉❥❛ ❛❞✐çã♦ s♦❜r❡ t♦❞♦s ♦s ♣♦ssí✈❡✐s ❡✈❡♥t♦s✱ ❞❡♥♦♠✐♥❛❞❛ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦✱ ❡①♣r❡ss❛ ❛ ❝❡rt❡③❛ ❡♠ s❡ ♦❜t❡r ✉♠ ❞♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✹ •Ú❧t✐♠❛ •❙❛✐r

✷✳✸✳ Pr♦❜❛❜✐❧✐❞❛❞❡s ❛ ♣♦st❡r✐♦r✐ ❡ ❞✐str✐❜✉✐çõ❡s

❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✶✴✻✱ q✉❡ ♣♦❞❡ s❡r ❛ ♣r✐♦r✐ ❛tr✐❜✉í❞❛ à ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ r❡s✉❧t❛❞♦ ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ ♥ã♦ ✈✐❝✐❛❞♦✱ ♣♦❞❡ s❡r ♦❜t✐❞❛✱ t❛♠❜é♠✱ ❛ ♣♦st❡r✐♦r✐✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✺ •Ú❧t✐♠❛ •❙❛✐r

❢❛❝❡ ✭i✮ ❢r❡q✳ ✭ni✮

✶ ✷✷

✷ ✶✻

✸ ✷✷

✹ ✷✷

✺ ✶✼

✻ ✷✶

❚❛❜❡❧❛ ✶✿ ❉✐str✐❜✉✐çã♦ ❞❡ ❢r❡qüê♥❝✐❛s ❞❛s ❢❛❝❡s ❞❡ ✉♠ ❞❛❞♦ ❡♠ ✶✷✵ ❧❛♥ç❛♠❡♥t♦s✳

❆ ❋✐❣✳ ✶ ♠♦str❛ ♦ ❤✐st♦❣r❛♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ à ❚❛❜✳ ✶✱ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ❢r❡qüê♥❝✐❛s (ni) ❞❡ ❝❛❞❛ ✉♠❛

❢❛❝❡s ✭i✮ ❞❡ ✉♠ ❞❛❞♦✱ r❡s✉❧t❛♥t❡ ❞❛ s✐♠✉❧❛çã♦ ❞❡N = 120❧❛♥ç❛♠❡♥t♦s✳

❉❡ss❡ ♠♦❞♦✱ ❛ ♠❡❞✐❞❛ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❧❛♥ç❛♠❡♥t♦s ❛✉♠❡♥t❛✱ ❛ ❢r❡qüê♥❝✐❛ r❡❧❛t✐✈❛ ✭fi✮ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡

✉♠❛ ❢❛❝❡ i✱

fi =

ni

N

❛♣r♦①✐♠❛✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❞❡ ✉♠ ♥ú♠❡r♦ ❞❡✜♥✐❞♦ ❡♥tr❡ 0❡ 1✱ ♥❡ss❡ ❝❛s♦✱ ✐❣✉❛❧ ❛ ✶✴✻✳

❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ ♥ú♠❡r♦ t♦t❛❧ N ❞❡ ♠❡❞✐❞❛s xi ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ x ❡♠ ✉♠ ❡①♣❡r✐♠❡♥t♦✱ ♦✉ q✉❡ ♦

♥ú♠❡r♦ni ❞❡ ♦❝♦rrê♥❝✐❛s ❞❡ ✉♠ ❡✈❡♥t♦ s❡❥❛ ✏s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✑✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛P(xi) ❞❛

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✻ •Ú❧t✐♠❛ •❙❛✐r

1 2 3 4 5 6

16 17 18 19 20 21 22

i

i

n

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✼ •Ú❧t✐♠❛ •❙❛✐r

❡ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❡✈❡♥t♦s ♦❜s❡r✈❛❞♦s✱ ♦✉ s❡❥❛✱ ❝♦♠♦ ♦ ❧✐♠✐t❡ ❞❛ ❢r❡qüê♥❝✐❛ r❡❧❛t✐✈❛✱

P(xi) = lim

N≫1fi = limN≫1 ni

N

❯♠❛ ✈❡③ q✉❡ ❛ t♦t❛❧✐❞❛❞❡ ❞♦s ❡✈❡♥t♦s ♦❜s❡r✈❛❞♦s s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦ 6

X

i=1

P(xi) = 1

♦ ❝♦♥❥✉♥t♦{P(xi)}s❛t✐s❢❛③ ♦ ❝r✐tér✐♦ ❞❡ ❑♦❧♠♦❣♦r♦✈ ♣❛r❛ ♣r♦❜❛❜✐❧✐❞❛❞❡s✳

❆ ❝♦♥❝❡♣çã♦ ❢r❡qü❡♥t✐st❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ❢♦✐ ❛❞♦t❛❞❛ ❡ s✐st❡♠❛t✐③❛❞❛ ♣♦r ✈♦♥ ▼✐s❡s✱ ❡♠ ✶✾✸✶ ❬✽✱ ✷✵✱ ✷✹✱ ✷✽❪✱ ❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ♠❡s♠♦ q✉❛♥❞♦ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❡✈❡♥t♦s ♣♦ssí✈❡✐s s❡❥❛ ✐♥✜♥✐t♦✱ ❜❛st❛♥❞♦ q✉❡ ❛ ❢r❡qüê♥❝✐❛ r❡❧❛t✐✈❛ ❛♣r♦①✐♠❡✲s❡ ❞❡ ✉♠ ❧✐♠✐t❡ ❡①♣❡r✐♠❡♥t❛❧✳

▲♦❣♦✱ ♦ ✈❛❧♦r ♠é❞✐♦ ✭µ✮ ❡ ❛ ✈❛r✐â♥❝✐❛ ✭σ2_{✮✱ ❛ss♦❝✐❛❞♦s ❛♦s ♣♦ssí✈❡✐s ✈❛❧♦r❡s ❞❡ ✉♠❛ ❣r❛♥❞❡③❛x✱ ❛ ♣❛rt✐r} ❞❡ ✉♠❛ ❛♠♦str❛ ❞❡ ♠❡❞✐❞❛s {xi}✮✱ ♣♦❞❡♠ s❡r ❡st✐♠❛❞♦s ♣♦r

              

µ= lim

N >>1x=

N

X

i=1

xi P(xi) =hxi

σ2 = lim

N >>1s 2 ₌

N

X

i=1

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✽ •Ú❧t✐♠❛ •❙❛✐r

❆ r❛✐③ q✉❛❞r❛❞❛ ♣♦s✐t✐✈❛ ❞❛ ✈❛r✐â♥❝✐❛✱ ❞❡♥♦t❛❞❛ ♣♦rσ✱ é ♦ ❝❤❛♠❛❞♦ ❞❡s✈✐♦✲♣❛❞rã♦✳

P❛r❛ ♦ ❝❛s♦ ❞❡ ❣r❛♥❞❡③❛s ❝✉❥❛s ✈❛r✐❛çõ❡s sã♦ ❤✐♣♦t❡t✐❝❛♠❡♥t❡ ❝♦♥tí♥✉❛s✱ ❝♦♠♦ ❛s ❝♦♦r❞❡♥❛❞❛s ✭x✮ ❞❡ ✉♠❛ ♣❡❞r❛ ❛t✐r❛❞❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ s♦❜r❡ ✉♠ ❝❛♥❛❧❡t❛✱ ♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦s✱ q✉❡ sã♦ ❛s ♠❡❞✐❞❛s ❞❛s ❝♦♦r❞❡♥❛❞❛s✱ ♣♦❞❡♠ s❡r ❛❣r✉♣❛❞♦s ❡♠ ✐♥t❡r✈❛❧♦s ❞✐s❝r❡t♦s ✭xk, xk+1✮✱ ❞❡ ❛♠♣❧✐t✉❞❡s ∆xk = xk+1 −xk✳ ✼ ❉❡ss❡ ♠♦❞♦✱

s❡ nk é ❛ ❢r❡qüê♥❝✐❛ ❞❛s ♠❡❞✐❞❛s ❛ss♦❝✐❛❞❛s ❛♦ ✐♥t❡r✈❛❧♦ ✭xk, xk+1✮✱ ♦❜té♠✲s❡ ♦ ❤✐st♦❣r❛♠❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ♠♦str❛❞♦ ♥❛ ❋✐❣✳✷✳

❆ss✐♠✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❞❛ ❝♦♦r❞❡♥❛❞❛x ❡♠ ✉♠ ❞❛❞♦ ✐♥t❡r✈❛❧♦ ✭xk, xk+1✮ é ❞❛❞❛ ♣❡❧❛ r❛③ã♦ ❡♥tr❡ ❛ ár❡❛ ✭nk✁∆xk✮ ❞❛ ♣❛rt❡ ❞♦ ❤✐st♦❣r❛♠❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❞❛❞♦ ✐♥t❡r✈❛❧♦ ❡ ❛ ár❡❛

t♦t❛❧

N

X

i=1

ni✁∆xi =A

❞♦ ❤✐st♦❣r❛♠❛✱ ♦✉ s❡❥❛✱

P(xk< x < xk+1) =

nk✁∆xk N

X

i=1

ni✁∆xi

= nk

A ✁∆xk =⇒

N

X

k=1

P(xk < x < xk+1) = 1

♦♥❞❡N é ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♠❡❞✐❞❛s✳

❆ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞♦s t❡r♠♦s ρ(xk) = nk/A ❛ss♦❝✐❛❞♦s ❛♦s ✐♥t❡r✈❛❧♦s ✭xk, xk+1✮ é ❞❡♥♦♠✐♥❛❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❢r❡qüê♥❝✐❛ ♥♦r♠❛❧✐③❛❞❛✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✶✾ •Ú❧t✐♠❛ •❙❛✐r

x

k

x xk+1

n

k

n

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✷✵ •Ú❧t✐♠❛ •❙❛✐r

◆♦ ❧✐♠✐t❡ ❞❡ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ♦❜s❡r✈❛çõ❡s ✭N _≫1✮✱ ♦s ✐♥t❡r✈❛❧♦s ✭xk, xk+1✮ ♦✉ ❝❧❛ss❡s ❞❡ ❢r❡qüê♥❝✐❛ ♣♦❞❡♠ s❡r t❛✐s q✉❡ ❛ ❛♠♣❧✐t✉❞❡ ∆xk ❞❡ ❝❛❞❛ ❝❧❛ss❡ s❡❥❛ tã♦ ♣❡q✉❡♥❛ q✉❛♥t♦ s❡ q✉❡✐r❛✳ ◆❡ss❛s ❝♦♥❞✐çõ❡s✱ ♦s

t❡r♠♦s ❧✐♠✐t❡s

ρ(x)|xk = lim ∆xk→0

P(xk < x < xk+1)

∆xk

= lim

N >>1 nk

A k= 1,2,3, . . . N ≫1

❞❡✜♥❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♥tí♥✉❛ ♥♦r♠❛❧✐③❛❞❛ ❞❡ ❢r❡qüê♥❝✐❛s r❡❧❛t✐✈❛s ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❞❛ ❝♦♦r❞❡♥❛❞❛ x✱ ❞❡♥♦♠✐♥❛❞❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ρ(x)✱ ♣❛r❛ ♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ❝♦♥tí♥✉♦s ❞❡ x✱ t❛❧ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ss♦❝✐❛❞❛ ❛ ♠❡❞✐❞❛ ❞❡x ❡♥tr❡ ❞♦✐s ✈❛❧♦r❡s a❡ bé ❞❛❞❛ ♣♦r

P(a < x < b) =

Z b

a

ρ(x)❞x

❡ _{Z x}

♠❛①

x_♠✐♥

ρ(x)❞x= 1

♦♥❞❡ x♠✐♥ ❡ x♠❛① sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ♠í♥✐♠♦ ❡ ♦ ♠á①✐♠♦ ✈❛❧♦r❡s ♣♦ssí✈❡✐s ♣❛r❛ ❛ ♠❡❞✐❞❛ ❞❡ x✱ ❡①♣r❡ss❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ♥♦r♠❛❧✐③❛çã♦ ❞❛ ❞✐str✐❜✉✐çã♦✳

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✷✶ •Ú❧t✐♠❛ •❙❛✐r

✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♥tí♥✉❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ρ(x)✱ t❛❧ q✉❡

Z

D

ρ(x)❞x= 1

♦ ✈❛❧♦r ♠é❞✐♦ ✭hx_i✮ ❞❡ x é ❞❛❞♦ ♣♦r

hx_i=

Z

D

xρ(x)❞x

❡ ❛ ✈❛r✐â♥❝✐❛ ✭σ2

x✮ ♣♦r

σ_x2 =

Z

D

(x_{− h}x_i)2ρ(x)❞x

❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rát✐❝♦✱ ❛ ✈❛r✐â♥❝✐❛ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r

σ2_x=hx2i − hxi2 ♦♥❞❡

hx2_i=

Z

D

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✷✷ •Ú❧t✐♠❛ •❙❛✐r

❆❧é♠ ❞❡ ❡st❛r ❛ss♦❝✐❛❞❛ à ❞✐str✐❜✉✐çã♦ ❞❡ ✈❛❧♦r❡s ❞❛ ✈❛r✐á✈❡❧x✱ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ρ(x)❞❡t❡r♠✐♥❛

t❛♠❜é♠ ❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ♦❝♦rrê♥❝✐❛ ❞♦s ✈❛❧♦r❡s ❞❡ q✉❛❧q✉❡r ♦✉tr❛ ❢✉♥çã♦f(x)✳ P♦r ❡①❡♠♣❧♦✱ s❡xr❡♣r❡s❡♥t❛ ❛s ♣♦ssí✈❡✐s ♠❡❞✐❞❛s ♣❛r❛ ♦ r❛✐♦ ❞❡ ✉♠ ❝ír❝✉❧♦✱ ♦s ♣♦ssí✈❡✐s ✈❛❧♦r❡s ♣❛r❛ ❛ ár❡❛f(x) =πx2 ❡st❛rã♦ ❛ss♦❝✐❛❞♦s t❛♠❜é♠ à ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ρ(x) ♣❛r❛ ❛s ♠❡❞✐❞❛s ❞❡ x✳ ❆ss✐♠✱ ♦ ✈❛❧♦r ♠é❞✐♦ ✭_hf_i✮ ❞❡ q✉❛❧q✉❡r ❢✉♥çã♦ ✭f✮ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ xs❡rá ❞❛❞♦ ♣♦r

hf(x)_i=

Z

D

f(x)ρ(x)❞x

❡✱ ❛ r❡s♣❡❝t✐✈❛ ✈❛r✐â♥❝✐❛ ✭σf✮✱ ♣♦r

σ2_f =

Z

D

(f _{− h}f_i)2ρ(x)❞x

❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡①♣❡r✐♠❡♥t❛❧✱ ❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✉t✐❧✐③❛❞❛s ♥♦s ❧❛❜♦r❛tór✐♦s ♣❡❧♦s ❢ís✐❝♦s ❞❡✈❡r✐❛♠ s❡r ❞❡t❡r♠✐♥❛❞❛s ❛ ♣♦st❡r✐♦r✐✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❛❧❡❛t♦r✐❡❞❛❞❡ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ♠❡❞✐çã♦ ♥ã♦ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ♦✉ ♣r❡✈✐st❛ ♣♦r q✉❛❧q✉❡r t❡♦r✐❛✳ ❊♥tr❡t❛♥t♦✱ ❞❡✈✐❞♦ ❛♦ ♣ró♣r✐♦ ❣r❛✉ ❞❡ ❞❡s❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣r♦❝❡ss♦✱ ♣♦❞❡♠ s❡r ❡st❛❜❡❧❡❝✐❞❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛ ♣r✐♦r✐ ✭❇✐♥♦♠✐❛❧✱ ●❛✉ss ❡ P♦✐ss♦♥✮ ❬✶✱ ✷✱ ✷✵✱ ✷✶✱ ✷✷✱ ✷✼✱ ✸✻✱ ✹✷✱✹✹✱ ✹✺❪ tã♦ ❣❡r❛✐s q✉❡ s❡r✈❡♠ ❞❡ ❢✉♥❞❛♠❡♥t♦s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s ❞❡ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❡ ♣❛r❛ ❛ t❡♦r✐❛ ❞♦s ❡rr♦s✳

❆ ❞❡✜♥✐çã♦ ❛ ♣r✐♦r✐ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ é t❛♠❜é♠ ❛♠♣❧❛♠❡♥t❡ ✉s❛❞❛ ♥❛ ❢✉♥❞❛♠❡♥t❛çã♦ ❞❡ ❡st✉❞♦s t❡ór✐❝♦s✱ ♥♦s q✉❛✐s sã♦ ✉t✐❧✐③❛❞♦s ❝♦♥❝❡✐t♦s ♣r♦❜❛❜✐❧íst✐❝♦s✱ ❝♦♠♦ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛ ❬✹✱✶✺✱✷✵✱

•❈❛♣❛ •❱♦❧t❛ •❆♥t❡r✐♦r •Pró①✐♠❛ •❚❡❧❛ ❝❤❡✐❛ •Pá❣✳ ✷✸ •Ú❧t✐♠❛ •❙❛✐r

♠✉✐t❛s ♣❛rtí❝✉❧❛s q✉❛s❡✲✐♥❞❡♣❡♥❞❡♥t❡s✱ sã♦ ❞❡❞✉③✐❞❛s ❡ ♠♦str❛♠✲s❡ ❝♦♠♣❛tí✈❡✐s ❝♦♠ ♦s ❝♦♠♣♦rt❛♠❡♥t♦s ❡①♣❡r✐♠❡♥t❛✐s ♦❜s❡r✈❛❞♦s ❡♠ ❞✐✈❡rs♦s s✐st❡♠❛s ❝♦♠♦ ♦s ❣❛s❡s ♠♦❧❡❝✉❧❛r❡s ❛ ❜❛✐①❛ ♣r❡ssã♦✱ ♦s ❡❧étr♦♥s ❡♠ ♠❡t❛✐s ❡ s❡♠✐❝♦♥❞✉t♦r❡s ❡✱ ❛✐♥❞❛✱ ❡♠ ❞✐✈❡rs♦s ❢❡♥ô♠❡♥♦s✱ ❝♦♠♦ ❛ ✈❛r✐❛çã♦ ❞♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦ ❞♦s só❧✐❞♦s ❛ ❜❛✐①❛s t❡♠♣❡r❛t✉r❛s ❡ ❛ r❛❞✐❛çã♦ ❞❡ ❝♦r♣♦ ♥❡❣r♦✳

❯♠❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s ❣❡r❛❧ ❞❛ ▼❡❝â♥✐❝❛ ❊st❛tíst✐❝❛ ✭●✐❜❜s ✲ ✶✾✵✶✮ ❬✶✶✱✶✷✱✶✸✱✶✹✱✶✺✱✶✼✱✶✾✱✷✵✱✷✸✱✷✺✱

✷✾✱✸✺✱ ✸✻✱ ✸✼✱ ✹✸❪✱ ❛♦ ✉t✐❧✐③❛r✱ t❛♠❜é♠✱ ♦ ❝♦♥❝❡✐t♦ ❛ ♣r✐♦r✐ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ♣❡r♠✐t❡ ♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❣❡r❛✐s ✭♠✐❝r♦❝❛♥ô♥✐❝❛✱ ❝❛♥ô♥✐❝❛ ❡ ❣r❛♥✲❝❛♥ô♥✐❝❛✮ ♥ã♦ ❛ss♦❝✐❛❞❛s ❛♣❡♥❛s ❛ ❣❛s❡s✱ ♣♦✐s sã♦ ❡①♣r❡ss❛s ❡♠ t❡r♠♦s ❞❡ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s q✉❡ ❝❛r❛❝t❡r✐③❛♠ q✉❛❧q✉❡r s✐st❡♠❛ ❞❡ ♣❛rtí❝✉❧❛s✳ ◆❡ss❡ ❝❛s♦✱ ♣❛r❛ ❝❛❞❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛✱ ❛ ♣❛rt✐r ❞❛s ❢♦r♠❛s ❣❡r❛✐s ❞❡ss❛s ❞✐str✐❜✉✐çõ❡s✱ é ♣♦ssí✈❡❧ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡s♣❡❝í✜❝❛✳

❆té ♦ ♣r✐♠❡✐r♦ q✉❛rt♦ ❞♦ sé❝✉❧♦ ❳❳✱ ❛ ✉t✐❧✐③❛çã♦ ♥❛ ❋ís✐❝❛ ❞♦ ❝♦♥❝❡✐t♦ ❛ ♣r✐♦r✐ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❡st❛✈❛ ❛ss♦❝✐❛❞❛ ❛♣❡♥❛s à ✐♠♣♦ss✐❜✐❧✐❞❛❞❡ ♣rát✐❝❛ ❞❛ ❝❛r❛❝t❡r✐③❛çã♦ s✐♠✉❧tâ♥❡❛ ❞♦ ❡st❛❞♦ ❞❡ t♦❞❛s ❛s ♣❛rtí❝✉❧❛s ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳ ❖✉ s❡❥❛✱ ♦ ♣r♦❜❧❡♠❛ r❡s✉❧t❛✈❛ ❞❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦s s✐st❡♠❛s ♦❜s❡r✈❛❞♦s ❬✷✵✱✷✺✱✸✻❪✳