Duarte Nuno da Silva Sousa MESTRADO EM MATEMÁTICA
julho | 2016
A
classe de distribuições de Panjer e a
modelação do risco coletivo Duarte Nuno da Silva Sousa
A classe de distribuições de Panjer e a modelação do risco coletivo
DISSERTAÇÃO DE MESTRADO
DIMENSÕES: 45 X 29,7 cm
PAPEL: COUCHÊ MATE 350 GRAMAS
IMPRESSÃO: 4 CORES (CMYK)
ACABAMENTO: LAMINAÇÃO MATE
NOTA*
Caso a lombada tenha um tamanho inferior a 2 cm de largura, o logótipo institucional da UMa terá de rodar 90º , para que não perca a sua legibilidade|identidade.
Caso a lombada tenha menos de 1,5 cm até 0,7 cm de largura o laoyut da mesma passa a ser aquele que consta no lado direito da folha.
Nome do Projecto/Relatório/Dissertação de Mestrado e/ou T
ese de Doutoramento |
Nome
do
Autor
Duarte Nuno da Silva Sousa
MESTRADO EM MATEMÁTICA
A classe de distribuições de Panjer e a
modelação do risco coletivo
DISSERTAÇÃO DE MESTRADO
ORIENTADOR
Júri:
Doutora Ana Maria Cortesão Pais Figueira da Silva Abreu
– Professora Auxiliar da Universidade da Madeira
Doutor Sílvio Filipe Velosa
– Professor Auxiliar da Universidade da Madeira
Doutora Sandra Maria Freitas Mendonça
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❡❧❛❜♦r❛çã♦ ❡ ❝♦♥❝❧✉sã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ s❡♠ ❛ ❝♦❧❛❜♦r❛çã♦ ❞❛s ♣❡ss♦❛s ❝♦♠ q✉❡♠ ❝♦♠♣❛rt✐❧❤♦ ♦ ♠❡✉ ❞✐❛ ❛ ❞✐❛✳ ❊✱ ♣♦r ❡ss❛ r❛③ã♦✱ q✉❡r♦ ❛♣r♦✈❡✐t❛r ❡st❡ ❡s♣❛ç♦ ♣❛r❛ ❛❣r❛❞❡❝❡r ❛ t♦❞❛s ❛s ♣❡ss♦❛s q✉❡✱ ❞✐r❡t❛ ❡ ✐♥❞✐r❡t❛✲ ♠❡♥t❡✱ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝❤❡❣❛❞❛ ❛ ❜♦♠ ♣♦rt♦ ❞❡st❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❡ ❝♦♠♦ ♥ã♦ ♣♦❞❡r✐❛ ❞❡✐①❛r ❞❡ s❡r✱ q✉❡r♦ ❛❣r❛❞❡❝❡r à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ❛ Pr♦❢❡ss♦r❛ ❉r✳a ❙❛♥❞r❛ ▼❛r✐❛ ❋r❡✐t❛s ▼❡♥❞♦♥ç❛✳ ➱ ✐♠♣♦ssí✈❡❧
❡①♣r❡ss❛r ❡♠ ♣❛❧❛✈r❛s ❛ ❞❡❞✐❝❛çã♦✱ ❛ ♣❛✐①ã♦ ❡ ❛ ♣❛❝✐ê♥❝✐❛ q✉❡ ❞❡♣♦s✐t♦✉ ❡♠ ♠✐♠ ❡ ♥❡st❛ ❞✐ss❡rt❛çã♦✳ ■♥ú♠❡r❛s ❢♦r❛♠ ❛s ❤♦r❛s q✉❡ ❞❡s♣❡♥❞❡✉ ❛ tr❛♥s♠✐t✐r✲♠❡ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❛ ❛❝♦♥s❡❧❤❛r✲♠❡ ❡ ❛ ♠♦t✐✈❛r✲♠❡ ❞❡ ♠♦❞♦ ❛ ♥ã♦ ❜❛✐①❛r ♦s ❜r❛ç♦s✳ ❊st❡✈❡ s❡♠♣r❡ ❞✐s♣♦st❛ ❛ ❛❥✉❞❛r✱ ❛ ❛❜❞✐❝❛r ❞♦ s❡✉ t❡♠♣♦ ❡ ❛❝r❡❞✐t♦✉ s❡♠♣r❡ ♥♦ s✉❝❡ss♦ ❞❡st❛ ❡♠♣r❡✐t❛❞❛✱ ❛té ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ❡✉ ♣ró♣r✐♦ ❞✉✈✐❞❛✈❛✳ P♦r ❡st❛s ❡ ♣♦r ♠✉✐t❛s ♦✉tr❛s r❛③õ❡s ❡st♦✉✲❧❤❡ ❡t❡r♥❛♠❡♥t❡ ❣r❛t♦✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦s ♠❡✉s ♣❛✐s ❡st♦✉✲❧❤❡s ♠✉✐t♦ ❛❣r❛❞❡❝✐❞♦ ♣❡❧♦s s❛❝rí✜❝♦s ❡ ♣r✐✈❛çõ❡s q✉❡ s✉♣♦rt❛r❛♠ ❛♦ ❧♦♥❣♦ ❞❡st❡s ❛♥♦s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝♦♥t✐♥✉❛r ❛ ❡st✉❞❛r✳ ❖s s❛❝r✐❢í❝✐♦s ❞✐ár✐♦s✱ ♦s ❝♦♥s❡❧❤♦s ❡ ❛s ♣❛❧❛✈r❛s ❞❡ ♠♦t✐✈❛çã♦ ❞❡r❛♠✲♠❡ ❢♦rç❛ ♣❛r❛ ❛❝❛❜❛r ❡st❛ ❞✐ss❡rt❛çã♦ ❡ ❞❡ ❝❡rt❛ ❢♦r♠❛✱ ♣r♦✈❛r q✉❡ ♦s s❡✉s ❡s❢♦rç♦s ♥ã♦ ❢♦r❛♠ ❡♠ ✈ã♦✳ P♦r ✐ss♦✱ ❛❣r❛❞❡ç♦✲❧❤❡s ♠✉✐t♦✳
❆♦s ♠❡✉s q✉❡r✐❞♦s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s ❡st♦✉✲❧❤❡s t❛♠❜é♠ ♠✉✐t♦ ❣r❛t♦✳ ❆♦ ❆♥tó♥✐♦ ●♦♠❡s✿ ♣❡❧♦s ♠♦♠❡♥t♦s ❥♦✈✐❛✐s q✉❡ ♣❛ss❛♠♦s q✉❡r ♥❛s ❛✉❧❛s q✉❡r ♥❛s ❛t✐✈✐❞❛❞❡s ❡①tr❛❝✉rr✐❝✉❧❛r❡s✱ ♣❡❧❛ s✉❛ ♣r♦♥t❛ ❞✐s♣♦s✐çã♦ ♣❛r❛ ❛❥✉❞❛r ❡ ♣❡❧❛ s✉❛ ❛♠✐③❛❞❡✳ ❆♦ ■✈♦ ❋❡rr❡✐r❛✿ ♣❡❧❛ s✉❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ♣❛r❛ ♠❡ ❛❥✉❞❛r ❡ ♣❛r❛ ♠❡ ❡s❝❧❛r❡❝❡r ❞ú✈✐❞❛s✱ ♣❡❧❛ s✉❛ ❛♠✐③❛❞❡ ❡ ♣❡❧❛ s✉❛ ♣♦st✉r❛ sér✐❛ ❡ ❡♠♣❡♥❤❛❞❛ q✉❡ s❡♠♣r❡ ❛❞♦t♦✉ ❡♠ q✉❛❧q✉❡r s✐t✉❛çã♦✱ s❡r✈✐♥❞♦ ❝♦♠♦ ✉♠ ♠♦❞❡❧♦ ❛ s❡❣✉✐r ❡ ❢❛✲ ③❡♥❞♦ ❞❡ ♠✐♠ ✉♠❛ ♣❡ss♦❛ ♠❡❧❤♦r✳ ❆♦ ❱ít♦r ❏❡s✉s✿ ♣❡❧❛ s✉❛ ♣♦st✉r❛ tr❛♥q✉✐❧❛ ✭às ✈❡③❡s✱ ❡♠ ❞❡♠❛s✐❛✮ ❡♠ ♠♦♠❡♥t♦s ❞❡ ♣r❡ssã♦✱ s❡r✈✐♥❞♦✲♠❡ ❞❡ ✐♥s♣✐r❛çã♦✱ ♣❡❧❛ s✉❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ♣❛r❛ ♠❡ ❛✉①✐❧✐❛r ❡ ♣❡❧❛ s✉❛ ❛♠✐③❛❞❡✳ ❆♦s r❡st❛♥t❡s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✿ ♣❡❧♦s ♠♦♠❡♥t♦s ♣❛ss❛❞♦s✱ ♣❡❧♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ❡①♣❡r✐ê♥❝✐❛s q✉❡ ♠❡ tr❛♥s♠✐t✐r❛♠ ❡ ♣❡❧❛ ❛♠✐③❛❞❡✳
❆❣r❛❞❡ç♦ t❛♠❜é♠ às ♠✐♥❤❛s ❛♠✐❣❛s ❞❡ ❧♦♥❣❛ ❞❛t❛ ❉❡✐s❡ ❋❛r✐❛✱ ❊st❡❢â♥✐❛ ❘✐❜❡✐r♦✱ ❏❡♥♥② ●♦♥ç❛❧✈❡s ❡ ▲✐❧✐❛♥❛ ❋❛r✐❛✱ q✉❡ ❡♠❜♦r❛ ♥ã♦ s❡♥❞♦ ❞❛ ♠❡s♠❛ ár❡❛ ❡ ♥ã♦ ♣❛rt✐❧❤❛♥❞♦ ♠♦♠❡♥t♦s ❞✐ár✐♦s ❝♦♠✐❣♦✱ ❞❡♠♦♥str❛r❛♠ s❡♠♣r❡ ♣r♦♥t❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ♣❛r❛ ♠❡ ❛♣♦✐❛r✱ ❛❝♦♥s❡❧❤❛r ❡ ❛❥✉❞❛r ❡♠ q✉❛❧q✉❡r ❛ss✉♥t♦ ♦✉ s✐t✉❛çã♦✳
✐✈ ❆❣r❛❞❡❝✐♠❡♥t♦s
◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ♥❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛✲ ❞é♠✐❝❛ ♣❡❧♦ s❛❜❡r q✉❡ ♠❡ tr❛♥s♠✐t✐r❛♠✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❧✐❝❡♥❝✐❛t✉r❛✱ ❛❧❣✉♥s ❞❡❧❡s ♠❛r❝❛♥❞♦✲♠❡ ♣r♦❢✉♥❞❛♠❡♥t❡ ❡ ❢❛③❡♥❞♦ ❞❡ ♠✐♠ ✉♠❛ ♣❡ss♦❛ ♠❡❧❤♦r✳
❆♦s r❡st❛♥t❡s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s ❛❣r❛❞❡ç♦✲❧❤❡s t❛♠❜é♠ ♣❡❧♦s ❝♦♥s❡❧❤♦s tr❛♥s✲ ♠✐t✐❞♦s✱ ♣❡❧♦s ♠♦♠❡♥t♦s ✈✐✈❡♥❝✐❛❞♦s ❡ ♣♦r ❢❛③❡r❡♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
❘❡s✉♠♦
❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❛♥❛❧✐s❛r ❛ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ P❛♥❥❡r✱ ❛❧❣✉♠❛s ❞❛s s✉❛s ❡①t❡♥sõ❡s ❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡st❛s ♥❛ ♠♦❞❡❧❛çã♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳
■♥✐❝✐❛❧♠❡♥t❡✱ sã♦ r❡❢❡r✐❞♦s ❛❧❣✉♥s ♠♦♠❡♥t♦s ❤✐stór✐❝♦s ✐♠♣♦rt❛♥t❡s ♥♦ ❡st✉❞♦ ❞❛ ❢❛♠í❧✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ P❛♥❥❡r ❡ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ❛❧❣✉♥s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s ♥♦s ú❧t✐♠♦s ✈✐♥t❡ ❛♥♦s s♦❜r❡ ❡st❛s t❡♠át✐❝❛s✳
❉❡♣♦✐s✱ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ❝♦♥❝❡✐t♦s ❡ ✐♥str✉♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ♥❛ ❝♦♥s✲ tr✉çã♦ ❞❛s ❡①t❡♥sõ❡s ❞❛ ❢❛♠í❧✐❛ ❞❡ P❛♥❥❡r ❡ ♥❛ ❝♦♥str✉çã♦ r❡❝✉rs✐✈❛ ❞❛s ❞✐str✐❜✉✐✲ çõ❡s ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳ ❚❛✐s ❝♦♥❝❡✐t♦s ❡ ✐♥str✉♠❡♥t♦s ✐♥❝❧✉❡♠ ❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ❛ ♠✐st✉r❛ ❡ ♠♦❞✐✜❝❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
❙❡❣✉✐❞❛♠❡♥t❡✱ sã♦ ❝❛r❛❝t❡r✐③❛❞❛s ❛s ❞✐str✐❜✉✐çõ❡s ❞✐s❝r❡t❛s ♣❡rt❡♥❝❡♥t❡s à ❝❧❛ss❡ ❞❡ P❛♥❥❡r✱ ❞✐str✐❜✉✐çõ❡s ❡ss❛s ❞❡♥♦♠✐♥❛❞❛s ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ❝♦♥t❛❣❡♠ ❜ás✐❝❛s✱ ❞❡✜♥✐❞❛s ❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r ❡ ❞✉❛s s✉❛s ❡①t❡♥sõ❡s ❡ ❛♣r❡s❡♥t❛❞❛s ❛s ❞✐str✐❜✉✐çõ❡s ♣❡rt❡♥❝❡♥t❡s ❛ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✳
❋✐♥❛❧♠❡♥t❡✱ é ❛♣r❡s❡♥t❛❞♦ ♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ ❞❡s✐❣♥❛❞❛♠❡♥t❡ ♦ ♠♦✲ ❞❡❧♦ ❝♦♠♣♦st♦ ❞❛s ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s✱ ❝✉❥❛s ❞✐str✐❜✉✐çõ❡s✱ ♥❡st❡ ❝❛s♦✱ sã♦ ❝♦♥str✉í❞❛s ❛tr❛✈és ❞♦ ♠ét♦❞♦ r❡❝✉rs✐✈♦✳ ❙ã♦ t❛♠❜é♠ ❡①♣♦st♦s ❞♦✐s ♠ét♦❞♦s ❞❡ ❝♦♥str✉çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❛r✐t♠ét✐❝❛s✳ ❆ ❞✐ss❡rt❛çã♦ t❡r♠✐♥❛ ❝♦♠ ❛ ❞❡❞✉çã♦ ❞❡ ❛❧❣✉♥s ♠♦❞❡❧♦s ♣❛rt✐❝✉❧❛r❡s ♣❛r❛ ♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ ♦❜t✐❞♦s ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦s ♣r♦❣r❛♠❛s ✐♥❢♦r♠át✐❝♦s ▼❛t❤❡♠❛t✐❝❛ ❡ ❘✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿ ❋❛♠í❧✐❛ ❞❡ P❛♥❥❡r✱ ❘✐s❝♦ ❝♦❧❡t✐✈♦✱ ▼ét♦❞♦ r❡❝✉rs✐✈♦✳
❆❜str❛❝t
❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ ❛♥❛❧②s❡ t❤❡ P❛♥❥❡r✬s ❝❧❛ss ♦❢ ❞✐s✲ tr✐❜✉t✐♦♥s✱ s♦♠❡ ♦❢ ✐ts ❡①t❡♥s✐♦♥s ❛♥❞ t❤❡✐r ✉s❡ ✐♥ t❤❡ ♠♦❞❡❧❧✐♥❣ ♦❢ ❝♦❧❧❡❝t✐✈❡ r✐s❦✳
■♥✐t✐❛❧❧②✱ s♦♠❡ ✐♠♣♦rt❛♥t ❤✐st♦r✐❝❛❧ ❡✈❡♥ts ✐♥ t❤❡ st✉❞② ♦❢ t❤❡ P❛♥❥❡r✬s ❢❛♠✐❧② ♦❢ ❞✐str✐❜✉t✐♦♥s ❛♥❞ ✐♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❝♦❧❧❡❝t✐✈❡ r✐s❦ ♠♦❞❡❧ ❛r❡ r❡❢❡rr❡❞✳ ❙♦♠❡ ❛rt✐❝❧❡s ♣✉❜❧✐s❤❡❞ ✐♥ t❤❡ ❧❛st t✇❡♥t② ②❡❛rs ❛❜♦✉t t❤❡s❡ s✉❜❥❡❝ts ❛r❡ ❛❧s♦ ♣r❡s❡♥t❡❞✳
❚❤❡♥✱ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❝❡♣ts ❛♥❞ t♦♦❧s ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ P❛♥❥❡r✬s ❢❛♠✐❧② ❡①t❡♥s✐♦♥s ❛♥❞ ✐♥ t❤❡ r❡❝✉rs✐✈❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ ❝♦❧✲ ❧❡❝t✐✈❡ r✐s❦ ♠♦❞❡❧ ❛r❡ ♣r❡s❡♥t❡❞✳ ❙✉❝❤ ❝♦♥❝❡♣ts ❛♥❞ t♦♦❧s ✐♥❝❧✉❞❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥✱ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❛♥❞ t❤❡ ♠✐①t✉r❡ ❛♥❞ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✳
◆❡①t✱ t❤❡ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥s t❤❛t ❜❡❧♦♥❣ t♦ t❤❡ P❛♥❥❡r✬s ❝❧❛ss ♦❢ ❞✐str✐❜✉✲ t✐♦♥s✱ ✇❤✐❝❤ ❛r❡ r❡❢❡rr❡❞ ❛s ❜❛s✐❝ ❝♦✉♥t✐♥❣ ❞✐str✐❜✉t✐♦♥s✱ ❛r❡ ❝❤❛r❛❝t❡r✐s❡❞✱ t❤❡ P❛♥❥❡r✬s r❡❝✉rs✐♦♥ ❛♥❞ t✇♦ ♦❢ ✐ts ❡①t❡♥s✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❛♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥s ❜❡❧♦♥❣✐♥❣ t♦ ❡❛❝❤ ♦♥❡ ♦❢ t❤❡♠ ❛r❡ ♣r❡s❡♥t❡❞✳
❋✐♥❛❧❧②✱ t❤❡ ❝♦❧❧❡❝t✐✈❡ r✐s❦ ♠♦❞❡❧ ✐s ♣r❡s❡♥t❡❞✱ ♥❛♠❡❧② t❤❡ ❛❣❣r❡❣❛t❡ ❧♦ss ♠♦✲ ❞❡❧✱ ✇❤♦s❡ ❞✐str✐❜✉t✐♦♥s✱ ✐♥ t❤✐s ❝❛s❡✱ ❛r❡ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ t❤❡ r❡❝✉rs✐✈❡ ♠❡t❤♦❞✳ ❚✇♦ ♠❡t❤♦❞s ❢♦r ❝♦♥str✉❝t✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐str✐❜✉t✐♦♥s ❛r❡ ❛❧s♦ ❞❡s❝r✐❜❡❞✳ ❚❤❡ ❞✐ss❡rt❛t✐♦♥ ❡♥❞s ✇✐t❤ t❤❡ ❞❡❞✉❝t✐♦♥ ♦❢ s♦♠❡ ♣❛rt✐❝✉❧❛r ♠♦❞❡❧s ❢♦r t❤❡ ❝♦❧❧❡❝t✐✈❡ r✐s❦✱ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ t❤❡ ♣r♦❣r❛♠s ▼❛t❤❡♠❛t✐❝❛ ❛♥❞ ❘✳
❑❡②✇♦r❞s✿ P❛♥❥❡r✬s ❢❛♠✐❧②✱ ❈♦❧❧❡❝t✐✈❡ r✐s❦✱ ❘❡❝✉rs✐✈❡ ♠❡t❤♦❞✳
❮♥❞✐❝❡
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✈
◆♦t❛çã♦ ①✈✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✺
✷✳✶ ❆ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸ ▼♦❞✐✜❝❛çã♦ ❡♠ ③❡r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✸ ❉✐str✐❜✉✐çõ❡s ❞❡ ❝♦♥t❛❣❡♠ ❜ás✐❝❛s ✶✼
✸✳✶ ❆ ❞✐str✐❜✉✐çã♦ ❜✐♥♦♠✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✷ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ P♦✐ss♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸ ❆ ❞✐str✐❜✉✐çã♦ ❜✐♥♦♠✐❛❧ ♥❡❣❛t✐✈❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✹ ❆ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ P❛♥❥❡r ❡ s✉❛s ❡①t❡♥sõ❡s ✷✼
✹✳✶ ❆ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ P❛♥❥❡r ♦✉(a, b,0) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✷ ❆ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s (a, b,1) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✸ ❆ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s (a, b, m)✱ m∈N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹✳✹ ❖✉tr❛s ❡①t❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✺ ❆ ♠♦❞❡❧❛çã♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ ✹✾
✺✳✶ ❖ ♠♦❞❡❧♦ ❝♦♠♣♦st♦ ❞❛s ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✷ ▼ét♦❞♦ r❡❝✉rs✐✈♦ ♥❛ ❝♦♥str✉çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞❛s ✐♥❞❡♠♥✐③❛çõ❡s
❛❣r❡❣❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✷✳✶ ◗✉❛♥❞♦ ♦ ✈❛❧♦r ❞❛ ✐♥❞❡♠♥✐③❛çã♦ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛
❞✐s❝r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✳✷✳✷ ◗✉❛♥❞♦ ♦ ✈❛❧♦r ❞❛ ✐♥❞❡♠♥✐③❛çã♦ é ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛
❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✺✳✷✳✸ ●❡♥❡r❛❧✐③❛çã♦ ❛ ❞✐str✐❜✉✐çõ❡s ♣❛r❛ ❛❧é♠ ❞❛ ❝❧❛ss❡(a, b, m)✱
m ∈N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
① ❮♥❞✐❝❡
✺✳✸ ❈♦♥str✉çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❛r✐t♠ét✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✺✳✸✳✶ ▼ét♦❞♦ ❞❡ ❛rr❡❞♦♥❞❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✺✳✸✳✷ ▼ét♦❞♦ ❞❡ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❧♦❝❛❧ ❞❡ ♠♦♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✻ ❖ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ ♥♦s ♣r♦❣r❛♠❛s ▼❛t❤❡♠❛t✐❝❛ ❡ ❘ ✼✾
✻✳✶ ❊①❡♠♣❧♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✻✳✷ ❊①❡♠♣❧♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✻✳✸ ❊①❡♠♣❧♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽
✼ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✾✸
❇✐❜❧✐♦❣r❛✜❛ ✾✺
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ N ❝♦♠ ❞✐str✐❜✉✐çã♦
Binomial(m, p) ✭♥❡st❡ ❝❛s♦✱ m= 10 ❡ p= 0.5✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✸✳✷ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ N ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡
P oisson(λ) ✭♥❡st❡ ❝❛s♦✱λ = 10✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✸ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ N ❝♦♠ ❞✐str✐❜✉✐çã♦
BinomialN egativa(r, p) ✭♥❡st❡ ❝❛s♦✱ r= 10 ❡ p= 0.5✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✹✳✶ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ❝♦♠ ❞✐str✐❜✉✐çã♦
BN T G(r, p) ✭♥❡st❡ ❝❛s♦✱r =−0.2 ❡p= 0.3✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹✳✷ ❱❛r✐❛çã♦ ❞❡ δ(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ p✱ q✉❛♥❞♦
X ⌢ BN T G(1, p)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✹✳✸ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ❝♦♠ ❞✐str✐❜✉✐çã♦ Log(p) ✭♥❡st❡ ❝❛s♦✱ p= 0.2✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✹ ❱❛r✐❛çã♦ ❞❡ δ(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡p✱ q✉❛♥❞♦ X ⌢ Log(p)✳✳ ✸✻
✹✳✺ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ❝♦♠ ❞✐str✐❜✉✐çã♦
BN E(l, r, p)✭♥❡st❡ ❝❛s♦✱ l = 5✱ r=−4.01❡ p= 0.1✮✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✻ ❱❛r✐❛çã♦ ❞❡ E(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ p✱ q✉❛♥❞♦
X ⌢ BN E(l,−l+ 0.5, p)✱ ♣❛r❛ l = 1 ❡ l= 11✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✹✳✼ ❱❛r✐❛çã♦ ❞❡ V ar(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ p✱ q✉❛♥❞♦
X ⌢ BN E(l,−l+ 0.5, p)✱ ♣❛r❛ l = 1 ❡ l= 11✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✽ ❱❛r✐❛çã♦ ❞❡ δ(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ p✱ q✉❛♥❞♦
X ⌢ BN E(l,−l+ 0.5, p)✱ ♣❛r❛ l = 1 ❡ l= 11✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✾ ❊①❡♠♣❧♦ ❞❡ ✉♠❛ ❢✳♠✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ❝♦♠ ❞✐str✐❜✉✐çã♦
LogE(l, p)✭♥❡st❡ ❝❛s♦✱ l = 4 ❡ p= 0.01✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✹✳✶✵ ❱❛r✐❛çã♦ ❞❡ E(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ p✱ q✉❛♥❞♦
X ⌢ LogE(l, p)✱ ♣❛r❛ l= 2 ❡l = 12✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹✳✶✶ ❱❛r✐❛çã♦ ❞❡ V ar(X) ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ p✱ q✉❛♥❞♦
X ⌢ LogE(l, p)✱ ♣❛r❛ l= 2 ❡l = 12✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✹✳✶✷ ❱❛r✐❛çã♦ ❞❡δ(X)❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡p✱ q✉❛♥❞♦X ⌢ LogE(l, p)✱ ♣❛r❛ l = 2 ❡ l= 12✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✶✸ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s (a, b, j)✱ j ♥❛t✉r❛❧✱ ♥♦
♣❧❛♥♦ (a, b) ✭❝♦♥s✐❞❡r❛♥❞♦ ♦s ♣❛râ♠❡tr♦s ✉t✐❧✐③❛❞♦s ♥❛s ❚❛❜❡❧❛s ✹✳✶✱ ✹✳✷ ❡ ✹✳✸✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
①✐✐ ▲✐st❛ ❞❡ ❋✐❣✉r❛s
✺✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✳❞✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ⌢ Exponencial(1) ❡ ❞❛ ❢✳♠✳♣✳ ❞❛ s✉❛ ❞✐s❝r❡t✐③❛çã♦ ♣❡❧♦ ♠ét♦❞♦ ❞♦ ❛rr❡❞♦♥❞❛♠❡♥t♦ ❝♦♠ h= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✺✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✳❞✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ⌢ Exponencial(1) ❡ ❞❛
❢✳♠✳♣✳ ❞❛ s✉❛ ❞✐s❝r❡t✐③❛çã♦ ♣❡❧♦ ♠ét♦❞♦ ❞♦ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ❧♦❝❛❧ ❞❡ ♠♦♠❡♥t♦s ❝♦♠ h= 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✻✳✶ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❞❡✜♥❡♠ ❛ ❢✳♠✳♣✳ ❡ ❛
❢✳❣✳♣✳ ❞❛ ✈✳❛✳N ⌢ P oisson(la)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵
✻✳✷ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❣❡r❛♠ ❛ ❢✳♠✳♣✳ ❞❡ X✳ ✳ ✽✶
✻✳✸ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❞❡✜♥❡♠ ❛ ❢✳♠✳♣✳ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱S✱ q✉❛♥❞♦ N ⌢ P oisson(10)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✻✳✹ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✳♠✳♣✳ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦
N ⌢ P oisson(10) ❡ ❝♦♠❛♥❞♦s q✉❡ ❡stã♦ ♥❛ s✉❛ ♦r✐❣❡♠ ♥♦ ♣r♦✲
❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✻✳✺ Pr♦❜❛❜✐❧✐❞❛❞❡s✱ Pr (S=k)✱ ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛❝✉♠✉❧❛❞❛s✱
Pr (S ≤k)✱ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦
N ⌢ P oisson(10) ❡k = 0,1, ...,27✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
✻✳✻ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ❘ q✉❡ ♣❡r♠✐t❡♠ ❛ ❝r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦N ⌢ P oisson(10)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸
✻✳✼ ❱❛❧♦r❡s ❞❡ Pr (S ≤k)✱ ❝♦♠ k = 0,1, ...,27✱ q✉❛♥❞♦
N ⌢ P oisson(10)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸
✻✳✽ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡✲ t✐✈♦✱ S✱ q✉❛♥❞♦ N ⌢ P oisson(10) ❡ ❝♦♠❛♥❞♦s q✉❡ ❡stã♦ ♥❛ s✉❛ ♦r✐❣❡♠ ♥♦ ♣r♦❣r❛♠❛ ❘✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✻✳✾ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❞❡✜♥❡♠ ❛ ❢✳♠✳♣✳ ❡ ❛
❢✳❣✳♣✳ ❞❛ ✈✳❛✳N ⌢ Log(p)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✻✳✶✵ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❞❡✜♥❡♠ ❛ ❢✳♠✳♣✳ ❞♦
♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱S✱ q✉❛♥❞♦ N ⌢ Log(0.2)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✻✳✶✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✳♠✳♣✳ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦
N ⌢ Log(0.2)❡ ❝♦♠❛♥❞♦s q✉❡ ❡stã♦ ♥❛ s✉❛ ♦r✐❣❡♠ ♥♦ ♣r♦❣r❛♠❛
▼❛t❤❡♠❛t✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✻✳✶✷ Pr♦❜❛❜✐❧✐❞❛❞❡s✱ Pr (S=k)✱ ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛❝✉♠✉❧❛❞❛s✱
Pr (S ≤k)✱ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱S✱ q✉❛♥❞♦N ⌢ Log(0.2)
❡k = 0,1, ...,15✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻
✻✳✶✸ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ❘ q✉❡ ♣❡r♠✐t❡♠ ❛ ❝r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦N ⌢ Log(0.2). ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✻✳✶✹ ❊rr♦ ❞❛ ❢✉♥çã♦ ❛❣❣r❡❣❛t❡❉✐st ❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ s♦❧✉çã♦ ♣❛r❛ ❡ss❡
♠❡s♠♦ ❡rr♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ✻✳✶✺ ❱❛❧♦r❡s ❞❡ Pr (S ≤k)✱ ❝♦♠ k = 0,1, ...,15✱ q✉❛♥❞♦ N ⌢ Log(0.2)✳ ✽✼ ✻✳✶✻ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦✲
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐✐✐
✻✳✶✼ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❞❡✜♥❡♠ ❛ ❢✳♠✳♣✳ ❡ ❛ ❢✳❣✳♣✳ ❞❛ ✈✳❛✳ N ⌢ Binomial(m, p)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✻✳✶✽ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❣❡r❛♠ ❛ ❢✳♠✳♣✳ ❞❡X✳ ✳ ✽✾ ✻✳✶✾ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛ q✉❡ ❞❡✜♥❡♠ ❛ ❢✳♠✳♣✳ ❞♦
♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦N ⌢ Binomial(10,0.5)✳ ✳ ✳ ✳ ✽✾ ✻✳✷✵ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❢✳♠✳♣✳ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦
N ⌢ Binomial(10,0.5) ❡ ❝♦♠❛♥❞♦s q✉❡ ❡stã♦ ♥❛ s✉❛ ♦r✐❣❡♠ ♥♦
♣r♦❣r❛♠❛ ▼❛t❤❡♠❛t✐❝❛✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ✻✳✷✶ Pr♦❜❛❜✐❧✐❞❛❞❡s✱ Pr (S =k)✱ ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❛❝✉♠✉❧❛❞❛s✱
Pr (S ≤k)✱ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦
N ⌢ Binomial(10,0.5)❡ k = 0,1, ...,17✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵
✻✳✷✷ ❈♦♠❛♥❞♦s ♥♦ ♣r♦❣r❛♠❛ ❘ q✉❡ ♣❡r♠✐t❡♠ ❛ ❝r✐❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ S✱ q✉❛♥❞♦ N ⌢ Binomial(10,0.5)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✻✳✷✸ ❱❛❧♦r❡s ❞❡ Pr (S ≤k)✱ ❝♦♠ k = 0,1, ...,17✱ q✉❛♥❞♦
N ⌢ Binomial(10,0.5)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✹✳✶ ▼❡♠❜r♦s ❞❛ ❝❧❛ss❡ ❞❡ P❛♥❥❡r ♦✉(a, b,0)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✷ ▼❡♠❜r♦s ❞❛ ❝❧❛ss❡ (a, b,1)✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✸ ▼❡♠❜r♦s ❞❛ ❝❧❛ss❡(a, b, m)♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛(a, b,0) (m ≥l≥1)✳ ✹✺
◆♦t❛çã♦
a b
Γ(a+1)
Γ(b+1)Γ(a−b+1)✱a ≥b≥0✳
❇◆❊(l, r, p) ❉✐str✐❜✉✐çã♦ ❜✐♥♦♠✐❛❧ ♥❡❣❛t✐✈❛ ❡st❡♥❞✐❞❛ ❝♦♠ ♣❛râ♠❡✲
tr♦s l∈N1✱ r∈(−l,−l+ 1) ❡ p∈(0,1)✳
❇◆❚●(r, p) ❉✐str✐❜✉✐çã♦ ❜✐♥♦♠✐❛❧ ♥❡❣❛t✐✈❛ tr✉♥❝❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛
❝♦♠ ♣❛râ♠❡tr♦sr >−1, r6= 0 ❡p∈(0,1)✳
Dist1≡Dist2 ❆s ❞✐str✐❜✉✐çõ❡s Dist1 ❡ Dist2 sã♦ ✐❣✉❛✐s✳
δ(N) ❮♥❞✐❝❡ ❞❡ ❞✐s♣❡rsã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛N✳
E[f(N)] =EN[f(N)] ❱❛❧♦r ❡s♣❡r❛❞♦ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ f(N)✱ s❡♥❞♦ N
✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛✳
❢✳❞✳♣✳ ❋✉♥çã♦✭õ❡s✮ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
❢✳❣✳♣✳ ❋✉♥çã♦✭õ❡s✮ ❣❡r❛❞♦r❛✭s✮ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✳
❢✳♠✳♣✳ ❋✉♥çã♦✭õ❡s✮ ♠❛ss❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
FX ❋✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ✈❛r✐á✈❡❧
❛❧❡❛tór✐❛ X✳
fX ❋✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tó✲
r✐❛ X✳
Γ ❋✉♥çã♦ ❣❛♠❛✳
✐✳✐✳❞✳ ■♥❞❡♣❡♥❞❡♥t❡s ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛s✳
LX ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦✲
❜❛❜✐❧✐❞❛❞❡ fX✱ s❡♥❞♦ X ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❛❜s♦✲
❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛✱ ♥ã♦ ♥❡❣❛t✐✈❛✳
▲♦❣(p) ❉✐str✐❜✉✐çã♦ ❧♦❣❛rít♠✐❝❛ ❝♦♠ ♣❛râ♠❡tr♦p∈(0,1)✳
①✈✐✐✐ ◆♦t❛çã♦
▲♦❣❊(l, p) ❉✐str✐❜✉✐çã♦ ❧♦❣❛rít♠✐❝❛ ❡st❡♥❞✐❞❛ ❝♦♠ ♣❛râ♠❡tr♦sl =
1,2,3, . . . ❡ p∈(0,1)✳
N={0,1,2, ...} ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
N1 ={1,2,3, ...} ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦s✐t✐✈♦s✳
PN ❋✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tó✲
r✐❛ N✳
Pr (A) Pr♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ❛❝♦♥t❡❝✐♠❡♥t♦ A✳
R ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
Sn
n P
i=1
Xi✱ s❡♥❞♦ X1, ..., Xn ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s✳
supA ❙✉♣r❡♠♦ ❞♦ ❝♦♥❥✉♥t♦ A✳
SX ❙✉♣♦rt❡ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ X✳
❚✳P✳❚✳ ❚❡♦r❡♠❛ ❞❛ Pr♦❜❛❜✐❧✐❞❛❞❡ ❚♦t❛❧✳
✈✳❛✳ ❱❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ✭✏✈✳❛✳✬s✑ ♥♦ ♣❧✉r❛❧✮✳
V ar(N) ❱❛r✐â♥❝✐❛ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ N✳
[x] P❛rt❡ ✐♥t❡✐r❛ ❞❡ x✳
X ⌢ Dist(parˆam) X t❡♠ ❞✐str✐❜✉✐çã♦ ✏❉✐st✑ ❝♦♠ ♣❛râ♠❡tr♦s ✏♣❛râ♠✑✳
XM ❱❛r✐á✈❡❧ ❛❧❡❛tór✐❛ q✉❡ r❡s✉❧t❛ ❞❛ ♠♦❞✐✜❝❛çã♦ ♥♦ ♣♦♥t♦
0 ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞✐s❝r❡t❛X✳
XT ❱❛r✐á✈❡❧ ❛❧❡❛tór✐❛ q✉❡ r❡s✉❧t❛ ❞❛ tr✉♥❝❛t✉r❛ ❛♦ ❝♦♥✲
❥✉♥t♦ N1 ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ ❞✐s❝r❡t❛X✳
X =d Y X ❡Y tê♠ ❛ ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✳
(#)
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❖ r✐s❝♦ ❞❡ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❛❝♦♥t❡❝✐♠❡♥t♦ ❛❧❡❛tór✐♦ ❡ ♠❡♥s✉rá✈❡❧ ♠♦♥❡t❛r✐❛✲ ♠❡♥t❡ é ✉♠❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♥st❛♥t❡ ❞❛s s❡❣✉r❛❞♦r❛s ✭♦✉ ❝♦♠♣❛♥❤✐❛s ❞❡ s❡❣✉r♦s✮✳ ❚❛❧ ♦❝♦rrê♥❝✐❛ é ✐♠♣♦ssí✈❡❧ ❞❡ ♣r❡✈❡r✱ ❝♦♥t✉❞♦ ❡st❛ ❛❢❡t❛✱ ❡ ♠✉✐t♦✱ ❛ ❢♦r♠❛ ❞❡ ♦♣❡r❛r ❞❛s s❡❣✉r❛❞♦r❛s✱ ♥♦♠❡❛❞❛♠❡♥t❡ ♥❛ q✉❛♥t✐❞❛❞❡ ❡ ✈❛❧♦r ❞❡ ✐♥❞❡♠♥✐③❛çõ❡s ❛ ♣❛❣❛r✳ P♦r t❛❧ ❢❛❝t♦✱ ❛q✉❛♥❞♦ ❞❛ ❝♦♥❝❡çã♦ ❞♦s ❝♦♥tr❛t♦s ❞❡ s❡❣✉r♦s ❡♥tr❡ ❛ s❡❣✉r❛❞♦r❛ ❡ ♦s s❡❣✉r❛❞♦s✱ ❡st❛s tê♠ ❞❡ ✐♥❝❧✉✐r ♥♦ ❝♦♥tr❛t♦ ✉♠❛ ❝♦♠♣❡♥s❛çã♦ ✜♥❛♥❝❡✐r❛ ✭♣ré♠✐♦ ❞❡ s❡❣✉r♦✮ q✉❡ ❛s ♠♦t✐✈❡ ❛ ❝♦♥t✐♥✉❛r ❛ ♦♣❡r❛r ❡ ♥ã♦ ❛❜r✐r ❢❛❧ê♥❝✐❛✳ ❆♦ ♠❡s♠♦ t❡♠♣♦✱ ♦ ❞✐t♦ ❝♦♥tr❛t♦ t❡♠ ❞❡ s❡r ❝♦♠♣❡t✐t✐✈♦ ❛ ♥í✈❡❧ ❞❡ ♣r❡ç♦s ❡ ❝♦♥❞✐çõ❡s ♦❢❡r❡❝✐❞❛s✱ ❞❡ ♠♦❞♦ ❛ ❛tr❛✐r ♦ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ s❡❣✉r❛❞♦s✳ P❡r❛♥t❡ t❛✐s ✈❛r✐á✈❡✐s✱ t♦r♥❛✲s❡ ♥✉♠ ❞❡s❛✜♦ ❛❧❝❛♥ç❛r ✉♠ ❝♦♥tr❛t♦ q✉❡ s❛t✐s❢❛ç❛ t♦❞❛s ❛s ♣❛rt❡s ❞♦ ❛❝♦r❞♦✳ ▼♦❞❡❧♦s ♠❛t❡♠át✐❝♦s tê♠ s✐❞♦ ❝♦♥str✉í❞♦s ❡ ❡st✉✲ ❞❛❞♦s ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ tr❛t❛r ♦ r✐s❝♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s ♥ã♦ ✐♠♣❧✐❝❛ q✉❡ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡❧♦s ♠♦❞❡❧♦s ❝♦✐♥❝✐❞❛♠ ❝♦♠ ♦s ✈❡r✐✜❝❛❞♦s ♥❛ r❡❛❧✐❞❛❞❡ ✭❞✐✜❝✐❧♠❡♥t❡ ❝♦✐♥❝✐❞❡♠✮✱ ✉♠❛ ✈❡③ q✉❡ ❛s s❡❣✉r❛❞♦✲ r❛s ♦♣❡r❛♠ ❡♠ ❛♠❜✐❡♥t❡s ❞✐♥â♠✐❝♦s✱ ❝♦♠♣❧❡①♦s✱ ✐♥❝♦♥tr♦❧á✈❡✐s ❡ ✐♠♣r❡✈✐sí✈❡✐s✳ P♦ré♠✱ ♦ s✉r❣✐♠❡♥t♦ ❞❡ t❛✐s ♠♦❞❡❧♦s s♦❜r❡ ♦ r✐s❝♦ tê♠ ❝♦♥tr✐❜✉í❞♦ ♣❛r❛ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ❞❛ ár❡❛ ❛t✉❛r✐❛❧ ❡ ❞♦s s❡❣✉r♦s ✭❝❢✳✱ ❡✳❣✳✱ ❬✷❪✮✳
❆ ♠♦❞❡❧❛çã♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ é ✉♠ ❞♦s t❡♠❛s ♣r✐♥❝✐♣❛✐s ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ❆ s✉❛ ✐♥tr♦❞✉çã♦ ❞❡✉✲s❡ ❡♠ ✶✾✵✸ ♣♦r ▲✉♥❞❜❡r❣ ✭❝❢✳ ❬✸✾❪ ❡ ❬✶❪✮ ♥❛ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛♠❡♥t♦ ✭❝❢✳ ❬✸✼❪✮✳ ❍❛r❛❧❞ ❈r❛♠ér ♣✉❜❧✐❝♦✉ ✉♠❛ r❡✈✐sã♦ ❞♦ tr❛❜❛❧❤♦ ❞❡ ▲✉♥❞❜❡r❣ ❡♠ ✶✾✷✻ ✭❝❢✳✱ ❬✼❪ ❡ ❬✸✾❪✮ ❡ ❡♠ ✶✾✸✵ ✐♥tr♦❞✉③✐✉ ♥♦✈♦s ❞❡s❡♥✈♦❧✈✐♠❡♥t♦s s♦❜r❡ ❡st❛ t❡♠át✐❝❛ ✭❝❢✳ ❬✽❪ ❡ ❬✹❪✮✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❇❧♦♠✱ ❛♥t✐❣♦ ❛❧✉♥♦ ❞❡ ❈r❛♠ér ✭❝❢✳ ❬✹❪✮✱ três ❞♦s tr❛❜❛❧❤♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡ ❈r❛♠ér ♥❛ ár❡❛ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s ❡♠ ✶✾✸✵ ✭❝❢✳ ❬✽❪✮✱ ✶✾✺✹ ✭❝❢✳ ❬✾❪✮ ❡ ✶✾✺✺ ✭❝❢✳ ❬✶✵❪✮✳
✷ ✶✳ ■♥tr♦❞✉çã♦
❖ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ t❡♠ ❢♦♠❡♥t❛❞♦ ❡st✉❞♦s ❡ ❞❡s❝♦❜❡rt❛s ❞❡ r❡s✉❧t❛❞♦s r❡❧❡✈❛♥t❡s ♣❛r❛ ❛❧é♠ ❞❛ ♣ró♣r✐❛ ár❡❛✳ ❯♠ ❡①❡♠♣❧♦ ❡♠ ❝♦♥❝r❡t♦ é ❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r✱ ✐♥tr♦❞✉③✐❞❛ ♥❛ ❞é❝❛❞❛ ❞❡ ✽✵ ❞♦ sé❝✉❧♦ ♣❛ss❛❞♦✱ s❡♥❞♦ ❡st❛ t❛♠❜é♠ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ❢♦❝♦s ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ❖ ♠♦❞❡❧♦ ❞❡ P❛♥❥❡r é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞✐s❝r❡t❛s q✉❡ é ❞❡✜♥✐❞❛ r❡❝✉rs✐✈❛♠❡♥t❡ ❡ q✉❡ ❛♣❛r❡❝❡✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ ✶✾✽✶ ✭❝❢✳ ❬✹✶❪✮✳ ❊st❛ ❢❛♠í❧✐❛ ❢♦✐✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ❣❡♥❡r❛❧✐③❛❞❛ ♣♦r ✈ár✐♦s ❛✉t♦r❡s ❝♦♠♦ ❙✉♥❞t ❡ ❏❡✇❡❧❧ ✭❝❢✳ ❬✺✸❪✮✱ P❛♥❥❡r ❡ ❲✐❧❧♠♦t ✭❝❢✳ ❬✹✹❪✮✱ ❙❝❤röt❡r ✭❝❢✳ ❬✹✾❪✮✱ ❙✉♥❞t ✭❝❢✳ ❬✺✶❪✮✱ ❍❡ss❡❧❛❣❡r ✭❝❢✳ ❬✸✷❪✮✱ ❍❡ss ❡t ❛❧✳ ✭❝❢✳ ❬✸✶❪✮ ❡ P❡st❛♥❛ ❡ ❱❡❧♦s❛ ✭❝❢✳ ❬✹✼❪✮✳
❙✉♥❞t✱ ❡♠ ✷✵✵✷ ✭❝❢✳ ❬✺✷❪✮✱ ♥✉♠ ❛rt✐❣♦ ❞❡ r❡✈✐sã♦ ✐♥❝♦♥t♦r♥á✈❡❧✱ ❡♥✉♠❡r❛ ♦s ♠ét♦❞♦s r❡❝✉rs✐✈♦s ♣r❡s❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛ ♣❛r❛ ✉♠❛ ❛✈❛❧✐❛çã♦ ❡①❛t❛ ❡ ❛♣r♦①✐✲ ♠❛❞❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s ✉♥✐✈❛r✐❛❞❛s ❡ ♠✉❧t✐✈❛r✐❛❞❛s✳ ◆❛s ❞✐str✐❜✉✐çõ❡s ❞❡ ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s ✉♥✐✈❛r✐❛❞❛s✱ ❙✉♥❞t ❝♦♥s✐❞❡r❛ ❞✐s✲ tr✐❜✉✐çõ❡s q✉❡ tê♠ ❝♦♠♦ s✉♣♦rt❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦✲♥❡❣❛t✐✈♦s ❡ ❝♦♥s✐❞❡r❛ t❛♠❜é♠ ❞✐str✐❜✉✐çõ❡s q✉❡ ♣❡r♠✐t❡♠ ✐♥❞❡♠♥✐③❛çõ❡s ❞❡ ✈❛❧♦r ♥❡❣❛✲ t✐✈♦✳ ❆❧é♠ ❞❡ ❛♣r❡s❡♥t❛r ❛s r❡❝✉rsõ❡s ❝♦♠ ♠❛✐s r❡♥♦♠❡ ❝♦♠♦ ❛ ❞❡ P❛♥❥❡r ❡ s✉❛s ❣❡♥❡r❛❧✐③❛çõ❡s✱ ❙✉♥❞t ❛♣r❡s❡♥t❛ t❛♠❜é♠ r❡❝✉rsõ❡s ♣❛r❛ ♠♦❞❡❧♦s ✐♥❞✐✈✐❞✉✲ ❛✐s✱ ❝♦♥s✐❞❡r❛♥❞♦ ♠ét♦❞♦s ❡①❛t♦s ❡ ♠ét♦❞♦s ❜❛s❡❛❞♦s ❡♠ ❛♣r♦①✐♠❛çõ❡s ✭❢♦r♥❡✲ ❝❡♥❞♦ t❛♠❜é♠ ❢ór♠✉❧❛s q✉❡ ❞❡✜♥❡♠ ❛s ❜❛rr❡✐r❛s ❞❡ ❡rr♦ ❞❡ss❛s ❛♣r♦①✐♠❛çõ❡s✮✳ ❖ ❛✉t♦r ❛❜♦r❞❛ ❛✐♥❞❛ ❛ ❡st❛❜✐❧✐❞❛❞❡✱ ❛ s♦❜r❡st✐♠❛çã♦ ❡ ❛ s✉❜❡st✐♠❛çã♦ ❞♦s r❡✲ s✉❧t❛❞♦s ♥✉♠ér✐❝♦s✳ ❆ ♣❛rt❡ ✜♥❛❧ ❞♦ s❡✉ tr❛❜❛❧❤♦ é ❞❡❞✐❝❛❞❛ às ❞✐str✐❜✉✐çõ❡s ❞❡ ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s ♠✉❧t✐✈❛r✐❛❞❛s✳
◆♦ sé❝✉❧♦ ❳❳■✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s r❛♠✐✜❝❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ ❡ ❞❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r ✭❝❢✳✱ ❡✳❣✳✱ ❬✺✹❪ ❡ ❬✹✽❪✮✳ ❙❡❣✉✐❞❛♠❡♥t❡ sã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s tr❛❜❛❧❤♦s r❡❝❡♥t❡s ♥❡st❛s ár❡❛s✳
✶✳ ■♥tr♦❞✉çã♦ ✸
❊①✐st❡♠✱ ♦❜✈✐❛♠❡♥t❡✱ ♦✉tr♦s ❛✉t♦r❡s q✉❡ r❡❛❧✐③❛♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ✐♥✈❡s✲ t✐❣❛çã♦ ❛trás r❡❢❡r✐❞♦✳ P♦r ❡①❡♠♣❧♦✱ ●ó♠❡③✲❉é♥✐③ ❡ ❈❛❧❞❡rí♥✲❖❥❡❞❛ ✭❝❢✳ ❬✷✸❪✮ ❞❡❞✉③❡♠ ❛ ❞✐str✐❜✉✐çã♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ q✉❛♥❞♦ ♦ ✈❛❧♦r ❞❛s ✐♥❞❡♠✲ ♥✐③❛çõ❡s t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❊r❧❛♥❣ ❡ ❛s ✐♥❞❡♠♥✐③❛çõ❡s ♦❝♦rr✐❞❛s ♣♦ss✉❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡ ▲✐♥❞❧❡② ❞✐s❝r❡t❛ ❣❡♥❡r❛❧✐③❛❞❛✳
◆♦t❡✲s❡ q✉❡ ♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ ♣♦❞❡ s❡r ✉s❛❞♦ ❡♠ ♦✉tr❛s ár❡❛s✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❡st✉❞♦ ❞♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ♣♦♣✉❧❛çõ❡s ✭❝❢✳✱ ❡✳❣✱ ❇r✐❧❤❛♥t❡ ❡t ❛❧✳ ❬✺❪✮ ❡ ♥♦ ❡st✉❞♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞♦ ♠á①✐♠♦ ❛❧❡❛tór✐♦ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s✱ ♦♥❞❡ ♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❛ ❢❛♠í❧✐❛ ❞❡ P❛♥❥❡r ✭❝❢✳✱ ❡✳❣✳✱ ▼❡♥❞♦♥ç❛ ❡t ❛❧✳ ❬✹✵❪✮✳
❆ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❡①❛t❛ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦ ♥ã♦ é s❡♠♣r❡ ♣♦ssí✈❡❧✱ ❞❛❞❛ ❛ s✉❛ ❝♦♠♣❧❡①✐❞❛❞❡✱ s❡♥❞♦✱ ♣♦r ✈❡③❡s✱ ♥❡❝❡ssár✐♦ r❡❝♦rr❡r ❛ ♠ét♦❞♦s ♥✉♠é✲ r✐❝♦s✳ ❊st❡ t❡♠❛ ♥ã♦ s❡rá ❛❜♦r❞❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ♠❛s ✈❛❧❡ ❛ ♣❡♥❛ r❡❢❡r✐✲❧♦ ♣❡❧♦ s❡✉ ❝r❡s❝✐♠❡♥t♦ r❡❝❡♥t❡ q✉❡ s❡ tr❛❞✉③✱ ❡♠ ♣❛rt❡✱ ♥♦ ♥ú♠❡r♦ ❞❡ ❛rt✐❣♦s ❡♥❝♦♥tr❛❞♦s s♦❜r❡ ♦ ♠❡s♠♦✳ ❈♦♠❡ç❛♠♦s ♣♦r r❡❢❡r✐r ♦ tr❛❜❛❧❤♦ ❞❡ ✷✵✶✵ ❞❡ ❙❤❡✈✲ ❝❤❡♥❦♦ ❬✺✵❪✱ q✉❡ ✐♥❝❧✉✐ ✉♠❛ r❡✈✐sã♦ ❡ ❝♦♠♣❛r❛çã♦ ❞❡ ✈ár✐♦s ♠ét♦❞♦s ♥✉♠ér✐❝♦s q✉❡ sã♦ ✉s❛❞♦s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳
◆♦ q✉❡ ❝♦♥❝❡r♥❡ à r❡❝✉rsã♦ ❞❡ P❛♥❥❡r✱ ❛✉t♦r❡s ❝♦♠♦ ❍✐♣♣ ✭❝❢✳ ❬✸✸❪✮✱ ❊♠❜r❡✲ ❝❤ts ❡ ❋r❡✐ ✭❝❢✳ ❬✶✹❪✮✱ ●✉❡❣❛♥ ❡ ❍❛ss❛♥✐ ✭❝❢✳ ❬✷✺❪✮✱ ●❡r❤♦❧❞ ❡t ❛❧✳ ✭❝❢✳ ❬✷✶❪✮ ❡ ❳✐❡ ❡t ❛❧✳ ✭❝❢✳ ❬✺✺❪✮✱ ♣♦r ❡①❡♠♣❧♦✱ ♣✉❜❧✐❝❛r❛♠ tr❛❜❛❧❤♦s ❡♥✈♦❧✈❡♥❞♦ t❛♠❜é♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ♥✉♠ér✐❝♦s à ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳ ❙❡❣✉❡✲s❡ ✉♠❛ ❞❡s❝r✐çã♦ ❜r❡✈❡ ❞❡ ❝❛❞❛ ✉♠ ❞♦s tr❛❜❛❧❤♦s r❡❢❡r✐❞♦s✳
❍✐♣♣ ✭❝❢✳ ❬✸✸❪✮✱ ♥♦ s❡✉ tr❛❜❛❧❤♦ ❞❡ ✷✵✵✻✱ ❞❡s❡♥✈♦❧✈❡ ✉♠ ❛❧❣♦r✐t♠♦ ❝❛♣❛③ ❞❡ s✐♠♣❧✐✜❝❛r ❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r q✉❛♥❞♦ ❛s ❞✐str✐❜✉✐çõ❡s ❞♦ ✈❛❧♦r ❞❛s ✐♥❞❡♠♥✐③❛✲ çõ❡s ♣❛❣❛s sã♦ ❞♦ t✐♣♦ ❢❛s❡✱ ✐✳❡✳✱ ❞✐str✐❜✉✐çõ❡s q✉❡ tê♠ ✉♠❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s r❛❝✐♦♥❛❧ ✭❝❢✳✱ ❡✳❣✳✱ ❬✶✶❪✱ ♣✳ ✺✶✵✮✳ ❊st❡ ❛❧❣♦r✐t♠♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ❛ ❞✐str✐❜✉✐çõ❡s ❞✐s❝r❡t❛s✱ ❝♦♥tí♥✉❛s ♦✉ ♠✐st❛s✳
❊♠❜r❡❝❤ts ❡ ❋r❡✐ ✭❝❢✳ ❬✶✹❪✮ ❝♦♠♣❛r❛♠ ❞✉❛s té❝♥✐❝❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✿ ❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r ❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r rá♣✐❞❛ ✭❡♠ ✐♥❣❧ês✱ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠✮✳ ❊st❡s ❛✉t♦r❡s ❛♣r❡s❡♥t❛♠ ❡st❡s ❞♦✐s ♠ét♦❞♦s r❡❝✉rs✐✈♦s✱ ❛♣♦♥t❛♥❞♦ ♦s ♣♦♥t♦s ❢♦rt❡s ❞❡ ❝❛❞❛ ✉♠✱ ✜♥❛❧✐③❛♥❞♦ ❝♦♠ ❡①❡♠♣❧♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❝♦♠♣❛r❛r ❛ ❡✜❝á❝✐❛ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s✳
✹ ✶✳ ■♥tr♦❞✉çã♦
❯♠ ❛♥♦ ❞❡♣♦✐s✱ ●❡r❤♦❧❞ ❡t ❛❧✳ ✭❝❢✳ ❬✷✶❪✮ ❞❡s❡♥✈♦❧✈❡♠ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r ❝❛♣❛③ ❞❡ ✉❧tr❛♣❛ss❛r ❛s ✐♥st❛❜✐❧✐❞❛❞❡s ♥✉♠ér✐❝❛s ❞❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r ♦r✐❣✐♥❛❧ ❡ q✉❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❛ ❛❧❣✉♥s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳
❋✐♥❛❧♠❡♥t❡✱ ❳✐❡ ❡t ❛❧✳ ✭❝❢✳ ❬✺✺❪✮✱ ❡♠ ✷✵✶✷✱ ✉t✐❧✐③❛♠ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❇❡r♥st❡✐♥ q✉❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ❞✐str✐❜✉✐çã♦ ❞♦ ✈❛❧♦r ❞❛s ✐♥❞❡♠♥✐③❛çõ❡sX é ❝♦♥✲ tí♥✉❛✱ ❝♦♥s❡❣✉✐♥❞♦ ❛ss✐♠ s✐♠♣❧✐✜❝❛r ♦s ❛❧❣♦r✐t♠♦s ♥✉♠ér✐❝♦s✱ ♦❜t❡r ❡q✉❛çõ❡s r❡❝✉rs✐✈❛s ❛♣r♦①✐♠❛❞❛s ❡ ❞❡t❡r♠✐♥❛r ♦ ❡rr♦ t❡ór✐❝♦ ❞❛ ❛♣r♦①✐♠❛çã♦✳
◆❡st❛ ❞✐ss❡rt❛çã♦ sã♦ ❛♥❛❧✐s❛❞❛s ❛ ❝❧❛ss❡ ❞❡ ❞✐str✐❜✉✐çõ❡s ❞❡ P❛♥❥❡r✱ ❛s ❡①t❡♥✲ sõ❡s ❞❛ ♠❡s♠❛✱ ♦❜t✐❞❛s ♣♦r ♠❡✐♦ ❞❛ ♠♦❞✐✜❝❛çã♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ❞♦ s✉♣♦rt❡ ❞❛s ❞✐str✐❜✉✐çõ❡s ❞❡ P❛♥❥❡r✱ ❡ ❛ ♠♦❞❡❧❛çã♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✳ ❊st❛ ❞✐ss❡rt❛çã♦ ❡stá ❞✐✈✐❞✐❞❛ ❡♠ s❡t❡ ❝❛♣ít✉❧♦s✱ s❡♥❞♦ ♦ ♣r❡s❡♥t❡ ♦ ♣r✐♠❡✐r♦ ❞❡❧❡s✳
◆♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ sã♦ r❡❢❡r✐❞♦s ♦s ❝♦♥❝❡✐t♦s ❡ ✐♥str✉♠❡♥t♦s ❢✉♥❞❛♠❡♥t❛✐s ♥❛ ❝♦♥str✉çã♦ ❞❛s ❡①t❡♥sõ❡s ❞❛ ❢❛♠í❧✐❛ ❞❡ P❛♥❥❡r ❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦s r❡s✉❧✲ t❛❞♦s ❞♦ q✉✐♥t♦ ❝❛♣ít✉❧♦✱ ❝♦♠♦ ❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ❛ tr❛♥s❢♦r✲ ♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ❛ ♠✐st✉r❛ ❡ ♠♦❞✐✜❝❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s✳
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛s ❞✐str✐❜✉✐çõ❡s ❞✐s❝r❡t❛s ♣❡rt❡♥❝❡♥t❡s à ❝❧❛ss❡ ❞❡ P❛♥❥❡r✱ q✉❡ ❞❡♥♦♠✐♥❛♠♦s ♣♦r ❞✐str✐❜✉✐çõ❡s ❞❡ ❝♦♥t❛❣❡♠ ❜ás✐❝❛s✳
❙❡❣✉✐❞❛♠❡♥t❡✱ ♥♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ sã♦ ❞❡✜♥✐❞❛s ❛ r❡❝✉rsã♦ ❞❡ P❛♥❥❡r ❡ ❞✉❛s s✉❛s ❡①t❡♥sõ❡s ❡ ❛♣r❡s❡♥t❛❞❛s ❛s ❞✐str✐❜✉✐çõ❡s ♣❡rt❡♥❝❡♥t❡s ❛ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✳
❖ q✉✐♥t♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞♦ ♠♦❞❡❧♦ ❞♦ r✐s❝♦ ❝♦❧❡t✐✈♦✱ ❞❡s✐❣♥❛❞❛♠❡♥t❡ ♦ ♠♦✲ ❞❡❧♦ ❝♦♠♣♦st♦ ❞❛s ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s✳ ❆s ❞✐str✐❜✉✐çõ❡s ❞❛s ✐♥❞❡♠♥✐③❛çõ❡s ❛❣r❡❣❛❞❛s sã♦ ❛q✉✐ ❝♦♥str✉í❞❛s ❛tr❛✈és ❞♦ ♠ét♦❞♦ r❡❝✉rs✐✈♦✳ ❙ã♦ t❛♠❜é♠ ❛♣r❡✲ s❡♥t❛❞♦s ❞♦✐s ♠ét♦❞♦s ❞❡ ❝♦♥str✉çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❛r✐t♠ét✐❝❛s✳
◆♦ s❡①t♦ ❝❛♣ít✉❧♦ s❡rã♦ ❝♦❧♦❝❛❞♦s ❡♠ ♣rát✐❝❛ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡❞✉③✐❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦s ♣r♦❣r❛♠❛s ✐♥❢♦r♠át✐❝♦s ▼❛t❤❡♠❛t✐❝❛ ❡ ❘✳
❈❛♣ít✉❧♦ ✷
❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s
❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s
❆s ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✭✈✳❛✳✬s✮ q✉❡ s✉r❣❡♠ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ sã♦✱ ❡ss❡♥✲ ❝✐❛❧♠❡♥t❡✱ ❞❡ ❞♦✐s t✐♣♦s✿ ❞❡ ❝♦♥t❛❣❡♠ ✭❞✐s❝r❡t❛s✱ t♦♠❛♥❞♦ ✈❛❧♦r❡s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s✮ ❡ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s ♥ã♦ ♥❡❣❛t✐✈❛s✳
◗✉❛♥❞♦X é ✉♠❛ ✈✳❛✳ ❞❡ ❝♦♥t❛❣❡♠✱ ❛ ❢✉♥çã♦ ♠❛ss❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✭❢✳♠✳♣✳✮ pk ✐❞❡♥t✐✜❝❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ k (k ∈N) ❛❝♦♥t❡❝✐♠❡♥t♦s ♦❝♦rr❡r❡♠✳ ❙❡ X ❢♦r
❛ ✈✳❛✳ q✉❡ r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ t❛✐s ❛❝♦♥t❡❝✐♠❡♥t♦s✱ t❡♠♦s ❡♥tã♦
X =
(
k, k= 0,1, . . . , pk= Pr (X =k)
✭✷✳✶✮
❡✱ ♣❛r❛ a, b∈R✱ a≤b✱ t❡♠♦s
Pr (X ∈[a, b]) = X
k:k∈[a,b]∩N
pk.
◗✉❛♥❞♦X é ✉♠❛ ✈✳❛✳ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❝♦♠ ❢✉♥çã♦ ❞❡♥s✐❞❛❞❡ ❞❡ ♣r♦✲ ❜❛❜✐❧✐❞❛❞❡ ✭❢✳❞✳♣✳✮ ❞❡ X✱ fX✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛♥t❡r✐♦r é ❞❡t❡r♠✐♥❛❞❛ ❛tr❛✈és ❞♦
✐♥t❡❣r❛❧ ❞❡ fX ♥♦ ✐♥t❡r✈❛❧♦ [a, b]✱ ✐✳❡✳✱
Pr (X ∈[a, b]) =
Z b
a
fX(x)dx,
♦♥❞❡ fX é ❛ ❢✳❞✳♣✳ ❞❡ X✳
❘❡❝♦r❞❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ s✉♣♦rt❡ ❞❡ ✉♠❛ ✈✳❛✳ q✉❡ ✉t✐❧✐③❛r❡♠♦s ❡♠ ❜r❡✈❡✿
❉❡✜♥✐çã♦ ✷✳✶ ❖ s✉♣♦rt❡ ❞❡ ✉♠❛ ✈✳❛✳ X ❞✐s❝r❡t❛ é ♦ ❝♦♥❥✉♥t♦
SX ={xk, k ∈N:pk = Pr(X =xk)>0}, ✭✷✳✷✮
✻ ✷✳ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s
s❡ X t✐✈❡r ❢✳♠✳♣✳ ❞❡✜♥✐❞❛ ♣♦r
X=
(
xk, k = 0,1, . . . ,
pk = Pr (X =xk)
;
♦✉✱
SX ={x∈R:fX(x)>0}, ✭✷✳✸✮
s❡ X ❢♦r ✉♠❛ ✈✳❛✳ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❝♦♠ ❢✳❞✳♣✳ fX✳
◆❛ s❡❝çã♦ s❡❣✉✐♥t❡ ❛♣r❡s❡♥t❛♠♦s ❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ ❝✉✲ ❥❛s ♣r♦♣r✐❡❞❛❞❡s ♥♦s ♣❡r♠✐t❡♠ ❡st✉❞❛r ❝♦♠ ♠❛✐s ❢❛❝✐❧✐❞❛❞❡ ❛s ✈✳❛✳✬s ❞✐s❝r❡t❛s✱ t❡♠❛ ❛❜♦r❞❛❞♦ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✱ ❡ ♥❛ s❡❝çã♦ s✉❜s❡q✉❡♥t❡ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ q✉❡ s❡rá ✉s❛❞❛ ♣❛r❛ ❡st✉❞❛r ❛s ✈✳❛✳✬s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s✳ ❊st❡ ❝❛♣ít✉❧♦ t❡r♠✐♥❛ ❝♦♠ ✉♠❛ s❡❝çã♦ ❞❡❞✐❝❛❞❛ à ♠♦❞✐✜❝❛çã♦ ❞❡ ✈✳❛✳✬s ❞❡ ❝♦♥t❛❣❡♠ ♥♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❞♦ s❡✉ s✉♣♦rt❡✳
✷✳✶ ❆ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s
❆ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ✭❢✳❣✳♣✳✮ P ❞❡ ✉♠❛ ✈✳❛✳ X é ❞❛❞❛ ♣♦r
P (z) =PX (z) = E zX
, ✭✷✳✹✮
♥♦s ♣♦♥t♦s z ❡♠ q✉❡ ❡①✐st❡ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ E zX
✳
◗✉❛♥❞♦ ✉♠❛ ✈✳❛✳ N é ❞✐s❝r❡t❛ ❝♦♠ ❢✳♠✳♣✳ ❞❡✜♥✐❞❛ ♣♦r ✭✷✳✶✮✱ ❛ s✉❛ ❢✳❣✳♣✳ é
P (z) = PN(z) =E zN
= +∞
X
k=0
pkzk. ✭✷✳✺✮
◆❡st❡ ❝❛s♦ ♣♦❞❡♠♦s t♦♠❛rz t❛❧ q✉❡ |z|<1✱ ❥á q✉❡✱ s❡ ❛ss✐♠ ❢♦r✱
E zN
≤
+∞
X
k=0
pk|z|k ≤
+∞
X
k=0
pk = 1✳
❆ ❢✳❣✳♣✳ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❣❡r❛r ♦s ♠♦♠❡♥t♦s ❞❛ ✈✳❛✳ N✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡
P′(1) =E(N) ✭✷✳✻✮
❡ q✉❡
✷✳ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✼
❞❡ ♦♥❞❡ r❡s✉❧t❛
V ar(N) = E[N(N −1)] +E(N)−E2(N)
= P′′(1) +P′(1)−[P′(1)]2. ✭✷✳✽✮
❆ ❞❡t❡r♠✐♥❛çã♦ ❞❡st❡s ❞♦✐s ♠♦♠❡♥t♦s ♣❡r♠✐t❡✲♥♦s ❝❛❧❝✉❧❛r ♦ í♥❞✐❝❡ ❞❡ ❞✐s✲ ♣❡rsã♦ ❞❛ ✈❛r✐á✈❡❧ ❛❧❡❛tór✐❛ N q✉❡ é ❞❛❞♦ ♣♦r ✭❝❢✳✱ ❡✳❣✳✱ ❬✸✹❪✱ ♣✳ 163✮
δ(N) = V ar(N)
E(N) . ✭✷✳✾✮
❆ ❢✳❣✳♣✳ ❞❡✈❡ ♦ s❡✉ ♥♦♠❡ ❛♦ ❢❛❝t♦ ❞❡ ♣❡r♠✐t✐r ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛s ♣r♦❜❛❜✐✲ ❧✐❞❛❞❡s q✉❡ ❞❡✜♥❡♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ ✈✳❛✳ ❞✐s❝r❡t❛ ❡♠ ❝❛✉s❛✳ ❉❡ ❢❛❝t♦✱ ♣❛r❛
z ∈(−1,1)❡ m∈N✱
P(m)(z) = d
m
dzmE z
N(2.5)
= d m dzm +∞ X k=0
pkzk
!
= +∞
X
k=m
dm
dzm pkz
k
= +∞
X
k=m
k(k−1). . .(k−m+ 1)zk−mpk
= +∞
X
k=m
k! (k−m)!z
k−mp
k =
+∞
X
k=0
(k+m)!
k! z
kp k+m
❡
P(m)(0) =m!pm✱
♦✉ s❡❥❛✱
pm =
P(m)(0)
m! ✱ ♣❛r❛ m∈N✳ ✭✷✳✶✵✮
❊st❡ r❡s✉❧t❛❞♦ ♣❡r♠✐t❡✲♥♦s ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❞❡ ✉♥✐❝✐❞❛❞❡✿
❚❡♦r❡♠❛ ✷✳✷ ❆ ❢✳❣✳♣✳ ❝❛r❛❝t❡r✐③❛ ❛s ✈✳❛✳✬s ❞✐s❝r❡t❛s✱ ❞❡ t❛❧ ♠♦❞♦ q✉❡✱ s❡ N ❡ M sã♦ ✈✳❛✳✬s ❞✐s❝r❡t❛s✱
PN =PM ⇔N
d
=M.
❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛❝t♦✱ ❞❛ ✐❣✉❛❧❞❛❞❡ ✭✷✳✶✵✮✱ t❡♠♦s
N =
(
k, k= 0,1, . . . ,
pk = Pr (N =k) = PN(k)(0)
k!
✱ M =
(
k, k= 0,1, . . . ,
p∗
k= Pr (M =k) = PM(k)(0)
✽ ✷✳ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s
❡✱ ♣♦rt❛♥t♦✱ ❝♦♠♦ PN = PM,Pr (N =k) = Pr (M =k), ♣❛r❛ k = 0,1, . . .✱ ✐✳❡✳✱
❛s ✈✳❛✳✬s ❞✐s❝r❡t❛s N ❡M tê♠ ❛ ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ ✭❛ ✐♠♣❧✐❝❛çã♦ ❝♦♥trár✐❛ ♥ã♦ ♥❡❝❡ss✐t❛ ❞❡ ❞❡♠♦♥str❛çã♦✮✳
◗✉❛♥❞♦ ❞✉❛s ✈❛r✐á✈❡✐s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞♦ ♣r♦❞✉t♦ ❞❡s✲ s❛s ✈❛r✐á✈❡✐s é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦s ✈❛❧♦r❡s ❡s♣❡r❛❞♦s ❞❡ss❛s ♠❡s♠❛s ✈❛r✐á✈❡✐s ✭❝❢✳ ❞❡♠♦♥str❛çã♦✱ ❡✳❣✳✱ ❡♠ ❬✹✻❪✮✳ ❊st❡ r❡s✉❧t❛❞♦ ❡ ♦ ❚❡♦r❡♠❛ ✷✳✷ ♣❡r♠✐t❡♠ ❛ ✉t✐❧✐③❛çã♦ ❞❛s ❢✳❣✳♣✳ ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛s ✈✳❛✳✬s q✉❡ r❡s✉❧t❛♠ ❞❛ s♦♠❛ ❞❡ ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s✿
❚❡♦r❡♠❛ ✷✳✸ ❙❡❥❛♠ ◆ ❡ ▼ ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s. ❊♥tã♦✱
PN+M =PNPM.
❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ q✉❡N ❡M sã♦ ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s ❡♥tr❡ s✐✱ P =zN
❡ Q=zM sã♦ t❛♠❜é♠ ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ▲♦❣♦✱ ❛♣❧✐❝❛♥❞♦ ❛s r❡❣r❛s ❞♦ ✈❛❧♦r
❡s♣❡r❛❞♦ ♥♦ q✉❡ ❝♦♥❝❡r♥❡ ❛♦ ♣r♦❞✉t♦ ❞❡ ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❡♠♦s q✉❡
PN+M(z) =E zN+M
=E zNzM=PN(z)PM (z),
♣❛r❛ z ∈ (−1,1)✳ ❉❡♠♦♥str❛✲s❡✱ ❛ss✐♠✱ q✉❡ ❛ ❢✳❣✳♣✳ ❞❛ s♦♠❛ ❞❡ ❞✉❛s ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s é ♦ ♣r♦❞✉t♦ ❞❛s ❢✳❣✳♣✳ ✐♥❞✐✈✐❞✉❛✐s ❞❛s r❡❢❡r✐❞❛s ✈✳❛✳✬s✳
✷✳✷ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
❆ ❢✳❣✳♣✳ ❛♣r❡s❡♥t❛❞❛ ♥❛ s❡❝çã♦ ❛♥t❡r✐♦r é ✉s❛❞❛ ♥♦r♠❛❧♠❡♥t❡ q✉❛♥❞♦ ❛ ✈✳❛✳ X
é ❞✐s❝r❡t❛✳ ◗✉❛♥❞♦ ❛ ✈✳❛✳ X é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛✱ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦
t✐♣♦ Pr (X =x)✱ ❝♦♠ x ∈ R✱ sã♦ ♥✉❧❛s ❡ ❛ ❢✳❣✳♣✳ ♣❡r❞❡ ✉♠❛ ❞❛s s✉❛s ♠❛✐s
✐♠♣♦rt❛♥t❡s ❝❛r❛❝t❡ríst✐❝❛s✳ ◆❡st❡ ❝❛s♦ ✉s❛♠♦s ♦✉tr❛s tr❛♥s❢♦r♠❛❞❛s ❝♦♠♦ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ♦✉ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱E eizX✱ ❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡
♠♦♠❡♥t♦s✱E ezX
✱ ♦✉ ♥♦ ❝❛s♦ ❞❛ ✈✳❛✳ X t❡r s✉♣♦rt❡ SX ⊆R+0✱ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ E e−zX
✱ q✉❡ ♣❛ss❛♠♦s ❛ ❞❡✜♥✐r ✭❝❢✳✱ ❡✳❣✳✱ ❬✹✺❪✱ ♣♣✳ ✸✷ ❛ ✸✹✮✳
❉❡✜♥✐çã♦ ✷✳✹ ❙❡❥❛ fX ❛ ❢✳❞✳♣✳ ❞❡ ✉♠❛ ✈✳❛✳ X ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ♥ã♦ ♥❡✲
❣❛t✐✈❛✳ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ fX é ❞❛❞❛ ♣♦r
LX(z) = L(fX) (z) =E e−zX
= +∞
Z
0
✷✳ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✾
♣❛r❛ ♦s ♣♦♥t♦s z t❛✐s q✉❡ E e−zX
<+∞✶✳
❖❜s❡r✈❛çã♦ ✷✳✺ ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❛ ❢✉♥çõ❡s q✉❡ ♥ã♦ sã♦ ❢✳❞✳♣✳✳ ▼♦str❛✲s❡ q✉❡ ♦ ✐♥t❡❣r❛❧ L(f) (z) =
+∞
R
0
e−zxf(x)dx ❡①✐st❡ ❞❡s❞❡
q✉❡ f s❡❥❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ✭♣♦r ♣❛rt❡s✮ ❡ ❡①✐st❛♠ a1, b > 0 ❡ a2 ∈ R t❛✐s
q✉❡ |f(x)|< a1ea2x, ♣❛r❛ t♦❞♦ ♦ x > b ✭❝❢✳✱ ❡✳❣✳✱ ❬✹✺❪✱ ♣✳ ✾✮✳
❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱LX✱ ♣♦❞❡ s❡r r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ❢✳❣✳♣✳✱ PX✳ ❘❡✲
❝♦r❞❛♥❞♦ ❞❡ (2.4) q✉❡
PX (z) = E zX
,
❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
LX(z) = E e−zX
=PX e−z
. ✭✷✳✶✷✮
❚❛❧ ❝♦♠♦ ❛❝♦♥t❡❝❡ ❝♦♠ ❛ ❢✳❣✳♣✳✱ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✐❞❡♥t✐✜❝❛ ❛ ❞✐s✲ tr✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ✈✳❛✳ ❡♠ ❝❛✉s❛✳
❉❛s ✈ár✐❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣♦ss✉✐ ✭❝❢✳✱ ❡✳❣✳✱ ❬✹✺❪✱ ♣♣✳ ✶✵ ❡ ✶✶✮✱ sã♦ ❛♣r❡s❡♥t❛❞❛s s❡❣✉✐❞❛♠❡♥t❡ ❛♣❡♥❛s ❛s ♥❡❝❡ssár✐❛s ❡♠ ❞❡♠♦♥s✲ tr❛çõ❡s ♣♦st❡r✐♦r❡s✳ ❈♦♥s✐❞❡r❡♠♦s X ❡ Y ✈✳❛✳✬s ♥ã♦ ♥❡❣❛t✐✈❛s ❝♦♠ ❢✳❞✳♣✳ fX
❡ fY✱ r❡s♣❡t✐✈❛♠❡♥t❡✱ ❡ ❝♦♠ ✈❛❧♦r❡s ❡s♣❡r❛❞♦s ✜♥✐t♦s✱ ♣❛r❛ ❛s q✉❛✐s é ♣♦ssí✈❡❧
❞❡t❡r♠✐♥❛r L(fX)❡ L(fY)✳ ❊♥tã♦✿
✐✳ Pr♦♣r✐❡❞❛❞❡ 1 ✭❞❡r✐✈❛❞❛ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✮✿
[L(fX)]
′ (z) =
+∞
Z
0
e−zx[−xfX(x)]dx, ✭✷✳✶✸✮
♦✉ s❡❥❛✱ [L(fX)]
′
(z) =L(g) (z)✱ ❝♦♠ g(x) = −xfX(x)✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ x, z >0✳ ❈♦♠♦
∂
∂z e
−zx
=−xe−zx
=xe−zx≤x
❡
Z +∞
−∞
xfX(x)dx <+∞,
✶❖ ❞♦♠í♥✐♦ ❞❡L
X ♣♦❞❡ s❡r ❡st❡♥❞✐❞♦ ❛♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦sz✱ ❝✉❥❛ ♣❛rt❡ r❡❛❧ é ♣♦s✐t✐✈❛✱
✶✵ ✷✳ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s
♣♦❞❡♠♦s ❛✜r♠❛r q✉❡
d dz +∞ Z 0
e−zxfX (x)dx
= +∞ Z 0 ∂ ∂ze
−zxf
X(x)dx
✭❝❢✳✱ ❡✳❣✳✱ ❬✻❪✱ ♣✳ 119✮✳ ❆ss✐♠✱ t❡♠♦s
[L(fX)]
′
(z) = d
dz +∞ Z 0
e−zxfX(x)dx
= +∞ Z 0 d dze
−zxf
X(x)dx
= +∞
Z
0
e−zx[−xfX(x)]dx
= L(g) (z), ❝♦♠ g(x) =−xfX(x).
✐✐✳ Pr♦♣r✐❡❞❛❞❡ 2✭tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ s♦♠❛ ❞❡ ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s✮✿
LX+Y (z) =LX(z)LY (z), ✭✷✳✶✹✮
s❡♥❞♦ X ❡ Y ✈✳❛✳✬s ✐♥❞❡♣❡♥❞❡♥t❡s✳
❉❡♠♦♥str❛çã♦✿ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ r❡s✉❧t❛ ❞❡ ✐♠❡❞✐❛t♦ ❞♦ ❚❡♦r❡♠❛ ✷✳✸ ❡ ❞❡
(2.12)✳ ❉❡ ❢❛❝t♦✱
LX+Y (z)
(2.12)
= PX+Y e−z
(❚❡♦r❡♠❛2.3)
= PX e−z
PY e−z
(2.12)
= LX (z)LY (z).
◆♦ ❝❛s♦ ❡♠ q✉❡ ❛s ✈✳❛✳✬s X ❡Y sã♦ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❛♥t❡r✐♦r ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
✐✐✐✳ Pr♦♣r✐❡❞❛❞❡ 3 ✭♣r♦❞✉t♦ ❞❛s tr❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ❞❡ ❞✉❛s ❢✉♥çõ❡s
❞❡♥s✐❞❛❞❡✮✳ ❙❡❥❛♠ fX ❡ fY ❛s ❢✳❞✳♣✳ ❞❡ ❞✉❛s ✈✳❛✳✬s X ❡ Y✳ ❆ss✐♠✱
L(fX)L(fY) (z) = +∞
Z
0
e−zx
x Z
0
fX (y)fY (x−y)dy
✷✳ ❙♦❜r❡ tr❛♥s❢♦r♠❛❞❛s ❡ ♠♦❞✐✜❝❛çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ✶✶
♦✉ s❡❥❛✱ L(fX)L(fY) (z) =L(f) (z)✱ ❝♦♠ f(x) = x R
0
fX(y)fY (x−y)dy✳
❉❡♠♦♥str❛çã♦✿ ❚♦♠❛♥❞♦ ❞✉❛s ✈✳❛✳✬s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s ❡ ✐♥❞❡♣❡♥✲ ❞❡♥t❡s✱ X❡Y✱ ♥ã♦ ♥❡❣❛t✐✈❛s✱ ❝♦♠ ❢✳❞✳♣✳fX ❡fY✱ r❡s♣❡t✐✈❛♠❡♥t❡✱ ♠♦str❛✲s❡ q✉❡✱
✭❝❢✳✱ ❡✳❣✳✱ ❬✹✻❪✱ ♣✳ 780✮✱
fX+Y (x) = Z +∞
−∞
fX (y)fY (x−y)dy. ✭✷✳✶✻✮
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱
L(fX) (z)L(fY) (z) (2.14)
=
+∞
Z
0
e−zxfX+Y (x)dx
(2.16) =
+∞
Z
0
e−zx
Z +∞
−∞
fX(y)fY (x−y)dy
dx.
❈♦♠♦ fX(y)fY (x−y)>0 só s❡ 0≤y≤x✱ t❡♠♦s
L(fX) (z)L(fY) (z) = +∞
Z
0
e−zx
x Z
0
fX(y)fY (x−y)dy
dx
= L(f) (z)✱ ❝♦♠ f(x) =
x Z
0
fX(y)fY (x−y)dy.
❏✉♥t❛♥❞♦ (2.13)❡ (2.15) ✈❡♠
L(fX) (z) [L(fY)]′(z) (2.15)
= L(f) (z), ✭✷✳✶✼✮
❝♦♠ f(x) =
x R
0
fX(y)g(x−y)dy ❡ g(x) = −xfY (x)✱ ✐✳❡✳✱
f(x) =
x Z
0
fX (y) [−(x−y)fY (x−y)]dy.
