Giới thiệu về tập ngẫu nhiên

➤➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❤➭ ♥é✐
tr➢ê♥❣ ➤➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥

❚r➬♥ ❚❤Þ ❍➯✐ ▲ý

●✐í✐ t❤✐Ư✉ ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥

❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ❦❤♦❛ ❤ä❝

❍➭ ◆é✐ ✲ ✷✵✶✹


➤➵✐ ❤ä❝ q✉è❝ ❣✐❛ ❤➭ ♥é✐
tr➢ê♥❣ ➤➵✐ ❤ä❝ ❦❤♦❛ ❤ä❝ tù ♥❤✐➟♥

❚r➬♥ ❚❤Þ ❍➯✐ ▲ý

●✐í✐ t❤✐Ư✉ ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥
❈❤✉②➟♥ ♥❣➭♥❤✿ ▲ý t❤✉②Õt ①➳❝ s✉✃t ✈➭ t❤è♥❣ ❦➟ t♦➳♥ ❤ä❝
▼➲ sè✿ ✻✵✳✹✻✳✶✺

❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ❦❤♦❛ ❤ä❝

◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿
❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ♥❤

❍➭ ◆é✐ ✲ ✷✵✶✹


▼ô❝ ❧ô❝

▼ë ➤➬✉

✶

✶✳ ❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t❐♣ ♥❣➱✉ ♥❤✐➟♥

✸

✶✳✶✳ ❑✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò ①➳❝ s✉✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸

✶✳✷✳ ▼ét ✈➭✐ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ t❤è♥❣ ❦➟ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷✳✶✳ ▼✐Ò♥ t✐♥ ❝❐② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷✳✷✳ ❚❤è♥❣ ❦➟ ❇❛②❡s ♠➵♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✷✳✸✳ P❤➞♥ tÝ❝❤ ❞÷ ❧✐Ư✉ t❤➠

✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷✳✹✳ ❚❤➠♥❣ t✐♥ ❞ù❛ tr➟♥ ♥❤❐♥ t❤ø❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✷✳✺✳ ▲✃② ♠➱✉ ①➳❝ s✉✃t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✷✳ ❈➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❤÷✉ ❤➵♥

✶✹

✷✳✶✳ ❚❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♣❤➞♥ ❜è ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✷✳✷✳ ❈➳❝ q✉❛♥ s➳t ❝ã ❣✐➳ trÞ t❐♣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✸✳ ❳➳❝ s✉✃t ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✻

✷✳✺✳ ❚❐♣ ➤ã♥❣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t➠♣➠ ❧✐➟♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✹

✷✳✹✳ P❤➞♥ ❜è ❡♥tr♦♣② ❝ù❝ ➤➵✐

✸✳ ❚❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➳❝ ✈✃♥ ➤Ò ❧✐➟♥ q✉❛♥

✶

✹✼


✸✳✶✳ ▼è✐ q✉❛♥ ❤Ư ✶✲✶ ❣✐÷❛ ❤➭♠ ♣❤➞♥ ❜è ✈➭ ❤➭♠ ♠❐t ➤é ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✸✳✷✳ ❚Ý❝❤ ♣❤➞♥ ❈❤♦q✉❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✾

✸✳✸✳ ➜➵♦ ❤➭♠ ❘❛❞♦♥ ✲ ◆✐❦♦❞②♠

✻✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑Õt ❧✉❐♥

✼✶

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✼✶

✷


▼ë ➤➬✉
▲ý t❤✉②Õt t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❧➭ t➢➡♥❣ ➤è✐ ♠í✐✳ ❈❤♦q✉❡t ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét
✈➭✐ ý t➢ë♥❣ t❤❡♥ ❝❤èt ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭♦ ♥➝♠ ✶✾✺✸✱ ❑❡♥❞❛❧❧ ♥➝♠ ✶✾✼✹
✈➭ ▼❛t❤❡r♦♥ ➤➲ ❝✉♥❣ ❝✃♣ ♥❤÷♥❣ ❝➡ së ♥Ị♥ ♠ã♥❣ ❝❤♦ ❧ý t❤✉②Õt ♥➭② ✈➭♦ ♥➝♠
✶✾✼✺✳ ❚➭✐ ❧✐Ư✉ ✈Ị ❧ý t❤✉②Õt t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ❝➳❝ ø♥❣ ❞ô♥❣ ❞➬♥ ❞➬♥ trë ♥➟♥
❝ã ý ♥❣❤Ü❛ ❦Ĩ tõ ➤ã✳ ▼➷❝ ❞ï ❝ã ♥❤÷♥❣ ❦❤ã ❦❤➝♥ tr➢í❝ ♠➽t✱ ❦❤➠♥❣ ❝❤Ø ❜ë✐ ✈×
tÝ♥❤ ♣❤ø❝ t➵♣ ❝đ❛ tí ợ ị trị t ò ì sù t❤✐Õ✉ t❤è♥ ❝➳❝
♠➠ ❤×♥❤ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❞Ơ ①ư ❧ý✱ t✉② ♥❤✐➟♥ ❦❤➠♥❣ ♣❤➯✐ ✈× t❤Õ ♠➭ ❧ý t❤✉②Õt
✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ ➤➢ỵ❝ ♥❤✐Ị✉ t➳❝ ❣✐➯ q✉❛♥ t➞♠✳
❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ♠✉è♥ ❣✐í✐ t❤✐Ư✉ tỉ♥❣ q✉➳t ✈Ò t❐♣ ♥❣➱✉
♥❤✐➟♥ ✈➭ ♥❣❤✐➟♥ ❝ø✉ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥✳
❉➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ❚❙✳ ễ ị ủ t ồ

rì ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❧ý t❤✉②Õt ①➳❝ s✉✃t
♠➭ t❛ ❝➬♥ ➤Õ♥ ➤Ĩ t❤➯♦ ❧✉❐♥ ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣ s❛✉✳ ❈❤Ø r❛
sù tå♥ t➵✐ ❝ñ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✱ ➤➷❝ ❜✐Öt ❧➭ tr♦♥❣ ❧ý t❤✉②Õt t❤è♥❣ ❦➟ t♦➳♥
❤ä❝✳
❈❤➢➡♥❣ ✷✿ ◆❣❤✐➟♥ ❝ø✉ tr➢ê♥❣ ❤ỵ♣ ❝đ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥

❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥✳ ➜➢❛ r❛ ♠➠ ❤×♥❤ ❈❆❘ ❝❤♦ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❞ù❛
✈➭♦ ❧ý t❤✉②Õt ✈Ò t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ➜Ò ❝❐♣ ➤Õ♥ ❜➭✐ t♦➳♥ ❡♥tr♦♣② ❝ù❝ ➤➵✐ ❧✐➟♥
q✉❛♥ ❝❤♦ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❚r×♥❤ ❜➭② ✈Ị t❐♣ ➤ã♥❣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t➠♣➠ t❤Ý❝❤
❤ỵ♣ ❝❤♦ ❧í♣ ❝➳❝ t❐♣ ❝♦♥ ➤ã♥❣ ❝đ❛

Rd ✳

❈❤➢➡♥❣ ✸✿ ❈❤Ø r❛ ♠è✐ q✉❛♥ ❤Ư ✶✲✶ ❣✐÷❛ ❤➭♠ ♣❤➞♥ ❜è ✈➭ ❤➭♠ ♠❐t
➤é ❝đ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❚r×♥❤ ❜➭② tÝ❝❤ ♣❤➞♥ ❈❤♦q✉❡t ✈➭ ➤➵♦ ❤➭♠ ❘❛❞♦♥
✲ ◆✐❦♦❞②♠ ❝ñ❛ ❝➳❝ ❤➭♠ t❐♣ ❦❤➠♥❣ ❝é♥❣ tÝ♥❤✳
▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ủ ễ
ị tỏ ò ết ➡♥ s➞✉ s➽❝ ✈➭ ❝❤➞♥ t❤➭♥❤ tí✐ t❤➬②✱ t❤➬② ➤➲

✶


ề t ó ữ ị ớ ú ❝❤♦ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❧➭♠
❧✉❐♥ ✈➝♥✳ ❈➯♠ ➡♥ ❝➳❝ t❤➬②✱ ❝➠ tr♦♥❣ ❑❤♦❛ ❚♦➳♥ ✲ ❈➡ ✲ ❚✐♥ ❤ä❝✱ tr➢ê♥❣ ➜➵✐
❤ä❝ ❑❤♦❛ ❤ä❝ ❚ù ♥❤✐➟♥ ✲ ➜➵✐ ❤ä❝ ◗✉è❝ ❣✐❛ ❍➭ ◆é✐ ➤➲ ➤é♥❣ ✈✐➟♥✱ q✉❛♥ t➞♠
✈➭ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ t trờ
ố ù t rt ợ ữ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ q✉ý
❜➳✉ ❝ñ❛ ❝➳❝ t❤➬②✱ ❝➠ ❣✐➳♦ ✈➭ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥
❤➡♥✳
❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥✦

❍➭ ◆é✐✱ ♥❣➭② ✵✶ t❤➳♥❣ ✵✶ ♥➝♠ ✷✵✶✹
❚➳❝ ❣✐➯
❚r➬♥ ❚❤Þ ❍➯✐ ▲ý

✷

❈❤➢➡♥❣ ✶✳

❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈➭ t❐♣ ♥❣➱✉ ♥❤✐➟♥

✶✳✶✳

❑✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ①➳❝ s✉✃t
P❤➬♥ ♥➭② tr×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ò ❧ý t❤✉②Õt ①➳❝ s✉✃t ♠➭ t❛ ❝➬♥ ➤Õ♥

➤Ĩ t❤➯♦ ❧✉❐♥ ✈Ị ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ ❝➳❝ s
ị ĩ

ì t ọ ệ tợ

ì t ọ ột é tử ♥❣➱✉ ♥❤✐➟♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✭Ω ✱

A✱ P ✮✱ tr♦♥❣ ➤ã✿
α✮ Ω ❧➭ ♠ét t❐♣✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❦❤➠♥❣ ❣✐❛♥ ♠➱✉ ❝đ❛ ♣❤Ð♣ t❤ư✳
β ✮ A ❧➭ ♠ét σ ✲➤➵✐ sè ✭❜✐Ĩ✉ ❞✐Ơ♥ ❝➳❝ ❜✐Õ♥ ❝è✮✱ tø❝ ❧➭ ✿
✭✐✮ Ω ∈ A
✭✐✐✮ ◆Õ✉ A ∈ A t❤× Ac ∈ A
✭✐✐✐✮ ◆Õ✉ An ∈ A ✈í✐ n ≥ 1 t❤×

An ∈ A✳
n≥1

❈➷♣ (Ω, A) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤♦ ➤➢ỵ❝✳

γ ✮ P : A → [0, 1] ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é ➤♦ ①➳❝ s✉✃t✱ tø❝ ❧➭ ✿
✐✮ P (Ω) = 1
✸


✐✐✮ ◆Õ✉ {An , n ≥ 1} ❧➭ ♠ét ❞➲② ữ ế ợ ủ
tử rê✐ ♥❤❛✉ tõ♥❣ ➤➠✐ ♠ét ✭Ai ∩Aj = ∅ ✈í✐ i=j tì

P(

An ) =

n1

P (An )
n1

í t ợ ❣ä✐ ❧➭ σ ✲ ❝é♥❣ tÝ♥❤ ❝đ❛ P
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷✳

✭P❤➬♥ tö ♥❣➱✉ ♥❤✐➟♥✮

❈❤♦ ✭Ω ✱ A✱ P ✮ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳ ▼ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X ❧➭ ♠ét ➳♥❤
①➵ tõ Ω tí✐ R s❛♦ ❝❤♦ X −1 (B(R)) ⊆ A✱ tø❝ ❧➭✱ ∀B ∈ B(R)✱ X −1 (B) ∈ A✱
♥ã✐ ❝➳❝❤ ❦❤➳❝✱ X ❧➭ ♠ét ➳♥❤ ①➵ A ✲ B(R) ✲ ➤♦ ➤➢ỵ❝✳
❚r♦♥❣ ➤ã

B(R)

❧➭

σ

✲ tr➢ê♥❣ ❇♦r❡❧ ➤➢ỵ❝ s✐♥❤ r❛ ❜ë✐ ❝➳❝ t❐♣ ♠ë ❝ñ❛

R✱

X −1 (B) = {ω : X(ω) ∈ B}✳
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✸✳

✭▲✉❐t ①➳❝ s✉✃t ❝đ❛ ❝➳❝ ♣❤➬♥ tö ♥❣➱✉ ♥❤✐➟♥✮

❈❤♦ ✭Ω ✱ A✱ P ✮ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈➭ (U, U) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤♦
➤➢ỵ❝✳ ➪♥❤ ①➵ X : Ω → U ❧➭ A ✲ U ✲ ➤♦ ➤➢ỵ❝✳ ▲✉❐t ①➳❝ s✉✃t ❝đ❛ X ❧➭ ➤é ➤♦
①➳❝ s✉✃t tr➟♥ U ➤➢ỵ❝ ➤Þ♥❤ ♥❣❤Ü❛ ❜ë✐ PX = P X −1
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✹✳

✭❍➭♠ ♣❤➞♥ ❜è ❝ñ❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✮

❈❤♦ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X : (Ω, A, P ) → (R, B(R)) ✈➭ ❧✉❐t ①➳❝ s✉✃t ❝ñ❛
♥ã✿ PX = P X −1 tr➟♥ B(R)✳ ❍➭♠ ♣❤➞♥ ❜è ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X

F : R [0, 1] ợ ị ĩ ❜ë✐✿
F (x) = PX ((−∞, x])
❚Ý♥❤ ❝❤✃t ✶✳

❍➭♠ F ♥➭② t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ s❛✉✿

✭✐✮ F ❧➭ ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ ❣✐➯♠✱ tø❝ ❧➭✱ ♥Õ✉ x ≤ y t❤× F (x) ≤ F (y)✱
✭✐✐✮

lim F (x) = 1 ❀ lim F (x) = 0

x +∞

x −∞

✭✐✐✐✮ F ❧➭ tụ tr R tứ ớ ỗ x ∈ R✱ F (x) = lim F (y) =
y

+

F (x )✱ ✈➭ ❝ã ❣✐í✐ ❤➵♥ tr➳✐ t➵✐ ♠ä✐ x ∈ R✳
✹

x


❚✃t ❝➯ ❝➳❝ ❤➭♠ tr➟♥

R t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ✭✐✮✱ ✭✐✐✮✱ ✭✐✐✐✮ ë tr➟♥ ❧➭ ❝➳❝ ❤➭♠

♣❤➞♥ ❜è ❝ñ❛ ❝➳❝ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥

B(R)✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❝ã ♠ét s♦♥❣ ➳♥❤ ❣✐÷❛ ❝➳❝

❤➭♠ t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t ✭✐✮✱ ✭✐✐✮✱ ✭✐✐✐✮ ë tr➟♥ ✈í✐ ❝➳❝ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳

B(R)✳

✭❍➭♠ ♣❤➞♥ ❜è ❝đ❛ ✈Ð❝ t➡ ♥❣➱✉ ♥❤✐➟♥✮

❈❤♦ X : (Ω, A, P ) → (Rd , B(Rd )) ✭X = (X1 , · · · , Xd ) ❧➭ ✈Ð❝ t➡ ♥❣➱✉ ♥❤✐➟♥

d ❝❤✐Ò✉✮✳ ❍➭♠ ♣❤➞♥ ❜è F ❝ñ❛ X ❧➭ ❤➭♠✿ F : Rd → [0, 1]
F (x) = P (X ≤ x) = P (X1 ≤ x1 , · · · , Xd ≤ xd ) = P X −1 ((−∞, x]), ∀x =
(x1 , · · · , xd ) ∈ Rd
❚Ý♥❤ ❝❤✃t ✷✳

❚õ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ P, F t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿

✭✐✮ 0 ≤ F (x) ≤ 1✱
✭✐✐✮

lim F (x1 , · · · , xd ) = 0 ✈í✐ Ýt ♥❤✃t ♠ét ❥ ♥➭♦ ➤ã✱ ✈➭

xj →−∞

lim F (x1 , · · · , xd ) = 1 ✈í✐ t✃t ❝➯ j = 1, 2, · · · , d✳

xj →+∞

✭✐✐✐✮ F ❧➭ ❧✐➟♥ tô❝ ♣❤➯✐ tr➟♥ Rd ✱ tø❝ ❧➭ lim F (y) = F (x), ∀x ∈ Rd .
y

✶✳✷✳

x

▼ét ✈➭✐ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ t❤è♥❣ ❦➟
P❤➬♥ ♥➭② ❝❤Ø r❛ sù tå♥ t➵✐ ❝ñ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✱ ➤➷❝ ❜✐Öt ❧➭ tr♦♥❣ ❧ý t❤✉②Õt

t❤è♥❣ ❦➟ t♦➳♥ ❤ä❝✳

✶✳✷✳✶✳

▼✐Ị♥ t✐♥ ❝❐②

❳Ðt ♠ét ♠➠ ❤×♥❤ t❤è♥❣ ❦➟ t❤❛♠ sè ❤ã❛✳

{f (x, θ) : x ∈ X ⊆ Rm , θ ∈
ϕ(θ)✳ X

⊆ Rd }✱

❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ❤➭♠ ♠❐t ➤é

X1 , X2 , · · · , Xn
♥❣➱✉ ♥❤✐➟♥

❝ñ❛

✈➭ ♠ét t❤❛♠ sè ♠➭ t❛ q✉❛♥ t➞♠

f (x, θ)✱ ❝❤♦ tr➢í❝ ♠ét ♠➱✉ ♥❣➱✉ ♥❤✐➟♥

X ✱ ❜➯♥ ❝❤✃t ủ sự ớ ợ ề t ó tì ♠ét t❐♣

C(X1 , X2 , · · · , Xn ) ♠➭ ❝❤ø❛ ϕ(θ0 )✱ θ0

❧í♥✳ ❈ơ t❤Ĩ✱ t❐♣ ♥❣➱✉ ♥❤✐➟♥ C(X1 , X2 , · · ·

✺

❧➭ t❤❛♠ sè t❤ù❝✱ ✈í✐ ①➳❝ s✉✃t

, Xn ) ❧➭ ♠ét t❐♣ t✐♥ ❝❐② ❝❤♦ ϕ(θ) ✈í✐ ♠ø❝


t✐♥ ❝❐②

1 − α ∈ (0, 1) ♥Õ✉ ∀θ ∈

❚r♦♥❣ ➤ã

: Pθ (ϕ(θ) ∈ C(X1 , X2 , · · · , Xn )) ≥ 1 − α✳

dPθ = f (x, θ)dx✳

❚r♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ➤➡♥ ❣✐➯♥✱ ✈✐Ư❝ ①➞② ❞ù♥❣ ♠✐Ị♥ t✐♥ ❝❐② tèt ♥❤✃t ❝❤♦

ϕ(θ) ❝ã t❤Ĩ

➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ ♠➭ ❦❤➠♥❣ ❝➬♥ sư ❞ơ♥❣ ❦❤➳✐ ♥✐Ư♠ ❤×♥❤ t❤ø❝ ✈Ị ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭
❝➳❝ ♣❤➞♥ ❜è ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳

❱Ý ❞ô✿ ❈❤♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X ❝ã ♣❤➞♥ ❜è ❝❤✉➮♥ N (µ, σ 2 )✱ ë ➤➞② θ = (µ, σ 2 )✳

√

n(Xn − µ)/V ❝ã ♣❤➞♥ ♣❤è✐ ❙t✉❞❡♥t ✈í✐ n − 1 ❜❐❝ tù
n
X
+
·
·
·
+
X
1
2
2
1
n
❞♦✱ tr♦♥❣ ➤ã Xn =
✈➭ V =
n
n − 1 i=1(Xi − Xn ) ❧➭ tr✉♥❣ ❜×♥❤
♠➱✉ ✈➭ ♣❤➢➡♥❣ s❛✐ ♠➱✉ ❤✐Ư✉ ❝❤Ø♥❤✱ ❞ù❛ tr➟♥ ♠ét ♠➱✉ ♥❣➱✉ ♥❤✐➟♥ X1 , · · · , Xn ợ
ét

() = à

ó

X ó ột t (1
) à ó tể t ợ ♥❤➢ s❛✉✿
n
n

t1 < V (Xn − µ) < t2 s❛♦ ❝❤♦ P (t1 < V (Xn − µ) < t2 ) = 1 − α✳
⇒ L(X1 , · · · , Xn ) < µ < U (X1 , · · · , Xn )
rót r❛ √
tõ

❚r♦♥❣ ➤ã✿

L = Xn −

V
√
t ❀
n 2

U = Xn −

V
√
t
n 1

√

n
(t1 , t2 ) s❛♦ ❝❤♦✿ P (t1 < V (Xn − µ) < t2 ) =
1 − α✳ ❑❤♦➯♥❣ t✐♥ ❝❐② tèt ♥❤✃t t ứ 1 ó tể ợ ị ĩ ♥❤➢ ❧➭ ❦❤♦➯♥❣
❍✐Ĩ♥ ♥❤✐➟♥✱ ❝ã ♥❤✐Ị✉ ➤✐Ĩ♠

t✐♥ ❝❐② ✈í✐ ➤é ❞➭✐ ♥❤á ♥❤✃t ✭❜✐Ĩ✉ ❞✐Ơ♥ ➤é ❝❤Ý♥❤ ①➳❝ ❝đ❛ ➢í❝ ❧➢ỵ♥❣ ❝❤♦

ϕ(θ)✮✳ ➜é ❞➭✐

V (t − t )✱ ♥➟♥
|S| = U − L = √
2
1
n
❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❝❤ä♥ t1 , t2 s❛♦ ❝❤♦ ❝ù❝ t✐Ĩ✉ ➤é ❞➭✐ ❦× ✈ä♥❣ E|S| ❝ñ❛ t❐♣ ❝♦♥ ♥❣➱✉
ë ➤➞② ❝❤Ý♥❤ ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ ❝ơ t❤Ĩ ❧➭✱

♥❤✐➟♥

S = [L, U ]✳ ❚r♦♥❣ ❝➳❝ ❝❤✐Ị✉ ❧í♥ ❤➡♥✱ ➤é ❞➭✐ ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ S

t❤❛② t❤Õ ❜ë✐ ❦❤è✐ ▲❡❜❡s❣✉❡

∧(S)✱

♥➟♥ t❛ sÏ ♣❤➯✐ tÝ♥❤ t♦➳♥

E∧(S)✳

➤✐Ị✉ ♥➭②✱ t❛ ❝➬♥ ➤Þ♥❤ ♥❣❤Ü❛ ❦❤➳✐ ♥✐Ư♠ ❝đ❛ ♠ét t tr

ợ

ể ợ

Rd

t ó tì

(S) ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✭❦❤➠♥❣ ➞♠✮✳
✶✳✷✳✷✳

❚❤è♥❣ ❦➟ ❇❛②❡s ♠➵♥❤

❚r➢í❝ ❦❤✐ ♥ã✐ ✈Ị ❤Ư ♣❤➢➡♥❣ ♣❤➳♣ ❇❛②❡s ♠➵♥❤✱ t❛ ➤➢❛ r❛ ♠ét ✈Ý ❞ơ ❝đ❛ tr➢ê♥❣
❤ỵ♣ ✧❝➳❝ ①➳❝ s✉✃t ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝✧✳

❱Ý ❞ô✿ ❈❤♦ X ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ {a, b, c, d} ⊆ R ✈í✐ t
ộ

f0 ỉ ợ ị




f0 (a) ≥ 0.4, f0 (b) ≥ 0.2, f0 (c) ≥ 0.2, f0 (d) ≥ 0.1
➜➷t

Pf 0

= {a, b, c, d}

❜✐Ĩ✉ t❤Þ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥

f0 (θ), A ⊆

Pf0 (A) =

(1.1)

ợ s r từ

f0

tứ



A

t

P

ể tị ớ tt ❝➯ ❝➳❝ ➤é ➤♦ ①➳❝ s✉✃t

t❛ ❝❤Ø ❜✐Õt r➺♥❣
❈❤ó ý r➺♥❣

F

tr➟♥

Pf

♠➭

f

t❤á❛ ♠➲♥ ✭✶✳✶✮✳ ❑❤✐ ➤ã✱

Pf0 ∈ P ✱ ✈➭ ❞♦ ➤ã F ≤ Pf0 ≤ G ✈í✐ F = inf P ✱ G = sup P ✳
P

✈➭

P

G ❧➭ ❧✐➟♥ ❤ỵ♣ ✈í✐ ♥❤❛✉ t❤❡♦ ♥❣❤Ü❛✿ F (A) + G(Ac ) = 1, A ⊆

❱× ✈❐②✱ ❝❤ó♥❣ t❛ ❝❤Ø ❝➬♥ ①Ðt ♠ét tr

F





G ò ó tể

ợ s r
t
ì

F

F

tr

ộ tí

2

ợ ị ĩ tr t ợ s tứ tự ữ ị

ữ t ❝ã ♠ét ➳♥❤ ①➵ ▼♦❜✐✉s

(2 , ⊆) ✭

❧➭

φ : 2 → R ①➳❝ ➤Þ♥❤ ❜ë✐✿
(−1)#(A\B) F (B)

φ(A) =
B⊆A

❚❛ ❝ã✱

F

t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t t❤❡♦ s❛✉ ✭①❡♠ ❝❤➢➡♥❣ ✷✮✿

✐✮ F (∅) = 0, F ( ) = 1✳
✐✐✮ ∀n ≥ 2,

✈➭

A1 , · · · , An ⊆

✱

n

i=1

❚õ ❝➳❝ tÝ♥❤ ❝❤✃t ♥➭② ❝ñ❛

(−1)#(I)+1 F (

Ai ) ≥

F(

F

j∈I

∅=I⊆{1,2,··· ,n}

s✉② r❛ r➺♥❣

Aj )

φ ❦❤➠♥❣ ➞♠ ✈➭

φ(A) = 1✳
A⊆

❉♦ ➤ã✱

φ ❧➭ ♠ét ❤➭♠ ❦❤è✐ ①➳❝ s✉✃t t❤❐t sù ❝đ❛ ♣❤➬♥ tư ♥❣➱✉ ♥❤✐➟♥ S

❝➳❝ ❣✐➳ trÞ tr♦♥❣

2

✱ ❝ơ t❤Ĩ ❧➭✿

φ(A) = P (S = A),
P❤➬♥ tö ♥❣➱✉ ♥❤✐➟♥

♥➭♦ ➤ã ♥❤❐♥

S

❧➭ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr

A



ớ ố

F ở ì

t é ị ➤➯♦ ▼♦❜✐✉s✱ t❛ ❝ã✿

φ(B)✳

F (A) =
B⊆A

❚õ sù ♣❤➞♥ tÝ❝❤ ë tr➟♥✱ ❜➭✐ t♦➳♥ ❣è❝ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

X

❝ã ♠❐t ➤é

❦❤➠♥❣ ❤♦➭♥ ❝❤Ø♥❤ ❝ã t❤Ĩ ➤➢ỵ❝ ❜✐Õ♥ ➤ỉ✐ t❤➭♥❤ ♠ét ❜➭✐ t♦➳♥ ❧✐➟♥ q✉❛♥ ➤Õ♥ t❐♣ ♥❣➱✉
♥❤✐➟♥

S

✈í✐ ✧♠❐t ➤é✧

φ ➤➲ ❜✐Õt✳

✼


φ : 2 → [0, 1]
φ({a}) = 0.4, φ({b}) = 0.2, φ({c}) = 0.2, φ({d}) = 0.1, φ( ) = 0.1,
✈➭

φ(A) = 0 ✈í✐ t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ A ❦❤➳❝ ❝ñ❛

✳

❑❤✐ ➤ã✱ t❛ ❝ã ❦Õt ❧✉❐♥✳
❑Õt ❧✉❐♥✿ ▼ét ❜➭✐ t♦➳♥ ✈Ò ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ♣❤➞♥ ❜è ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝ ❝ã t❤Ĩ ➤➢ỵ❝

❜✐Õ♥ ➤ỉ✐ t❤➭♥❤ ♠ét ❜➭✐ t♦➳♥ ✈Ị t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❝ã ♣❤➞♥ ❜è ❝❤Ý♥❤ ①➳❝✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱
❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❞➢ê♥❣ ♥❤➢ ➤➢❛ r❛ ♠ét ♠➠ ì tí ợ t ề
ế ♥❤✐➟♥ ♠➭ ❝ã ①➳❝ s✉✃t ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝✱ ➤➷❝ ❜✐Öt ❧➭ tr♦♥❣ t❤è♥❣ ❦➟ ❇❛②❡s
♠➵♥❤✳
◗✉❛♥ ➤✐Ĩ♠ ❝đ❛ ❦Õt ❧✉❐♥ ❇❛②❡s ♠➵♥❤✿ ❳Ðt ♠ét ♠➠ ❤×♥❤ t❤è♥❣ ❦➟ ❝ã ❞➵♥❣ {F (x|θ), θ

} ❝❤♦ ✈Ð❝t➡ ♥❣➱✉ ♥❤✐➟♥ X
✈➭

♥➭♦ ➤ã✱ tr♦♥❣ ➤ã

F (x|θ) ❜✐Ĩ✉ t❤Þ ❤➭♠ ♣❤➞♥ ❜è t❤❡♦ θ✱

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t❤❛♠✳ P❤➢➡♥❣ ♣❤➳♣ ❇❛②❡s ❜➽t ➤➬✉ ❜➺♥❣ ✈✐Ö❝ ❣✐➯ sư r➺♥❣ ❝ã

♠ét ➤é ➤♦ ①➳❝ s✉✃t t✐➟♥ ♥❣❤✐Ư♠

π

tr➟♥ ❦❤➠♥❣ ợ

( , ( ))

tr tự tế tì ì t ệ ề ợ ①Ø✳ ❉♦ ➤ã✱ ❦Õt ❧✉❐♥ ❇❛②❡s
♠➵♥❤ ❧✐➟♥ q✉❛♥ ➤Õ♥ sù ♣❤➞♥ tÝ❝❤ ♠➭ t✐➟♥ ♥❣❤✐Ư♠ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ t❐♣ ❝đ❛ ❝➳❝ t✐➟♥
♥❣❤✐Ư♠✳ ❈ơ t❤Ĩ✱ sù ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝ tr♦♥❣ ệ ị rõ t ệ sẽ ợ
rõ t q✉❛ ✈✐Ư❝ ①Ðt ♠ét ❧í♣ ➤é ➤♦ ①➳❝ s✉✃t
✧➤ó♥❣✧

π0 ✳ ❚❐♣ ❝đ❛ ❝➳❝ t✐➟♥ ♥❣❤✐Ư♠ P

P

tr➟♥

( , σ( )) ❝❤ø❛ t✐➟♥ ♥❣❤✐Ư♠

❝➯♠ s✐♥❤ ❝➳❝ ❜❛♦ ❤×♥❤ tr➟♥ ✈➭ ❞➢í✐ tr➟♥

π0 ✿

L(A) = inf π(A) ❀ U (A) = sup π(A) ❀ ∀A ∈ σ( )✳
P

❑❤✐ ➤ã✱ ❦Õt ❧✉❐♥ sÏ ❞ù❛ tr➟♥
tr➟♥(

✶✳✷✳✸✳

P

L✳

❈❤ó ý r➺♥❣

L(·) ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ➤é ➤♦ ①➳❝ s✉✃t

, σ( ))✱ ❜ë✐ ✈× ♥ã ❦❤➠♥❣ ❝é♥❣ tÝ♥❤✳
P❤➞♥ tÝ❝❤ ữ ệ t

ữ ệ t ột trờ ợ ể ❤×♥❤ ♠➭ ❝➳❝ q✉❛♥ s➳t ❧➭ ❝➳❝ t❐♣ t❤❛② ✈×
❝➳❝ ➤✐Ĩ♠ tr♦♥❣ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➱✉✳ ❑❤✐ ❞÷ ❧✐Ư✉ ❝ã s➼♥ ❧➭ ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝ ❤♦➷❝
❞♦ sù s❛✐ sãt ❝ñ❛ ♣❤➢➡♥❣ ♣❤➳♣ t❤✉ t❤❐♣ ❞÷ ❧✐Ư✉ ✭✈Ý ❞ơ ♥❤➢ tÝ♥❤ ❦❤➠♥❣ ❝❤Ý♥❤ ①➳❝ ❝đ❛
❝➳❝ ❞ơ♥❣ ❝ơ ➤♦✮ ❞➱♥ ➤Õ♥ ❞÷ ệ ó t ợ t ọ ữ ệ t r
trờ ợ t ì ố ❣➳♥ ❝➳❝ ❣✐➳ trÞ ❞✉② ♥❤✃t ❝❤♦ ❝➳❝ q✉❛♥ s➳t✱ t❛ ♥➟♥

✽

∈


➤➢❛ r❛ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ♣❤Ð♣ t❤ư ♥❣➱✉ ♥❤✐➟♥ ❝❤Ý♥❤ ❧➭ ❝➳❝ t❐♣ ❝♦♥ ♠➭ ❝❤ø❛ ❝➳❝ ❣✐➳ trÞ
q✉❛♥ s➳t ✧➤ó♥❣✧✳ ❙❝❤r❡✐❜❡r ➤➲ ➤➢❛ r❛ ❤Ư ❞➭♥ tỉ♥❣ q✉➳t ❝❤♦ ❝➳❝ q✉❛♥ s➳t ❝ã ❣✐➳ trÞ t❐♣
♠➭ sư ❞ơ♥❣ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳
❈ơ t❤Ĩ✱ ❝❤♦ X ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ❝➳❝ ❦Õt q✉➯ ❦❤➠♥❣ q✉❛♥ s➳t ➤➢ỵ❝ X1 , X2 , Ã Ã Ã
ớ ỗ

Xj tồ t t❐♣ Sj

s❛♦ ❝❤♦

Xj ∈ Sj , j = 1, 2, · · · , n✳

❚r♦♥❣ ➤ã

Sj , j =

1, 2, · · · , n ❧➭ ❝➳❝ ❦Õt q✉➯ ❝ã t❤Ĩ q✉❛♥ s➳t ➤➢ỵ❝ ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ S ✳
t❛ ♥ã✐

X

❧➭ ♠ét ❜é ❝❤ä♥ ❤➬✉ ❝❤➽❝ ❝❤➽♥ ❝ñ❛

❚❛ ①Ðt ❜➭✐ t♦➳♥ s✉② ❧✉❐♥ t❤è♥❣ ❦➟ ✈Ò

, Xn ✳

❑❤✐ ➤ã

S✳

X ✿ ➛í❝ ❧➢ỵ♥❣ ❝➳❝ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❝đ❛ X ✱

❜➭✐ t♦➳♥ ♥➭② sÏ ❞ù❛ tr➟♥ ❞÷ ❧✐Ư✉ ❝đ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥

Sj , j = 1, 2, · · · , n✳

➜Ĩ ♥❣❤✐➟♥ ❝ø✉ ❜➭✐ t♦➳♥ ➢í❝ ❧➢ỵ♥❣ tr➟♥✱ t❛ ❝➬♥ ♠ét ❧ý t❤✉②Õt ❝❤➷t ❝❤Ï ✈Ò ❝➳❝ t❐♣ ♥❣➱✉
♥❤✐➟♥✱ ➤➷❝ ❜✐Ưt ❧➭ ❝➳❝ ♣❤➞♥ ❜è ❝đ❛ ❝❤ó♥❣✳
❚r♦♥❣ tr➢ê♥❣ ợ tự tế tí ữ ệ t t tể q st ợ í
trị ủ ♠ét ♠➱✉ ♥❣➱✉ ♥❤✐➟♥

q✉❛♥ s➳t ♥➭② tr♦♥❣ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥

X1 , · · · , Xn ❝ñ❛ X ✱ t❛ ❝è ❣➽♥❣ ➤Þ♥❤ ✈Þ ❝➳❝

Sj , j = 1, · · · , n ♠➭ ❝ã t❤Ó q✉❛♥ s➳t ợ

ó rt ề ể ề ỗ ➤➢❛ r❛ ♠ét sù ❧➭♠ t❤➠ ❞÷ ❧✐Ư✉ ❦❤➳❝
♥❤❛✉✳ ❈➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥
❧➭♠ t❤➠

S

❝đ❛

X

✈í✐

S

Sj

➤➢ỵ❝ ①❡♠ ♥❤➢ ❧➭ ♠ét ♠➱✉ ♥❣➱✉ ♥❤✐➟♥ rót r❛ tõ ♠ét sù

❧➭ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥✳ ❑❤✐ ➤ã✱ t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ sù ❧➭♠ t❤➠

♥❤➢ s❛✉✿
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✻✳

▼ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ S ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét sù ❧➭♠ t❤➠ ❝ñ❛ X

♥Õ✉ S ❝❤ø❛ X ❤➬✉ ❝❤➽❝ ❝❤➽♥✱ ❝ã ♥❣❤Ü❛ ❧➭ X ❧➭ ♠ét ❜é ❝❤ä♥ ❤➬✉ ❝❤➽❝ ❝❤➽♥
❝đ❛ S ✳
❈❤ó ý r➺♥❣

X

➤➲ ➤➢ỵ❝ ❝❤♦ tr➢í❝✱ ✈➭

S ❧➭ ♠ét ♠➠ ❤×♥❤ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❝❤♦ X ✳

▼ét ì ữ í sự t ❤×♥❤ ❈❆❘✱ tr♦♥❣ ➤ã ❈❆❘ ❧➭ ✈✐Õt t➽t ❝đ❛
✧❝♦❛rs❡♥♥✐♥❣ ❛t r❛❞♦♠✧✱ ❝ã ♥❣❤Ü❛ ❧➭ sù ❧➭♠ t❤➠ ♥❣➱✉ ♥❤✐➟♥✳ ❚❛ sÏ ♥❣❤✐➟♥ ❝ø✉ ♠➠
❤×♥❤ ♥➭② tr♦♥❣ ❝❤➢➡♥❣ s❛✉✳

❱Ý ❞ơ✿ ❈❤♦ U ⊆ R ❧➭ ♠✐Ị♥ ❣✐➳ trÞ ❝đ❛ X ✱ ✈➭ {A1 , · · · , Ak } ❧➭ ♠ét ♣❤➞♥ ❤♦➵❝❤ ✭➤♦
➤➢ỵ❝✮ ❝đ❛

U ✳ ❳Ðt s➡ ➤å ❧➭♠ t❤➠✳
S : (Ω, A, P ) → {A1 , · · · , Ak }✳
✾


●✐➯ sö r➺♥❣ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❝❤➢❛ ❜✐Õt ❝ñ❛

f (x|θ), θ ∈

✳

X

❝ã ❞➵♥❣ t❤❛♠ sè ❤ã❛✱ tø❝ ❧➭

Sj , j = 1, 2, · · · , n ❧➭ ♠ét ♠➱✉ ➤é❝ ❧❐♣ ✈➭ ❝ã ❝ï♥❣ ♣❤➞♥ ♣❤è✐ ✈í✐

S ✳ ❑❤✐ ➤ã✱ ❤➭♠ ❤ỵ♣ ❧ý ❞ù❛ tr➟♥ ♠ét ♠➱✉ ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥ S

❧➭✿

n

L(θ|S1 , · · · , Sn ) =

f (x|θ)dx
j=1 S

j

❉♦ ➤ã✱ ❝➠♥❣ t❤ø❝ ➢í❝ ❧➢ỵ♥❣ ❤ỵ♣ ❧ý ❝ù❝ ➤➵✐ ❝đ❛
❞ơ♥❣

S1 , · · · , Sn ✳

θ

❝ã t❤Ĩ ➤➢ỵ❝ tÝ♥❤ ♠➭ ❝❤Ø ❝➬♥ sư

❚✉② ♥❤✐➟♥✱ sù ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ♠➱✉ ❧í♥ ❝đ❛ ❝➠♥❣ t❤ø❝

➢í❝ ❧➢ỵ♥❣ ❝ị♥❣ ②➟✉ ❝➬✉ ❝➳❝ ❦❤Ý❛ ❝➵♥❤ ♣❤➞♥ ❜è ❝đ❛ ♠➠ ❤×♥❤ t❐♣ ♥❣➱✉ ♥❤✐➟♥

✶✳✷✳✹✳

S✳

❚❤➠♥❣ t✐♥ ❞ù❛ tr➟♥ tứ

ệ tí ữ ệ t ó trò q✉❛♥ trä♥❣ tr♦♥❣ q✉➳ tr×♥❤ t❤✉ t❤❐♣
t❤➠♥❣ t✐♥ ❞ù❛ tr➟♥ ♥❤❐♥ t❤ø❝ ❝đ❛ ❝♦♥ ♥❣➢ê✐✳

❱Ý ❞ơ ✶✿ ❈❤ó♥❣ t❛ ❦❤➠♥❣ t❤Ĩ ❜➺♥❣ ♠➽t t❤➢ê♥❣ ♠➭ ➤♦ ➤➢ỵ❝ ♠ét ❝➳❝❤ ❝❤Ý♥❤
ữ ị trí ó ì t❛ ❝è ❣➽♥❣ rót r❛ t❤➠♥❣ t✐♥ ❝ã Ý❝❤ ❜➺♥❣
❝➳❝❤ ①❡♠ ①Ðt ♠ét ✈➭✐ s➡ ➤å ➤➡♥ ❣✐➯♥ tr➟♥ ♠✐Ò♥ ❣✐➳ trÞ ❝đ❛ ❝➳❝ ♣❤Ð♣ ➤♦✱ ♥❤➢ ♠ét ♣❤➞♥
❤♦➵❝❤ ❝đ❛ ♥ã✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❦❤✐ t❛ ❦❤➠♥❣ t❤Ĩ ➤♦ ➤➢ỵ❝ ♠ét ❝➳❝❤ ❝❤Ý♥❤ ①➳❝ ❝➳❝ ❣✐➳
trÞ ❝đ❛ ❜✐Õ♥

♥➭♦ ➤ã ♠➭ t❛ q✉❛♥ t➞♠✱ t❛ ❧➭♠ t❤➠ ♥ã✱ ♥❤➢ sư ❞ơ♥❣ t❐♣ ♥❣➱✉ ♥❤✐➟♥

X

♥➭♦ ➤ã s❛♦ ❝❤♦

S

P (X ∈ S) = 1 ➤Ĩ rót r❛ t❤➠♥❣ t✐♥ ✈Ị X ✳

❱Ý ❞ơ ✷✿ ▼ét t❤➠♥❣ t✐♥ ❞ù❛ tr➟♥ ♥❤❐♥ t❤ø❝ ❝ã ❞➵♥❣ ✧❍➯✐ tr❰✧ ❝ã ❝✃✉ tró❝ t❤❡♦ s❛✉✿
❇✐Õ♥ ❝➡ ❜➯♥ ♠➭ t❛ q✉❛♥ t➞♠ ❧➭

X = t✉ỉ✐ ✭❝đ❛ ❍➯✐✮ ✈í✐ ♠✐Ị♥ trị U = [0, 100]

ý ệ ữ tr ❦❤➠♥❣ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ râ r➭♥❣ ✈Ị ❜✐➟♥✱ ✈➭ ❞♦ ó ó tể ợ
ì ó ột t ờ ủ
trị ủ

X

tể q st ợ í

trị q✉❛♥ s➳t t❤❛② t❤Õ✳ ●✐➳ trÞ ♠ê
t❤Ĩ✱ ❦❤➠♥❣ ♣❤➯✐ ❝đ❛
♠ê✧ ❝ñ❛

U

U ✱ tø❝ ❧➭ ♥❤➢ ♠ét ❤➭♠ A : U [0, 1]

X

A =tr

ì

ợ ột

A tr tự tế ột tr ữ trị ờ ó

❝ñ❛ sù ❧➭♠ t❤➠

S ♥➭♦ ➤ã ❝ñ❛ X ✳ ❱Ý ❞ơ✿ ▼ét ✧♣❤➞♥ ❤♦➵❝❤

❝ã t❤Ĩ ❧➭ ✧r✃t tr❰✧✱ ✧tr❰✧✱ ✧tr✉♥❣ t✉æ✐✧✱ ✧❣✐➭✧✱ ♥➟♥

S

❧➭ ♠ét t❐♣ ♠ê ✭♥❣➱✉

♥❤✐➟♥✮✱ ❝ã ♥❣❤Ü❛ ❧➭ ♠ét ♣❤➬♥ tö ♥❣➱✉ ♥❤✐➟♥ ♥❤❐♥ ❝➳❝ t❐♣ ❝♦♥ ♠ê ủ

U

trị

ét trờ ợ ệt ủ t❐♣ ♥❣➱✉ ♥❤✐➟♥✱ ➤ã ❧➭ ❝➳❝ s➡ ➤å ❧➭♠ t❤➠ tr♦♥❣ q✉➳ tr×♥❤

✶✵


t❤✉ t❤❐♣ t❤➠♥❣ t✐♥ ❞ù❛ tr➟♥ ♥❤❐♥ t❤ø❝✳ ➜Ó tr➳♥❤ ❝➳❝ ❝❤✐ t✐Õt t➠♣➠ tr♦♥❣ ♣❤➬♥ ♥➭②✱ ❣✐➯
sư r➺♥❣ ♠✐Ị♥ trị

U

ột t ữ

t ủ ột ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
tr♦♥❣

U ✳ ❈❤♦ A ⊆ U

s✉✃t t❤❐t sù

s✉✃t ♥➭② ❧➭

A ➤➢ỵ❝ ❣ä✐ ❧➭ ①➯② r❛ ♥Õ✉ X(ω) ∈ A✳ ◆❤➢♥❣✱

A✳ ◆Õ✉ S(ω) ⊆ A t❤× A ①➯② r❛✳ ❉♦ ➤ã✱ t❛ ➤Þ♥❤

A ①➯② r❛ ❜➺♥❣ P (S ⊆ A) ♠➭ ①➳❝ s✉✃t ♥➭② ♥❤á ❤➡♥ ①➳❝

A ①➯② r❛ ❧➭ P (X ∈ A)✱ ✈× X

▼➷t ❦❤➳❝✱ ♥Õ✉

♥❤❐♥ ❣✐➳ trị

X() ỉ ó tể q st ợ S() tì râ r➭♥❣

t❛ ❦❤➠♥❣ ❝❤➽❝ ❝❤➽♥ ✈Ị sù ①➯② r❛ ❝đ❛
❧➢ỵ♥❣ ♠ø❝ ➤é t✐♥ ❝❐② r➺♥❣

❧➭ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ ♠➭ ❧➭ ♠ét

X ✱ ❝ã ♥❣❤Ü❛ ❧➭ P (X ∈ S) = 1✱ X

❧➭ ♠ét ❜✐Õ♥ ❝è✱

♥Õ✉ t❛ ❦❤➠♥❣ t❤Ó q✉❛♥ s➳t ➤➢ỵ❝

S

S(ω) ∩ A = ∅✱

P (S ∩ A = ∅)✳

❧➭ ♠ét ❜é ❝❤ä♥ ❤➬✉ ❝❤➽❝ ❝❤➽♥ ❝đ❛

t❤× ❝ã ❦❤➯



X

A

S

r t ị ợ

ột ộ ọ ❤➬✉ ❝❤➽❝ ❝❤➽♥ ❝ñ❛

S

♥➟♥

{X ∈ A} ⊆ {S ∩ A = ∅}✱ s✉② r❛ P (X ∈ A) ≤ P (S ∩ A = ∅)✳
❑❤✐ ➤ã✱ t❛ ❝ã ❝➞✉ ❤á✐ ❧➭✿ ❈ã tå♥ t➵✐ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥
s❛♦ ❝❤♦ ❤➭♠ t❐♣

S ✱ ♠➭ ❧➭ ♠ét ❧➭♠ t❤➠ ❝ñ❛ X ✱

Π : 2U [0, 1] ợ ị ĩ ở (A) = P (S ∩ A = ∅)✱

❧➭ ♣❤✐Õ♠ ❤➭♠ ♥ö❛ t❤ù❝✱ ❝ã ♥❣❤Ü❛ ❧➭✱ ♥Õ✉ ❜✐Õt

Π(A) ✈➭ Π(B) t❤× ❝ã ị ợ

(A B) ú t sẽ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❝❤➢➡♥❣ s❛✉ ➤Ĩ ❝ã ➤➢ỵ❝ ❝➞✉ tr➯
❧ê✐ ❧➭ ❦❤➻♥❣ ➤Þ♥❤✳

✶✳✷✳✺✳

▲✃② ♠➱✉ ①➳❝ s✉✃t

❚r➢í❝ ❤Õt✱ ➤Ĩ ❝ã t❤Ĩ ể ợ ì s ệ st ó ❧✐➟♥ ❤Ư ✈í✐ t❐♣
♥❣➱✉ ♥❤✐➟♥✱ t❛ ①Ðt ✈Ý ❞ơ s❛✉✿

❱Ý ❞ô✿ ❈❤♦ t❐♣ U ❝ã |U | = N ✱ ➤Ĩ ❝❤ä♥ ♥❣➱✉ ♥❤✐➟♥ ♠ét t❐♣ ❝♦♥ ❝đ❛ U ✈í✐ ❦Ý❝❤ ❝ì
❧➭

n ❝❤♦ tr➢í❝✱ t❛ ❝ã t❤Ĩ t➵♦ r❛ ♠ét ♠➠ ❤×♥❤ ♥❣➱✉ ♥❤✐➟♥ ➤Ị✉ ✈í✐ ①➳❝ s✉✃t
N
n

N!
n!(N −n)! ✱ tứ ỗ t ủ
1
ọ ố N ✳
n

➤ã

=

U

1
N ✱ tr♦♥❣
n

❝ã ❦Ý❝❤ t❤➢í❝ ♥ ➤Ị✉ ❝ã ❦❤➯ ♥➝♥❣ ợ

ừ ó t ó ị ĩ
ị ĩ

(, A, P ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t ✈➭ U ❧➭ ♠ét

t❐♣ ❤÷✉ ❤➵♥✳ ▲✃② ♠➱✉ ①➳❝ s✉✃t s✐♥❤ r❛ ột t S ợ ị ĩ
tr (Ω, A, P ) ♥❤❐♥ ❝➳❝ ❣✐➳ trÞ ❧➭ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ U ✳
✶✶


S : Ω → 2U ✈í✐ S(w) = A, w ∈ Ω, A ⊆ U
▼ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ ♥❤➢ ✈❐② ợ ọ ột t ữ ộ
st ủ ó ợ ị rõ tr t ũ t❤õ❛ ❝ñ❛ 2U ✳ ❈➳❝ ➤é ➤♦ ①➳❝ s✉✃t ❦❤➳❝ ♥❤❛✉
❞➱♥ ➤Õ♥ ❝➳❝ ♣❤➢➡♥❣ ➳♥ ❧✃② ♠➱✉ ❦❤➳❝ ♥❤❛✉✱ ❞♦ ➤ã t❛ ❝➬♥ ♣❤➯✐ ①❡♠ ①Ðt ❤➭♠ ♣❤➞♥ ❜è
❝ñ❛ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳
❈❤♦ ❤➭♠

f : 2U → [0, 1] ❝ã

f (A) = 1✱ ✈í✐ f (A) ❧➭ ①➳❝ s✉✃t ❝đ❛ ✈✐Ư❝ ❝❤ä♥
A⊆U

t❐♣ ❝♦♥

A ❝đ❛ U ✳ ▼ét ❤➭♠ f

s✉✃t✳ ▼➱✉

♥❤➢ ✈❐② ➤➢ỵ❝ ❝♦✐ ♥❤➢ ❧➭ ♠ét ♣❤➢➡♥❣ ➳♥ ❧✃② ♠➱✉ ①➳❝

A ➤➢ỵ❝ ❝❤ä♥ ❝❤Ý♥❤ ❧➭ ❦Õt q✉➯ ❝đ❛ ♣❤Ð♣ t❤ư ♥❣➱✉ ♥❤✐➟♥ ♠➭ t❤ù❝ ❤✐Ö♥ t❤❡♦

♣❤➢➡♥❣ ➳♥ ❧✃② ♠➱✉ ①➳❝ s✉✃t

f✳

P❤➢➡♥❣ ➳♥ ❧✃② ♠➱✉ ①➳❝ s✉✃t

f

❧➭ ❤➭♠ ✧♠❐t ➤é✧ ①➳❝ s✉✃t ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥

S ✱ tø❝

❧➭✱

f (A) = P (S = A), ∀A ⊆ U
❚❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ ❤➭♠ ♣❤đ ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ ♥❤➢ s❛✉✿

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✽✳

❈❤♦ S ❧➭ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❝ã ♠❐t ➤é ①➳❝ s✉✃t ❧➭ f ✳ ❍➭♠

π : U → [0, 1] ✈í✐ π(u) =

f (A) = P (u ∈ S) ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ♣❤đ ❝đ❛
A:A u

t❐♣ ♥❣➱✉ ♥❤✐➟♥ S ✳
●✐➳ trÞ

π(u) ❧➭ ①➳❝ s✉✃t ♠➭ u ①✉✃t ❤✐Ö♥ tr♦♥❣ ♣❤Ð♣ ❧✃② ♠➱✉ ➤➢ỵ❝ ✧rót✧ ♥❣➱✉

♥❤✐➟♥ ✈➭ ♣❤ơ t❤✉é❝ ✈➭♦ ♠❐t ➤é

f✳

❱Ý ❞ơ✿ ❚✐Õ♣ tơ❝ ✈Ý ❞ơ tr➢í❝✱ t❐♣ ♥❣➱✉ ♥❤✐➟♥ S ❜✐Ĩ✉ t❤Þ ✈✐Ư❝ ❝❤ä♥ ♠➱✉ ❝ã ❝ì n ❝❤♦
tr➢í❝✱ ➤é st

PS : E [0, 1]

ớ

PS

ủ

S

ợ ị ĩ t❤❡♦ s❛✉✿

P (A) ❀ A ⊆ 2U ❀

PS (A) =

tr♦♥❣ ➤ã

E

❧➭ t❐♣ ❧ị② t❤õ❛

A∈A

❝đ❛

2U

✈➭

A ∈ 2U
P (A) =


1

 N

|A| = n

♥Õ✉

|A| = n.

n


0
❑❤✐ ➤ã✱ t❛ ❝ã t❐♣ ♥❣➱✉ ♥❤✐➟♥

♥Õ✉

S ♥❤❐♥ ❣✐➳ trÞ ❧➭ ❝➳❝ ♠➱✉✱ ✈➭ PS

❧➭ ➤é ➤♦ ①➳❝ st ủ t

S


ợ ị ĩ ở tr


▼ét ♣❤Ð♣ tÝ♥❤ q✉❛♥ trä♥❣ ✈Ị ❤➭♠ ♣❤đ
♥❤✐➟♥
❈❤♦

π

➤ã ❧➭ ❦ú ✈ä♥❣ ❝đ❛ ❧ù❝ ❧➢ỵ♥❣ ❝đ❛ t❐♣ ♥❣➱✉

S✳

|A| ❧➭ ❧ù❝ ❧➢ỵ♥❣ ❝đ❛ t❐♣ A✱ ❦❤✐ ➤ã✱ ✈Ị ♠➷t ❤×♥❤ t❤ø❝
|A|f (A) =

E|S| =
A⊆U

f (A) =
A⊆U u:u∈A

f (A) =
u∈U A:A u

❑Õt q✉➯ ♥➭② ❧➭ ♠ét tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❝đ❛ ❝➠♥❣ t❤ø❝ ❘♦❜❜✐♥s✳

✶✸

π(u)
u∈U


❈❤➢➡♥❣ ✷✳

❈➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ❤÷✉ ❤➵♥

❚r♦♥❣ ❧ý t❤✉②Õt ①➳❝ s✉✃t ❝ỉ ➤✐Ĩ♥✱ ❜❛♥ ➤➬✉ t❛ ①Ðt ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ trị
tr ữ từ ó ➤✐ ♠ë ré♥❣ ①Ðt ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ ❝➳❝ ❦❤➠♥❣
❣✐❛♥ ①➳❝ s✉✃t tỉ♥❣ q✉➳t ❞ù❛ ✈➭♦ ❤Ư t✐➟♥ ➤Ị ❑♦❧♠♦❣♦r♦✈✳ ❚➢➡♥❣ tù ♥❤➢ ✈❐②✱ t❛ ❝ò♥❣ sÏ

①Ðt t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ tr➢í❝ ✈➭ tõ ❝➳❝ ❦Õt q✉➯ t❤✉ ➤➢ỵ❝ ❝đ❛
t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ ❝➳❝ ❦❤➠♥❣ ữ t ó tể ợ sẽ t ➤➢ỵ❝ tr♦♥❣
❝➳❝ ❦❤➠♥❣ ❣✐❛♥ trõ✉ t➢ỵ♥❣ ❤➡♥✳

✷✳✶✳

❚❐♣ ♥❣➱✉ ♥❤✐➟♥ ✈➭ ♣❤➞♥ ❜è ❝đ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥
❚r♦♥❣ s✉èt ❝❤➢➡♥❣ ♥➭②✱

❝đ❛

U

➤➢ỵ❝ ❦ý ệ ở

U

ột t ữ ự ợ ủ ột t❐♣ ❝♦♥

A

|A| ❤♦➷❝ #(A)✳ (Ω, A, P ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ①➳❝ s✉✃t✳

❚❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ t❐♣ ♥❣➱✉ ữ s
ị ĩ

ột t ữ trị tr 2U ột

X : Ω → 2U s❛♦ ❝❤♦✿

X −1 ({A}) = {ω ∈ Ω : X(ω) = A} ∈ A,

∀A ⊆ U

❈❤ó ý✿
✐✮ ❘â r➭♥❣✱ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ X ❧➭ ♠ét ♣❤➬♥ tö ♥❣➱✉ ♥❤✐➟♥ ♥Õ✉ (2U , E) ❧➭
✶✹


❦❤➠♥❣ ❣✐❛♥ ➤♦ ➤➢ỵ❝✱ ✈í✐
✭❞♦

E

❧➭ t❐♣ ❧ị② t❤õ❛ ❝đ❛

∀A ∈ E, X −1 (A) =

2U ✱ ❝ã ♥❣❤Ü❛ ❧➭ X −1 (E) ⊆ A

X −1 ({A})✮✳
A∈A

✐✐✮ ◆❤➢ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❝đ❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤÷✉ ❤➵♥✱ ➤✐Ị✉ ❦✐Ư♥ ✈Ị tí
ợ tr

E

tr

tết ể ị ĩ t ẽ ❧✉❐t ①➳❝ s✉✃t

PX

❝đ❛

X

q✉❛ ❝➳❝ ❣✐➳ trÞ ①➳❝ s✉✃t tr➟♥ ✧❝➳❝ t❐♣ ➤➡♥✧ ✲ ❝➳❝ t❐♣ ♥❤❐♥ ♠ét ❣✐➳ trÞ✳

❈ơ t❤Ĩ✱
ó

X

f

f : 2U [0, 1]

ợ ị ở

♠ét ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t t❤❐t sù tr➟♥

f (A) = P (X = A)✱
2U ✱

♥❣❤Ü❛ ❧➭

f ≥ 0

f (A) = P (X ∈ 2U ) = PX (2U ) = 1❀ ∀A ∈ E, PX (A) =
A⊆U

✈➭

f (A)✳
A∈A

❉ù❛ ✈➭♦ ❤➭♠ ♣❤➞♥ ❜è ❝ñ❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✱ t❛ sÏ ➤Þ♥❤ ♥❣❤Ü❛ râ r➭♥❣ ❦❤➳✐
♥✐Ư♠ ❤➭♠ ♣❤➞♥ ❜è ❝❤♦ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥✳
➜Þ♥❤ ❧ý ✷✳✶✳✶✳

❈❤♦ X ❧➭ ♠ét t❐♣ ♥❣➱✉ rỗ tr U sử

F : 2U [0, 1] ợ ị ở F (A) = P (X ⊆ A)✳ ❑❤✐ ➤ã F t❤á❛ ♠➲♥
❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿
✭✐✮ F (∅) = 0, F (U ) = 1
✭✐✐✮ ❱í✐ ❜✃t ❦ú sè k ≥ 2 ✱ ✈➭ A1 , · · · , Ak ❧➭ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ U ✱ t❛ ❝ã✿
k

(−1)|I|+1 F (

Aj ) ≥

F(
j=1

❈❤ø♥❣ ♠✐♥❤✳

❚Ý♥❤ ❝❤✃t

✭✷✳✶✮

i∈I

∅=I⊆{1,··· ,k}

✭✐✮

Ai )

✭✐✮

❧➭ ❤✐Ĩ♥ ♥❤✐➟♥ t❤❡♦ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý✳

F (∅) = P (X ⊆ ∅) = P (X = ) = 0 X

rỗ

F (U ) = P (X ⊆ U ) = 1
✭✐✐✮

❈❤♦

B ⊆ U✱

B⊆

➤➷t

J(B) = {i : i = 1, · · · , k

s❛♦ ❝❤♦

B ⊆ Ai }✱

♥❣❤Ü❛ ❧➭

Ai ✳
i∈J(B)
k

❘â r➭♥❣✿

F(
j=1

f (B) ≥

Aj ) =
B⊆

f (B)
B⊆U,J(B)=0

k

Aj

j=1

✶✺

✭✶✮


(1)|B|+1 = 1

A,

ì ớ t ỳ t ữ rỗ

=BA
ụ ớ

A = J(B) t ó
(1)|I|+1 f (B)

f (B) =

B⊆U,J(B)=∅ ∅=I⊆J(B)

B⊆U,J(B)=∅

(−1)|I|+1

=

∅=I⊆J(B)

f (B)
B⊆U,I⊆J(B)

(−1)|I|+1

=
∅=I⊆{1,2,··· ,k}

f (B)
B⊆

Ai
i∈I

(−1)|I|+1 F

=
⇒

(2)

i∈I

∅=I⊆{1,2,··· ,k}
❚õ ✭✶✮ ✈➭ ✭✷✮

Ai

➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤

❚Ý♥❤ ❝❤✃t ✭✐✮ ✈➭ ✭✐✐✮ ❝đ❛ ❤➭♠

F

tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶ ë tr➟♥ ♠✐➟✉ t➯ ❤➭♠

♣❤➞♥ ❜è ❝ñ❛ ❝➳❝ t rỗ í t ỳ ❤➭♠ t❐♣

F : 2U → [0, 1]

t❤á❛ ♠➲♥ ✭✐✮✱ ✭✐✐✮ ❝❤Ý♥❤ ❧➭ ♠ét ❤➭♠ ♣❤➞♥ ❜è ❝ñ❛ ♠ét t❐♣

♥❣➱✉ ♥❤✐➟♥ rỗ ớ trị tr
ị ĩ

2U

ột ❤➭♠ F : 2U → [0, 1] t❤á❛ ♠➲♥ ❝➳❝ tí t

ủ ị ý ợ ọ ❧➭ ❤➭♠ ♣❤➞♥ ❜è tr➟♥ 2U ✳
❚❛ t❤✃② r➺♥❣ ❤➭♠ ♣❤➞♥ ❜è

F (A) = P (X ⊆ A)

❧➭ ➤➡♥ ➤✐Ö✉

❧➭ ✷✲➤➡♥ ➤✐Ư✉✱ ❝ã ♥❣❤Ü❛ ❧➭

F

❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥

(A ⊆ B ⇒ F (A) ≤ F (B))✱

X✱

♥❣❤Ü❛ ❧➭

✈➭ ❤➡♥ ♥÷❛

F

∀A, B ⊆ U ✿ F (A∪B) ≥ F (A)+F (B)−F (A∩B)✳

◆❤➢♥❣✱ ♥❣➢ỵ❝ ❧➵✐ ♠ét ❤➭♠ ❜✃t ❦ú

F : 2U → [0, 1]

❝ã t❤Ĩ ❦❤➠♥❣ ❝➬♥ t❤✐Õt ❧➭ ➤➡♥ ➤✐Ư✉✱ ✈× ❝❤♦

♠➭ ❧➭ ✷✲➤➡♥ ➤✐Ö✉

A ⊆ B ✱ t❛ ❝ã✿

F (B) = F (A ∪ (B\A)) ≥ F (A) + F (B\A) − F (∅)
❉♦ ➤ã✱ ♥Õ✉

F (∅) = 0✱

♥❣❤Ü❛ ❧➭

➤➡♥ ệ ế tì

F

F ()

ột trị ỏ t ❝đ❛

F

t❤×

F

❧➭

❝ã t❤Ĩ ❦❤➠♥❣ ➤➡♥ ➤✐Ư✉ ✭✈➭ ❞♦ ➤ã ♥ã ❦❤➠♥❣ ♣❤➯✐

❧➭ ♠ét ❤➭♠ ♣❤➞♥ ❜è ❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥ ♥➭♦ ➤ã✮✳

✶✻


◆Õ✉

F

❧➭ ❤➭♠ ♣❤➞♥ ❜è ❝đ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥

X

✈í✐ ♠❐t ➤é

f ✱ t❤× râ r➭♥❣

F (∅) = f (∅) = P (X = ∅)✳ ❚õ ♥❛② trë ➤✐ t❛ sÏ ớ ỉ ét t
rỗ ĩ ❧➭✱ ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥

X

♠➭ ❝ã

F (∅) = f (∅) = 0✳

➜✐Ị✉ ♥➭② sÏ ❝❤♦ ♣❤Ð♣ ❝❤ó♥❣ t❛ ❝ã ♠ét ị ĩ rõ r ề
ố trừ tợ
ú ý r➺♥❣ ❤➬✉ ❤Õt ❝➳❝ t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr♦♥❣ ❝➳❝ ứ ụ t
rỗ í ụ ♥Õ✉

F = inf P

✭♥❣❤Ü❛ ❧➭✱

P

∀A ⊆ U, F (A) = inf{P (A) : P ∈ P}✮

❍➡♥ ♥÷❛✱ ♥Õ✉ t❐♣ ♥❣➱✉ ♥❤✐➟♥
➤ã

❧➭ ♠ét ❧í♣ ❝➳❝ ➤é ➤♦ ①➳❝ s✉✃t tr➟♥

(P (Y ∈ X) = 1)✱

X

❝ã

(U, 2U )✱ t❤×
F (∅) = 0✳

❧➭ ♠ét ❧➭♠ t❤➠ ❝đ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

t❤× ❝➬♥ t❤✐Õt ♣❤➯✐ ó

X

Y



ột t

rỗ
ú t sẽ ỉ r r➺♥❣ ♥Õ✉

F

❧➭ ❤➭♠ ♣❤➞♥ ❜è tr➟♥

❝ñ❛ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ rỗ
st

ó tr

U

ĩ tồ t ♠ét

(Ω, A, P ) ✈➭ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ ✭❦❤➳❝ rỗ X

tr

U

F (A) = P (X A), A U ✳ ➜Ĩ ❧➭♠ ➤➢ỵ❝ ➤✐Ị✉ ♥➭②✱ t❛ ❝❤Ø ❝➬♥ ❝❤Ø

s❛♦ ❝❤♦
r❛ r➺♥❣

X

2U ✱ t❤× ♥ã ❧➭ ❤➭♠ ♣❤➞♥ ❜è

F

❝ã ❞➵♥❣✿

F (A) =

f (B)
B⊆A

tr♦♥❣ ➤ã

f : 2U → [0, 1] ❧➭ ♠ét ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t tr➟♥ 2U ✳

➜Þ♥❤ ❧ý ✷✳✶✳✷✳

◆Õ✉ F : 2U → [0, 1] t❤á❛ ♠➲♥✿

✐✮ F (∅) = 0, F (U ) = 1
✐✐✮ ∀k ≥ 2 ✱ ✈➭ A1 , · · · , Ak ❧➭ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ U ✱
k

F(

(−1)|I|+1 F (

Aj ) ≥

j=1

t❤× ∀A ⊆ U,

F (A) =

Ai )

i∈I

∅=I⊆{1,··· ,k}

f (B)
B⊆A

❱í✐ f : 2U → [0, 1] t❤á❛ ♠➲♥✿ ❛✮ f (·) ≥ 0 ✈➭
❜✮

f (B) = 1✳
B⊆U

✶✼


ứ



f : 2U [0, 1] ợ ị ♥❣❤Ü❛ ❜ë✐✿
(−1)|A\B| F (B)✱

f (A) =
B⊆A
❑❤✐ ➤ã

f

❧➭ ❦❤➠♥❣ ➞♠✳

❚❤❐t ✈❐②✱ t❤❡♦

(i) : f (∅) = F (∅) = 0✱

✈➭ t❤❡♦ ❝➳❝❤ ①➞② ❞ù♥❣ t❤×

f ({u}) =

F ({u}) ≥ 0✳
❈❤♦

A⊆U

✈í✐

|A| ≥ 2✱ ➤➷t A = {u1 , · · · , un }✱

Ai = A\{ui }, i = 1, · · · , n
❑❤✐ ➤ã✿

n

n

f (A) = F (A) −

n−1

F (Ai ∩ Aj ) + · · · + (−1)

F (Ai ) +
i=1

i
Aj

F
i=1

j=i

n

≥ 0 ✭❞♦ A =

Ai ✮
i=1
n

(−1)|I|+1 F

Ai ≥

F (A) = F
i=1

Ai
i∈I

∅=I⊆{1,··· ,n}

n

Ai = ∅✮

✭❝❤ó ý r➺♥❣
❱❐②

❜✮

f (·) ≥

i=1
0✳

❚❛ ❝ã✿

(−1)|B\C| F (C) =

f (B) =
B⊆A

B⊆A C⊆B

C⊆B⊆A

◆Õ✉

C = A t❤× ❜✐Ĩ✉ t❤ø❝ ❝✉è✐ ❜➺♥❣ F (A)✳

◆Õ✉

C =A

B

♠➭

A\C ✱
♠➲♥

t❤×

A\C

C ⊆ B ⊆ A

❝ã

✭✈×

B

♠➭ sè t❐♣ ❝♦♥ ❝đ❛

C⊆B⊆A

2|A\C|

t❐♣ ❝♦♥✱ ❞♦ ➤ã ❝ã ♠ét sè ❝❤➼♥ ❝➳❝ t❐♣ ❝♦♥

❜➺♥❣ ❤ỵ♣ ❝đ❛

A\C

(−1)|B\C| F (C)

❧➭

2|A\C|

C

✈í✐ ♠ét tr♦♥❣ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛

❧➭ ♠ét sè ❝❤➼♥ ♥➟♥ sè t❐♣

❧➭ ❝❤➼♥✮✱ ❝❤Ý♥❤ ①➳❝ t❤× ♠ét ♥ư❛ sè t❐♣

❝➳❝ ♣❤➬♥ tư✳ ❑❤✐ ➤ã ♠ét ♥ư❛ ❝đ❛ ❝➳❝ sè
✈➭ ♠ét ♥ư❛ ❧➭ ❜➺♥❣

t❤á❛

❝ã ♠ét sè ❝❤➼♥

(−1)|B\C| ✱ ✈í✐ C ⊆ B ⊆ A✱ ❜➺♥❣ 1
(−1)|B\C| F (C) = 0

−1✳ ❱× ✈❐②✱ ỗ C = A
CBA



B

B


✈í✐ tỉ♥❣ ➤➢ỵ❝ ❧✃② tr➟♥

B ✳ ❙✉② r❛
f (B) = F (A)
B⊆A

➜➷❝ ❜✐Öt

1 = F (U ) =

f (B)✱
B⊆U

❉♦ ➤ã

f

❧➭ ♠ét ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t tr➟♥

2U ✳

❈❤ó ý✿
❛✮

◆Õ✉

F

❝ã ❞➵♥❣

F (A) =

f (B)

tì

f

ợ ị ĩ t

BA

F



(1)|A\B| F (B)

f (A) =

B⊆A

❜✮

➜✐Ị✉ ❦✐Ư♥ ✭✐✐✮ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ✭ ❤♦➷❝ ị ý ợ ọ tí

ệ ủ ✈➠ ❤➵♥ ❤♦➷❝ ➤➡♥ ➤✐Ư✉ ✈➠ ❤➵♥✳ ◆Õ✉
tr➟♥

U ✱ t❤× F

F

❧➭ ♠ét ➤é ➤♦ ①➳❝ s✉✃t

❧➭ ➤➡♥ ➤✐Ö✉ ✈➠ ❤➵♥ t❤❡♦ ➤➻♥❣ t❤ø❝ P♦✐♥❝❛r❡✬✳

❈❤ó ý r➺♥❣ ❝➳❝ ❤➭♠ ♣❤➞♥ ❜è✱ ♥❤➢ ❝➳❝ ❤➭♠ t❐♣✱ ♥❤×♥ ❝❤✉♥❣ ❧➭ ❦❤➠♥❣ ❝é♥❣
tÝ♥❤✳ ❚r➟♥ t❤ù❝ tÕ✱
▼ét ❤➭♠

F

∀A ⊆ U ✱ P (X ⊆ A) + P (X ∩ Ac = ∅) = 1✳

♠➭ t❤á❛ ♠➲♥ ✭✐✐✮ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ❝❤Ø ✈í✐ ♠ét sè

tr➢í❝ ó tì ợ ọ
ột t

F

k

k2



ệ

❧➭ ✷✲ ➤➡♥ ➤✐Ư✉✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ❜✃t ❦ú

A, B ⊆ U ✱ F (A) +

F (B) ≤ F (A ∩ B) + F (A ∪ B)✱ ❝ị♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ❧å✐ ❜ë✐ sù ❣✐è♥❣ ♥❤❛✉
✈í✐ ❝➳❝ ❤➭♠ ❧å✐ tr♦♥❣ ♣❤➞♥ tÝ❝❤ t❤ù❝✳
◆❤➢ ➤➲ ➤➢ỵ❝ ❝❤ó ý tr➢í❝✱ ❤➭♠
♥❤á ♥❤✃t ❝đ❛

F

tr➟♥

2U ✱

F

ë ➤➞②

✷✲ ➤➡♥ ➤✐Ư✉ ❧➭ ➤➡♥ ➤✐Ư✉ ♥Õ✉

F (∅) = 0✳

F (∅) ❧➭ ❣✐➳ trÞ

❉♦ ➤ã✱ ❝➳❝ ❤➭♠ ố

t ệ
r trờ ợ ữ ❤➵♥✱ t❛ ❝ị♥❣ ❝ã t❤Ĩ ♠➠ t➯ ➤➢ỵ❝ ❧✉❐t ①➳❝ st ủ
t rỗ t ột ♥✐Ư♠ ➤è✐ ♥❣➱✉ t❤➠♥❣ q✉❛ ❝➳❝
❤➭♠ ❦❤➯ ♥➝♥❣✳
➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✶✶✳

❈❤♦ X ❧➭ ♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ U ✱ t❛ ➤Þ♥❤ ♥❣❤Ü❛

♣❤✐Õ♠ ❤➭♠ ❦❤➯ ♥➝♥❣ ❝đ❛ X ❧➭ ❤➭♠ t❐♣✿
✶✾


T : 2U → [0, 1], T (A) = P (X ∩ A = ∅)
❱×

T (A) = 1 − F (Ac )✱ ✈í✐ F

t➯ ❧✉❐t ①➳❝ s✉✃t
❚Ý♥❤ ❝❤✃t ✸✳

PX

❝đ❛

X

X

❧➭ ❤➭♠ ♣❤➞♥ ❜è ❝đ❛

♥➟♥

T

❝ị♥❣ ♠✐➟✉

t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✶✳✷✳ ❚❛ ❝ã✿

❍➭♠ t❐♣ T ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿

α) T (∅) = 0, T (U ) = 1
β) ∀k ≥ 2✱ ✈➭ A1 , · · · , Ak tr♦♥❣ 2U ✱
k

(−1)|I|+1 T

Aj ≤

T
j=1

Ai
i∈I

∅=I⊆{1,··· ,k}

❈❤ø♥❣ ♠✐♥❤✳ α) T (∅) = 1 − F (U ) = 0; T (U ) = 1 − F (∅) = 1✳

β)
k

(−1)|I|+1 F

Aj ≥

F
j=1

Ai

✭➜Þ♥❤ ❧ý ✷✳✶✳✷✮

i∈I

∅=I⊆{1,··· ,k}

T (A) = 1 − F (Ac )
k

k

k

Aj = 1 − F

⇒T

Aj

c

j=1

j=1

j=1

Acj

=1−F

(−1)|I|+1 F

≤1−

Aci
i∈I

∅=I⊆{1,··· ,k}

(−1)|I|+1 F

≤1−

Ai
i∈I

∅=I⊆{1,··· ,k}

(−1)|I|+1 1 − T

≤1−

(−1)|I|+1 +

≤1−

Ai
i∈I

∅=I⊆{1,··· ,k}

∅=I⊆{1,··· ,k}

≤

c

(−1)|I|+1 T
∅=I⊆{1,··· ,k}

(−1)|I|+1 T

Ai
i∈I

∅=I⊆{1,··· ,k}

✷✵

Ai
i∈I


❈ơ t❤Ĩ✱ t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ râ r➭♥❣ ♣❤✐Õ♠ ❤➭♠ ❦❤➯ ♥➝♥❣ ❝đ❛ ❝➳❝ t❐♣ ♥❣➱✉
♥❤✐➟♥ ♥❤➢ s❛✉✿
➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✶✷✳

▼ét ❤➭♠ t❐♣ T : 2U → [0, 1] ❧➭ ♠ét ♣❤✐Õ♠ ❦❤➯ ♥➝♥❣ ❝ñ❛

♠ét t❐♣ ♥❣➱✉ ♥❤✐➟♥ ♥➭♦ ➤ã ♥Õ✉ ♥ã t❤á❛ ♠➲♥✿

α) T (∅) = 0, T (U ) = 1
β) ∀k ≥ 2✱ ✈➭ A1 , · · · , Ak tr♦♥❣ 2U ✱
k
j=1

T

T

β)

t❤á❛ ♠➲♥

Ai )

i∈I

∅=I⊆{1,··· ,k}

❚Ý♥❤ ❝❤✃t
tr➢í❝✱

(−1)|I|+1 T (

Aj ) ≤

T(

➤➢ỵ❝ ❣ä✐ ❧➭ tÝ♥❤ ❝❤✃t t❤❛② ♣❤✐➟♥ ❜❐❝ ✈➠ ❤➵♥✳ ❱í✐

β)

➤➢ỵ❝ ❣ä✐ ❧➭

k

F✱

✲ t❤❛② ♣❤✐➟♥✳ ❚➢➡♥❣ tù ♥❤➢

k

❝❤♦

❤➭♠ t❐♣

❧➭ ➤➡♥ ➤✐Ö✉✳

❚❤❐t ✈❐②✱ ❝❤♦

A ⊆ B ⇒ Ac ⊇ B c

⇒ F (Ac ) ≥ F (B c ) ⇒ 1 − T (A) ≥ 1 − T (B) ⇒ T (B) ≥ T (A)✳
❱Ý ❞ô ✶✿
❈❤♦

✭❚❐♣ ♠ø❝ ♥❣➱✉ ♥❤✐➟♥✮

α : Ω → [0, 1] ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ♣❤➞♥ ❜è ➤Ò✉ ✈➭ ❤➭♠ ϕ : U →

[0, 1]✳

❳Ðt

X : Ω → 2U

✈í✐

X(ω) = {u ∈ U : ϕ(u) ≥ α(ω)}✳

❑❤✐ ➤ã✱

X

❧➭ ➤♦ ➤➢ỵ❝ ✭t❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ò✉ ♥➭② ❜➺♥❣ ❝➳❝❤ ➤➷t ❧➵✐ t➟♥ ❝❤♦ ❝➳❝ ♣❤➬♥ tư
❝đ❛

U

➤Ĩ

ui ≤ uj ⇒ ϕ(ui ) ≤ ϕ(uj )✮✱

♠✐Ị♥ ❣✐➳ trÞ ❝đ❛

❤♦➭♥ t♦➭♥ ❜ë✐ ❜❛♦ ❤➭♠ t❐♣✳ ➜Ĩ ➤Þ♥❤ râ ♠❐t ➤é ①➳❝ s✉✃t

f

X

➤➢ỵ❝ s➽♣ t❤ø tù

❝đ❛

X

tr➟♥

2U ✱ ❝❤♦

t❤✉❐♥ t✐Ư♥ t❛ sư ❞ơ♥❣ ❦❤➳✐ ♥✐Ư♠ ❝đ❛ ❝➳❝ ❤➭♠ ♣❤➞♥ ❜è✳
➜➷t

F : 2U → [0, 1] ✈í✐ F (A) = P (X ⊆ A) = P (X ∩ Ac = ∅)✱ ⇒ F (A) =

1 − P (X ∩ Ac = ∅) = 1 − P {ω : α(ω) ≤ max
ϕ(u)} = 1 − max
ϕ(u) ✭α
c
c
A

A

❝ã ♣❤➞♥ ❜è ➤Ò✉✮✳
❚❛ ❝ã✿

F (A) =

f (B)✱ ❞♦ ➤ã t❛ ❝➬♥ ❜✐Ĩ✉ ❞✐Ơ♥ f

t❤❡♦

F✳

B⊆A

❱Ý ❞ơ ✷✿

❚✐Õ♣ tơ❝ ❱Ý ❞ơ ✶ ✈í✐

➤✴♥

F (A) = 1 − maxc ϕ(u) = 1 − T (Ac )

❚❛ t❤✃② r➺♥❣ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t

f

u∈A
❝ñ❛ t❐♣ ♥❣➱✉ ♥❤✐➟♥

(−1)|A\B| F (B) =

f (A) =
B⊆A

✷✶

X

➤➢ỵ❝ ❝❤♦ ❜ë✐

(−1)|A\B| [1 − maxc ϕ(u)]
B⊆A

u∈B