Luận văn thạc sĩ đánh giá độ tin cậy của hệ thống sử dụng mô hình rủi ro tỷ lệ cox

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆

❇ò✐ ❇→ ▼↕♥❤

✣⑩◆❍ ●■⑩ ✣❐ ❚■◆ ❈❾❨ ❈Õ❆ ❍➏ ❚❍➮◆●
❙Û ❉Ö◆● ▼➷ ❍➐◆❍ ❘Õ■ ❘❖ ❚✚ ▲➏ ❈❖❳

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❍➔ ◆ë✐ ✲ ✷✵✶✾


✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆

❇ò✐ ❇→ ▼↕♥❤

✣⑩◆❍ ●■⑩ ✣❐ ❚■◆ ❈❾❨ ❈Õ❆ ❍➏ ❚❍➮◆●
❙Û ❉Ö◆● ▼➷ ❍➐◆❍ ❘Õ■ ❘❖ ❚✚ ▲➏ ❈❖❳
❈❤✉②➯♥ ♥❣➔♥❤✿ ▲➼ t❤✉②➳t ①→❝ s✉➜t ✈➔ t❤è♥❣ ❦➯ t♦→♥ ❤å❝
▼➣ sè✿



ữớ ữợ ồ
P❍❸▼ ✣➐◆❍ ❚Ò◆●

❍➔ ◆ë✐ ✲ ✷✵✶✾

✐
▲í✐ ❝↔♠ ì♥
✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ tổ ổ ữủ
sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ ❚❙✳ P❤↕♠ ✣➻♥❤ ❚ị♥❣✱ ●✐↔♥❣ ✈✐➯♥
❚r÷í♥❣ ✤↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚æ✐ ①✐♥
❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t
❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ♣❤á♥❣ ✣➔♦ t↕♦✱ ỡ ồ
qỵ t ổ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ❑ ✶✼ ✲ ✶✾ ✭✷✵✶✼ ✲ ✷✵✶✾✮ ❚r÷í♥❣ ✤↕✐
❤å❝ ❦❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t
♥❤ú♥❣ ❦✐➳♥ tự qỵ ụ ữ t tổ ❤♦➔♥ t❤➔♥❤ ❦❤â❛
❤å❝✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ t tợ ỳ
ữớ ổ ở ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳
❚æ✐ ①✐♥ ❝↔♠ ì♥ sü ❤é trđ ❝õ❛ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ tr♦♥❣ ✤➲ t➔✐
◗●✳✶✽✳✵✸ tr♦♥❣ t♦➔♥ ❜ë q✉→ tr➻♥❤ ❧➔♠ ▲✉➟♥ ✈➠♥✳
❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦
❍➔ ◆ë✐✱ t❤→♥❣ ✳✳✳ ♥➠♠ ✷✵✳✳✳
◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥

❇ị✐ ❇→ ▼↕♥❤


❉❛♥❤ ♠ö❝ ❤➻♥❤
✶✳✶
✶✳✷
✶✳✸

✶✳✹
✶✳✺
✶✳✻
✶✳✼
✶✳✽
✶✳✾
✶✳✶✵
✶✳✶✶
✶✳✶✷
✶✳✶✸

❇✐➳♥ tr↕♥❣ t❤→✐ ✈➔ t✉ê✐ t❤å ❝õ❛ ♠ët ✤è✐ t÷đ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ♣❤➙♥ ❜è ①→❝ s✉➜t F (t) ✈➔ ❤➔♠ ♠➟t ✤ë ①→❝ s✉➜t f (t) ✳
❍➔♠ t✐♥ ❝➟② R(t) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❜➟❝ t❤❛♥❣ λ(t) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❜➟❝ t❤❛♥❣ λ(t) t✐➺♠ ❝➟♥ ✤➳♥ ♠ët ✤÷í♥❣ ❝♦♥❣ ❧✐➯♥ tư❝ ✳
❱à tr➼ ❝õ❛ ▼❚❚❋✱ tm ✈➔ tmode ❝õ❛ ♠ët ♣❤➙♥ ♣❤è✐ ✳ ✳
ỡ ỗ ố tố ữủ ❦➳t ♥è✐ ♥è✐ t✐➳♣ ✈➔ s♦♥❣ s♦♥❣ ✳
❍➺ t❤è♥❣ ❝➜✉ tró❝ ❤é♥ ❤đ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤➙♥ ♣❤è✐ ♠ơ (µ = 1) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ♠➟t ✤ë ①→❝ s✉➜t ❝õ❛ ♣❤➙♥ ♣❤è✐ ●❛♠♠❛✱ µ = 1 ✳ ✳ ✳ ✳
❍➔♠ t✐♥ ❝➟② ❝õ❛ ♣❤➙♥ ♣❤è✐ ●❛♠♠❛✱ µ = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ rõ✐ r♦ ❝õ❛ ♣❤➙♥ ♣❤è✐ ●❛♠♠❛✱ µ = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ♠➟t ✤ë ①→❝ s✉➜t ❝õ❛ ♣❤➙♥ ♣❤è✐ ❲❡✐❜✉❧❧ ✈ỵ✐ ♠ët sè ❣✐→
trà ❝õ❛ t❤❛♠ sè ❤➻♥❤ ❞↕♥❣ α ✈➔ β = 1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✹ ❍➔♠ t➾ ❧➺ rõ✐ r♦ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❲❡✐❜✉❧❧✱ β = 1✳ ✳ ✳ ✳ ✳ ✳ ✳

ỡ ỗ ố q tr P tử ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ t✐♥ ❝➟② ð ❝ị♥❣ 170 0 ❈ ✈ỵ✐ ✹ ♠ù❝ ✤✐➺♥ →♣ ❦❤→❝ ♥❤❛✉ ✳ ✳
❍➔♠ t✐♥ ❝➟② ð ❝ò♥❣ 180 0 ❈ ✈ỵ✐ ✹ ♠ù❝ ✤✐➺♥ →♣ ❦❤→❝ ♥❤❛✉ ✳ ✳
❍➔♠ t✐♥ ❝➟② ð ❝ị♥❣ 350 V ✈ỵ✐ ✷ ♠ù❝ ♥❤✐➺t ✤ë ❦❤→❝ ♥❤❛✉ ✳ ✳
❍➔♠ ♠➟t ✤ë ð ❝ò♥❣ 170 0 ❈ ✈ỵ✐ ✹ ♠ù❝ ✤✐➺♥ →♣ ❦❤→❝ ♥❤❛✉ ✳ ✳
❍➔♠ ♠➟t ✤ë ð ❝ị♥❣ 350 V ✈ỵ✐ ✷ ♠ù❝ ♥❤✐➺t ✤ë ❦❤→❝ ♥❤❛✉ ✳ ✳
❍➔♠ ♣❤➙♥ ♣❤è✐ ð ❝ò♥❣ 170 0 ❈ ✈ỵ✐ ✹ ♠ù❝ ✤✐➺♥ →♣ ❦❤→❝ ♥❤❛✉
❍➔♠ ♣❤➙♥ ♣❤è✐ ð ❝ị♥❣ 350 V ✈ỵ✐ ✷ ♠ù❝ ♥❤✐➺t ✤ë
ỗ t ữ ừ q s→t ❦❤æ♥❣ ❜à ❦✐➸♠ ❞✉②➺t

✐✐

✷✺
✷✺


❉❛♥❤ ♠ö❝ ❜↔♥❣
✷✳✶
✷✳✷
✷✳✸

❑➳t q✉↔ t❤û ♥❣❤✐➺♠ ❝❤♦ t✉ê✐ t❤å ❝õ❛ tư ✤✐➺♥
P❤➛♥ ❞÷ t÷ì♥❣ ù♥❣ ✸✷ q✉❛♥ s→t ❦❤ỉ♥❣ ❜à ❦✐➸♠

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
❞✉②➺t ✳ ✳ ✳ ✳ ✺✷

❚✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ❝õ❛ tö ✤✐➺♥ ð ❝→❝ ♠ù❝ ✤✐➺♥ →♣ ✈➔ ♥❤✐➺t
✤ë

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✐✐✐


▼ư❝ ❧ư❝
●✐ỵ✐ t❤✐➺✉ ❧✉➟♥ ✈➠♥

✶

✶ ▼ỉ ❤➻♥❤ ✤ë t✐♥ ❝➟②

✻

✶

✷

✸

❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶
❇✐➳♥ tr↕♥❣ t❤→✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷
❚✉ê✐ t❤å ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸
❍➔♠ t✐♥ ❝➟② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹
❍➔♠ t➾ ❧➺ rõ✐ r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✺

❚✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻
❑✐➸♠ ❞✉②➺t ✈➔ ❝❤➦t ❝öt ❞ú ❧✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✣ë t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶
❍➺ t❤è♥❣ ♥è✐ t✐➳♣ ✈➔ s♦♥❣ s♦♥❣ ✳ ✳ ✳ ✳

tố ỗ k ✤÷đ❝ ❧➜② r❛ tø n ❤➺ ❝♦♥
▼ët sè ♣❤➙♥ ♣❤è✐ t✉ê✐ t❤å t❤÷í♥❣ ❣➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✶
P❤➙♥ ♣❤è✐ ♠ô ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✷
P❤➙♥ ♣❤è✐ ●❛♠♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✳✸
P❤➙♥ ♣❤è✐ ❲❡✐❜✉❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ìợ ữủ t sè ❝õ❛ ♣❤➙♥ ♣❤è✐ t✉ê✐ t❤å ✳ ✳ ✳

✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳

✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳
✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✷ ▼æ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① ✭P❍▼✮
✶

✷

▼æ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶
▼æ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① ✭P❍▼✮
✶✳✷
▼ët sè ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳
ìợ ữủ t số tr ổ

ìợ ữủ t số ổ t✛ ❧➺
r♦ ❲❡✐❜✉❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐✈

✻
✻
✼
✽
✾
✶✷
✶✹

✶✻
✶✼
✶✾
✷✵
✷✵
✷✶
✷✹
✷✻

✷✾
✳ ✳
✳ ✳
✳ ✳
✳ ✳
rõ✐
✳ ✳

✳ ✳ ✳ ✳
✳ ✳ ✳ ✳
✳ ✳ ✳ ✳
✳ ✳ ✳ ✳
r♦ ❈♦①
✳ ✳ ✳ ✳

✳ ✳
✳ ✳
✳ ✳
✳ ✳
✈ỵ✐
✳ ✳

✳ ✳
✳ ✳
✳ ✳
✳ ✳
rõ✐
✳ ✳

✳
✳
✳
✳

✷✾
✷✾
✸✵
✸✷

✳ ✸✷







Pữỡ s ừ số ữợ ữủ
P t➼❝❤ ♣❤➛♥ ❞÷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❙ü ♣❤ị ❤đ♣ ❝õ❛ ♠æ ❤➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
Ù♥❣ ❞ö♥❣ ♣❤➙♥ t➼❝❤ ✤ë t✐♥ ❝➟② ❝õ❛ tö ✤✐➺♥

✺✳✶
❈♦① P❍▼ ❝❤♦ tö ✤✐➺♥ ✳ ✳ ✳ ✳ ✳

ìợ t t số ✳ ✳ ✳ ✳ ✳ ✳
✺✳✸
❈→❝ ✤↕✐ ❧÷đ♥❣ ✤➦❝ tr÷♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺✳✹
P❤➛♥ ❞÷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✺✳✺
❚✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

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✸✹
✸✺
✸✻
✸✼
✹✵
✹✶
✹✹
✺✷
✺✸

✺✽

✈


●✐ỵ✐ t❤✐➺✉ ❧✉➟♥ ✈➠♥

✶✳ ▲➼ ❞♦ ❝❤å♥ ✤➲ t➔✐
✣ë t✐♥ ❝➟② ❧➔ ①→❝ s✉➜t ♠➔ ♠ët ✤ì♥ ✈à ❤❛② ❤➺ t❤è♥❣ s➩ t❤ü❝ ❤✐➺♥ ❝❤ù❝ ♥➠♥❣
❞ü ✤à♥❤ ❝õ❛ ♠➻♥❤ ❝❤♦ ✤➳♥ ♠ët t❤í✐ ✤✐➸♠ ♥➔♦ ✤â tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ sû
❞ư♥❣ ❝ư t❤➸✳ ❱✐➺❝ ①→❝ ✤à♥❤ ✤ë t✐♥ ❝➟② ♥❤÷ ❧➔ ①→❝ ✤à♥❤ ❝❤➜t ❧÷đ♥❣ ❝õ❛ t❤✐➳t
❜à t❤❡♦ t❤í✐ ❣✐❛♥ ❤❛② ❝❤➼♥❤ ❧➔ ①→❝ ✤à♥❤ t✉ê✐ t❤å ❝õ❛ s↔♥ ♣❤➞♠ ✭t❤✐➳t ❜à✮✳
❑❤✐ ✤→♥❤ ❣✐→ ✤ë t✐♥ ❝➟② ❝õ❛ s↔♥ ♣❤➞♠ ✤÷đ❝ t❤✐➳t ❦➳✱ t❛ ♣❤↔✐ ①❡♠ ①➨t ❝❤✐
t✐➳t ✈➲ ❝→❝ tr÷í♥❣ ❤đ♣ ❣➙② ❧é✐ s↔♥ ♣❤➞♠ ✈➔ ❝ì ❝❤➳ t❤➜t ❜↕✐ ❤ä♥❣ ❤â❝ ❝õ❛
q✉→ tr➻♥❤ sû ❞ö♥❣ s↔♥ ♣❤➞♠✳ ❈→❝ s t tữớ ữ r ợ
ở t ❝➟② ❝❤➼♥❤ t❤ù❝ ❤♦➦❝ ❦❤æ♥❣ ❝❤➼♥❤ t❤ù❝ ❝õ❛ s↔♥ ♣❤➞♠ tt
ỳ õ t ỗ tứ q ự ợ s
tữỡ tỹ t tr♦♥❣ ❝æ♥❣ ♥❣❤✐➺♣✱ ❝→❝ ②➯✉ ❝➛✉ ❝õ❛ ❦❤→❝❤ ❤➔♥❣
❤♦➦❝ ♠ët ♠♦♥❣ ♠✉è♥ ❝↔✐ t❤✐➺♥ ✤ë t✐♥ ❝➟② ❤✐➺♥ ❝â ❝õ❛ s↔♥ ♣❤➞♠✳
❚r♦♥❣ t❤è♥❣ ❦➯✱ t❤í✐ ❣✐❛♥ ❤♦↕t ✤ë♥❣ ❝õ❛ ♠ët t❤✐➳t ❜à s↔♥ ♣❤➞♠ ✤÷đ❝ ❣å✐
❧➔ t✉ê✐ t❤å ❤❛② t❤í✐ ❣✐❛♥ sè♥❣ ❝õ❛ t❤✐➳t ❜à ✤÷đ❝ ❝♦✐ ❧➔ ♠ët ❜✐➳♥
ợ ố strt ử t ữ ♣❤è✐ ♠ô✱ ♣❤➙♥
♣❤è✐ ❲❡✐❜✉❧❧✳✳✳ ✣➸ ✤→♥❤ ❣✐→ t✉ê✐ t❤å ❝õ❛ t❤✐➳t ❜à s↔♥ ♣❤➞♠ ❦❤✐ t❤❛② ✤ê✐
❝→❝ ②➳✉ tè t→❝ ✤ë♥❣ ♥❤÷ ♥❤✐➺t ✤ë✱ ✤ë ➞♠✱ ✤ë r✉♥❣✱ sè❝ ♥❤✐➺t✱ ✳✳✳ ❝→❝ ❦ÿ s÷
t❤÷í♥❣ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ t❤û ♥❣❤✐➺♠ t➠♥❣ tè❝ tr♦♥❣ ♣❤á♥❣ t❤➼ ♥❣❤✐➺♠
✭❛❝❝❡❧❡r❛t❡❞ ❧✐❢❡ t❡st✐♥❣✮✳
❈æ♥❣ tr➻♥❤ ❧➔♠ ♥➯♥ t➯♥ t✉ê✐ ❝õ❛ ●❙✳ ❉❛✈✐❞ ❈♦① ❧➔ ❜➔✐ ❜→♦ ✏❘❡❣r❡ss✐♦♥
♠♦❞❡❧s ❛♥❞ ❧✐❢❡✲t❛❜❧❡s✑ ✤÷đ❝ ❝ỉ♥❣ ❜è tr➯♥ t➟♣ s❛♥ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❘♦②❛❧
❙t❛t✐st✐❝❛❧ ❙♦❝✐❡t② ♥➠♠ ✶✾✼✷✳ ❇➔✐ ❜→♦ ❝õ❛ ●❙✳ ❈♦① ❝❤♦ ✤➳♥ ♥❛② ✭s❛✉ ✹✽
♥➠♠✮ ✤➣ ❝â ❤ì♥ ✹✺✱✵✵✵ tr➼❝❤ ❞➝♥✦ ❇➔✐ ❜→♦ ♥➔② ✤÷đ❝ ✤→♥❤ ❣✐→ ❧➔ ♠ët tr
ổ tr ờ t t tứ trữợ ♥❛②✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ✤â✱ ỉ♥❣
♠ỉ t↔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤ ❝→❝ ❞ú ❧✐➺✉ sè♥❣ ❝á♥ t❤❡♦ ♠æ ❤➻♥❤ ỗ




q ổ s ữủ t ữợ t❤✉➟t ♥❣ú ✏❈♦①✬s ♣r♦♣♦rt✐♦♥❛❧

❤❛③❛r❞s ♠♦❞❡❧✑✳
▼æ ❤➻♥❤ ♥➔② ♣❤ê ❜✐➳♥ tr♦♥❣ ❤➛✉ ❤➳t ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝✱ tø ② ❦❤♦❛ ✤➳♥
①➣ ❤ë✐ ❤å❝ ✈➔ ❦❤♦❛ ❤å❝ ❦➽ t❤✉➟t✳ ❈❤➥♥❣ ❤↕♥✱ tr♦♥❣ ② ❦❤♦❛✱ ♠ỉ ❤➻♥❤ ✤÷đ❝
→♣ ❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ sü ữ ừ tố ữ tờ ợ t
❝❛♦✱ ❝➙♥ ♥➦♥❣✱✳✳✳ ✤➳♥ t✉ê✐ t❤å ❝õ❛ ❝→❝ ❜➺♥❤ ♥❤➙♥ ✉♥❣ t❤÷ s❛✉ ❦❤✐
✤÷đ❝ ✤✐➲✉ trà❀ tr♦♥❣ ❦❤♦❛ ❤å❝ ①➣ ❤ë✐✱ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ →♣ ❞ư♥❣ ♠ỉ ❤➻♥❤
✤➸ ♥❣❤✐➯♥ ự tớ số ừ ủ ỗ tø ❦❤✐ ❦➳t ❤ỉ♥
✤➳♥ ❧ó❝ ❧✐ ❞à❀ tr♦♥❣ ❦❤♦❛ ❤å❝ ❦➽ t❤✉➟t✱ ♠ỉ ❤➻♥❤ ❝â t❤➸ ✤÷đ❝ →♣ ❞ư♥❣ ✤➸
♥❣❤✐➯♥ ❝ù✉ sü ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❝→❝ t→❝ ♥❤➙♥ ✤➳♥ ✤ë t✐♥ ❝➟② ❝õ❛ ♠→② ♠â❝✳
❱➻ ✈➟②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ✧✣→♥❤ ❣✐→ ✤ë t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣ sû ❞ư♥❣
♠ỉ ❤➻♥❤ rõ✐ r♦ t✛ ❧➺ ❈♦①✧✳

✷✳ ▼ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
▲✉➟♥ ✈➠♥ ✤✐ t➻♠ ❤✐➸✉ ♠æ ❤➻♥❤ t✛ ❧➺ rõ✐ r ự ử ổ ợ
trữớ ủ rừ r♦ ❲❡✐❜✉❧❧ ✤➸ ❦✐➸♠ ❝❤ù♥❣ sü ♣❤ö t❤✉ë❝ ❝õ❛ ♥❤✐➺t ✤ë ✈➔
✤✐➺♥ →♣ ✤➳♥ t✉ê✐ t❤å ❝õ❛ tö ✤✐➺♥ t❤õ② t✐♥❤✳ ❚r➯♥ ❝ì sð ✤â✱ ✤÷❛ r❛ ❞ü ❜→♦
t✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ❝õ❛ tö ✤✐➺♥ t❤õ② t✐♥❤ ð ❝→❝ ♠ù❝ ♥❤✐➺t ✤ë ✈➔ ✤✐➺♥ →♣
❦❤→❝ ♥❤❛✉✳

✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
✸✳✶✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉
▼ỉ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① ✈ỵ✐ rõ✐ r♦ ❲❡✐❜✉❧❧ ✈➔ ù♥❣ ❞ư♥❣ ♠ỉ ❤➻♥❤ ✈➔♦ ♣❤➙♥
t➼❝❤ ✤ë t✐♥ ❝➟② ❝õ❛ tư ✤✐➺♥ t❤õ② t✐♥❤✳

✸✳✷✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ✤ë t✐♥ ❝➟② ❝õ❛ tö ✤✐➺♥ ❜➡♥❣ ❝→❝ t❤û ♥❣❤✐➺♠ t➠♥❣ tè❝ tr♦♥❣
♣❤á♥❣ t❤➼ ♥❣❤✐➺♠✱ ✤→♥❤ ❣✐→ t✉ê✐ t❤å ❝õ❛ tö ✤✐➺♥ t❤❡♦ ❝→❝ ❜✐➳♥ ↔♥❤ ❤÷ð♥❣
✭♥❤✐➺t ✤ë ✈➔ ✤✐➺♥ →♣✮✳

✷

✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
❙û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ t❤è♥❣ ❦➯ t❤❛♠ sè ✈ỵ✐ ♠ỉ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① tr÷í♥❣
❤đ♣ rõ✐ r♦ ❲❡✐❜✉❧❧ ✤➸ ✤→♥❤ ❣✐→ ✤ë t✐♥ ❝➟② ❝õ❛ tư ✤✐➺♥✳

✺✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ❝õ❛ ❧✉➟♥ ✈➠♥
✺✳✶✳ Þ ♥❣❤➽❛ ❦❤♦❛ ❤å❝
▲✉➟♥ ✈➠♥ ✤✐ t➻♠ ❤✐➸✉ q✉→ tr➻♥❤ ❲❡✐❜✉❧❧ P❍▼ ✈➔ ù♥❣ ❞ư♥❣ t❤➔♥❤ ❝ỉ♥❣ ♠æ
❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① ✤➸ ♣❤➙♥ t➼❝❤ ✤ë t✐♥ ❝➟② ❝õ❛ tö ✤✐➺♥ t❤õ② t✐♥❤✳ ❑➳t
q✉↔ ❝õ❛ ♠ö❝ ✺ tr♦♥❣ ❈❤÷ì♥❣ ✷ ✤➣ ❜ê s✉♥❣ ❝❤♦ ❝→❝ ❝ì sð ❧➼ ❧✉➟♥ tø ✤â ❣â♣
♣❤➛♥ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦✐➸♠ ❝❤ù♥❣ ✤➸ ❦❤➥♥❣ ✤à♥❤ r➡♥❣✿ ✧◆❤✐➺t ✤ë ✈➔ ✤✐➺♥
→♣ ❝â ữ ợ tờ tồ ừ tử từ t tờ tồ õ
ữợ t ở tỷ õ ữợ t
ụ r ❝→❝❤ t✐➳♣ ❝➟♥ ❤é trđ ❝❤♦ ✈✐➺❝ sû ❞ư♥❣ ❲❡✐❜✉❧❧ P❍▼
✤➸ ❞ü ✤♦→♥ ❝→❝ ✤➦❝ t➼♥❤ ✤ë t✐♥ ❝➟② ❝õ❛ ❝→❝ t❤✐➳t ❜à ✤✐➺♥ tû ♥â✐ ❝❤✉♥❣ ❦❤✐
❝â ✤÷đ❝ ❜ë ❞ú ❧✐➺✉ t❤û ♥❣❤✐➺♠✳

✺✳✷✳ Þ ♥❣❤➽❛ t❤ü❝ t✐➵♥
❚ø ❦➳t q✉↔ ❝õ❛ ▲✉➟♥ ✈➠♥ ❦❤➥♥❣ ✤à♥❤ r➡♥❣✿ ❚✉ê✐ t❤å ❝õ❛ tö õ
ữợ t ở õ ữợ t ữớ ũ õ t
ỹ ồ ổ tr÷í♥❣ ❧➔♠ ✈✐➺❝ ❝â ♥❤✐➺t ✤ë t❤➼❝❤ ❤đ♣ ✈➔ ✤✐➲✉ ❝❤➾♥❤ ✤✐➺♥ →♣
❤ñ♣ ❧➼ ✤➸ ✈ø❛ t✐➺♥ ❧ñ✐ ❝❤♦ ✈✐➺❝ sû ❞ư♥❣ ❝ơ♥❣ ♥❤÷ ❦➨♦ ❞➔✐ ❤ì♥ t✉ê✐ t❤å ❝õ❛
tư ✤✐➺♥✳

✻✳ ❚â♠ t➢t
❚➻♠ ❤✐➸✉ ♠æ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ự ử ổ ợ trữớ ủ
rừ r ❲❡✐❜✉❧❧ ♣❤➙♥ t➼❝❤ ✤ë t✐♥ ❝➟② ❝õ❛ ❜ë sè ❧✐➺✉✿✧❚❤í✐ ❣✐❛♥ sè♥❣ ❝õ❛ ✻✹
tö ✤✐➺♥ t❤õ② t✐♥❤✧✳ ✣➸ ❣✐↔✐ q✉②➳t ✤÷đ❝ ❝→❝ ✈➜♥ ✤➲ ✤➣ ♥➯✉✱ ▲✉➟♥ ✈➠♥ ✤÷đ❝
tr➻♥❤ ❜➔② t❤➔♥❤ ✷ ❝❤÷ì♥❣✿

✸


❈❤÷ì♥❣ ✶✳ ▼ỉ ❤➻♥❤ ✤ë t✐♥ ❝➟②
❚r➻♥❤ ❜➔② ✸ ♠ư❝ ❝❤➼♥❤✿ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥✱ ✤ë t✐♥ ❝➟② ❝õ❛ tố
ởt số ố tờ tồ tữớ

ã

❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥✿ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❜✐➳♥ tr↕♥❣
t❤→✐❀ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t✉ê✐ t❤å T ✈➔ ❝→❝ ✤➦❝ tr÷♥❣ ✭❤➔♠ t✐♥
❝➟②✱ ❤➔♠ t➾ ❧➺ rõ✐ r♦✱ t❤í✐ ❣✐❛♥ t❤➜t ❜↕✐ tr✉♥❣ ❜➻♥❤✮❀ ❦✐➸♠
❞✉②➺t ✈➔ ❝❤➦t ❝ưt ❞ú ❧✐➺✉✳

• ✣ë t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣✿ ❚r➻♥❤ ❜➔② tê♥❣ q✉❛♥ ✈➲ ✤ë t✐♥
❝➟② ❝õ❛ ❤➺ t❤è♥❣ ✈➔ ❝ư t❤➸ ❤â❛ ✈ỵ✐ tr÷í♥❣ ❤đ♣ ❤➺ t❤è♥❣ ✤÷đ❝
❦➳t ♥è✐ ♥è✐ t✐➳♣ ✈➔ ❤➺ t❤è♥❣ ✤÷đ❝ ❦➳t ♥è✐ s♦♥❣ s♦♥❣❀ ❦❤→✐ ♥✐➺♠
✈➔ ✤ë t✐♥ ừ trú ỗ k ữủ r tứ n

ã

ởt số ố tờ tồ tữớ ❣➦♣✿ ❚r➻♥❤ ❜➔② ✸
♣❤➙♥ ♣❤è✐ sè♥❣ sât t❤÷í♥❣ ✤÷đ❝ sû ❞ö♥❣ ❝❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
t✉ê✐ t❤å T ❧➔✿ P❤➙♥ ♣❤è✐ ụ ố ố
ữỡ ữợ ❧÷đ♥❣ t❤❛♠ sè ❝õ❛ ♣❤➙♥ ♣❤è✐
t✉ê✐ t❤å✳

❈❤÷ì♥❣ ✷✳ ▼ỉ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① ✭P❍▼✮
❚r➻♥❤ ❜➔② ✺ ♠ö❝ ❝❤➼♥❤✿ ổ ữợ ữủ t số tr ổ

t ♣❤➛♥ ❞÷✱ sü ♣❤ị ❤đ♣ ❝õ❛ ♠ỉ ❤➻♥❤ ✈➔ ù♥❣ ❞ư♥❣ ♣❤➙♥ t➼❝❤ ✤ë t✐♥ ❝➟②
❝õ❛ tư ✤✐➺♥✳

• ▼ỉ ❤➻♥❤✿ ❚r➻♥❤ ❜➔② ♠æ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦①✳ ❙❛✉ õ ử t
õ ổ ợ trữớ ủ rừ r♦ ❤➡♥❣ sè ✭λ0 (t) = λ✮
✈➔ tr÷í♥❣ ❤đ♣ rõ✐ r 0 (t) = t1
ã ìợ ữủ t số tr ổ r ữợ ữủ
t sè ♠ỉ ❤➻♥❤ t✛ ❧➺ rõ✐ r♦ ❈♦① ✈ỵ✐ rõ✐ r ữỡ
s ừ số ữợ ữủ
ã P t ữ r ữ
ã ❙ü ♣❤ị ❤đ♣ ❝õ❛ ♠ỉ ❤➻♥❤✿ ❚r➻♥❤ ❜➔② ✸ t❤è♥❣ ❦➯ ✤➸ ❦✐➸♠
tr❛ sü ♣❤ị ❤đ♣ ❝õ❛ ♠ỉ ❤➻♥❤✳ ❈→❝ t❤è♥❣ ❦➯ ✤÷đ❝ ❦➸ ✤➳♥ ❧➔✿

✹


❚❤è♥❣ ❦➯ ❦✐➸♠ ✤à♥❤ ✭t➾ sè ❤ñ♣ ❧➼✮✱ t❤è♥❣ ❦➯ ❙❝♦r❡ ✈➔ t❤è♥❣
❦➯ ❲❛❧❞✳

• Ù♥❣ ❞ư♥❣ ♣❤➙♥ t➼❝❤ ✤ë t✐♥ ❝➟② ❝õ❛ tư ✤✐➺♥✿ ❚❤✐➳t ❧➟♣
♠ỉ ❤➻♥❤ ❈♦① ✈ỵ✐ t✛ ❧➺ rõ✐ r♦ ❲❡✐❜✉❧❧ ❦✐➸♠ ❝❤ù♥❣ sü ↔♥❤ ❤÷ð♥❣
❝õ❛ ♥❤✐➺t ✤ë ✈➔ ✤✐➺♥ →♣ ✤➳♥ t✉ê✐ t❤å ❝õ❛ tö ✤✐➺♥ t❤õ② t✐♥❤✳

✺


❈❤÷ì♥❣ ✶
▼ỉ ❤➻♥❤ ✤ë t✐♥ ❝➟②
❈❤÷ì♥❣ ✶✱ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ tê♥❣ q✉❛♥ ✈➲✿ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t✉ê✐ t❤å
✈➔ ❝→❝ ✤➦❝ tr÷♥❣✱ ✤ë t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣ ✈➔ ♠ët sè ♣❤➙♥ ♣❤è✐ t✉ê✐ t❤å

t❤÷í♥❣ ❣➦♣✳

✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥
✶✳✶ ❇✐➳♥ tr↕♥❣ t❤→✐
❚r↕♥❣ t❤→✐ ❝õ❛ ♠ët ✤è✐ t÷đ♥❣ t↕✐ t❤í✐ ✤✐➸♠ t✱ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ X(t)✿

X(t) =

1 ♥➳✉ ✤è✐ t÷đ♥❣ ❤♦↕t ✤ë♥❣ t↕✐ t❤í✐ ✤✐➸♠ t❀
0 ♥➳✉ ✤è✐ t÷đ♥❣ t❤➜t ❜↕✐ t↕✐ t❤í✐ ✤✐➸♠ t.

❇✐➳♥ tr↕♥❣ t❤→✐ ❝õ❛ ♠ët ✤è✐ t÷đ♥❣ ✤÷đ❝ ♠✐♥❤ ❤å❛ ♥❤÷ ❍➻♥❤ ✶✳✶ ✈➔ t❤÷í♥❣
❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳

❍➻♥❤ ✶✳✶✿ ❇✐➳♥ tr↕♥❣ t❤→✐ ✈➔ t✉ê✐ t❤å ❝õ❛ ♠ët ✤è✐ t÷đ♥❣

✻


✶✳✷ ❚✉ê✐ t❤å
❚✉ê✐ t❤å ❝õ❛ ♠ët ✤è✐ t÷đ♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ tø ❦❤✐ ✤è✐
t÷đ♥❣ ❤♦↕t ✤ë♥❣ ❝❤♦ ✤➳♥ ❦❤✐ ❧➛♥ ✤➛✉ t✐➯♥ ✤è✐ t÷đ♥❣ t❤➜t ❜↕✐✳ ✣➦t t = 0
❧➔ ✤✐➸♠ ❜➢t ✤➛✉✳ ❚✉ê✐ t❤å ❧➔ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✱ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ T ✳ ▼è✐
❧✐➯♥ ❤➺ ❣✐ú❛ ❜✐➳♥ tr↕♥❣ t❤→✐ X(t) ✈➔ t✉ê✐ t❤å T ✤÷đ❝ ❜✐➸✉ t❤à ♥❤÷ ❍➻♥❤
1.1✳
❚✉ê✐ t❤å ❦❤ỉ♥❣ ♣❤↔✐ ❧ó❝ ♥➔♦ ❝ơ♥❣ ✤÷đ❝ ✤♦ ❜➡♥❣ t❤í✐ ❣✐❛♥ ♥❤÷ tr♦♥❣ ❧à❝❤✳
◆â ❝â t❤➸ ✤÷đ❝ ✤♦ ❜➡♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ t❤í✐ ❣✐❛♥ t ỡ

ã
ã

ã
ã

ố
ố
ố
ố

õ t ữủ ❤➔♥❤❀
❦✐✲❧ỉ✲♠❡t ❧→✐ ①❡❀
✈á♥❣ q✉❛② ❝õ❛ ê ✤ï trư❝❀
❝❤✉ ❦➻ ❝õ❛ ♠ët ✤è✐ t÷đ♥❣ ❧➔♠ ✈✐➺❝ ✤à♥❤ ❦➻✳

❚ø ♥❤ú♥❣ ✈➼ ❞ư tr➯♥ ♥❤➟♥ t❤➜② r➡♥❣✱ t✉ê✐ t❤å T t❤÷í♥❣ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
rí✐ r↕❝✳ ❚✉② ♥❤✐➯♥✱ ❝â t❤➸ ①➜♣ ①➾ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ ❜ð✐ ❜✐➳♥ ♥❣➝✉
♥❤✐➯♥ ❧✐➯♥ tư❝✳ ❱➻ ✈➟②✱ tr♦♥❣ ▲✉➟♥ ✈➠♥ s➩ ❧✉æ♥ ①➨t r➡♥❣ t✉ê✐ t❤å T ❧➔ ♠ët
❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧✐➯♥ tö❝✳ ❑➼ ❤✐➺✉ F (t) ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ✈➔ f (t)
❧➔ ❤➔♠ ♠➟t ✤ë ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t✉ê✐ t❤å T ✳ ❑❤✐ ✤â✿
t

F (t) = Pr (T ≤ t) =

f (u) du ợ t > 0



0

F (t) ữủ ❤✐➸✉ ❧➔ ①→❝ s✉➜t ♠ët ✤è✐ t÷đ♥❣ t❤➜t ❜↕✐ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥
(0, t]✳

❍➔♠ ♠➟t ✤ë ①→❝ s✉➜t f (t) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿
f (t) =

d
F (t + ∆t) − F (t)
F (t) = lim
∆t→0
dt
∆t
Pr (t < T ≤ t + ∆t)
= lim
∆t→0
∆t

✭✶✳✷✮

❱ỵ✐ ∆t ♥❤ä t❤➻ ❝ỉ♥❣ t❤ù❝ (1.2) ❝â t t ữợ
Pr (t < T t + ∆t) ≈ f (t) · ∆t
❍➔♠ ♣❤➙♥ ❜è ①→❝ s✉➜t F (t) ✈➔ ❤➔♠ ♠➟t ✤ë ①→❝ s✉➜t f (t) ✤÷đ❝ ❜✐➸✉ t❤à
♥❤÷ ❍➻♥❤ 1.2

✼


❍➻♥❤ ✶✳✷✿ ❍➔♠ ♣❤➙♥ ❜è ①→❝ s✉➜t F (t) ✈➔ ❤➔♠ ♠➟t ✤ë ①→❝ s✉➜t f (t)
✶✳✸ ❍➔♠ t✐♥ ❝➟②
❍➔♠ t✐♥ ❝➟② ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿

R(t) = 1 − F (t) = Pr (T > t) ✈ỵ✐ t > 0

✭✶✳✸✮

❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ (1.1)✱ ❝ỉ♥❣ t❤ù❝ (1.3) ❝â t❤➸ t ữợ
t

R(t) = 1

+

f (u) du =
0

f (u) du

✭✶✳✹✮

t

R(t) ❧➔ ①→❝ s✉➜t ♠ët ✤è✐ t÷đ♥❣ ❦❤ỉ♥❣ t❤➜t ❜↕✐ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ (0, t]✳
◆â✐ ❝→❝❤ ❦❤→❝✱ R(t) ❧➔ ①→❝ s✉➜t ♠ët ✤è✐ t÷đ♥❣ sè♥❣ sât tr♦♥❣ ❦❤♦↔♥❣ t❤í✐
❣✐❛♥ (0, t] ✈➔ ✈➝♥ ❤♦↕t ✤ë♥❣ t↕✐ t❤í✐ ✤✐➸♠ t✳ ❍➔♠ t✐♥ ❝➟② R(t) ❝á♥ ✤÷đ❝
❣å✐ ❧➔ ❤➔♠ sè♥❣ sât ✈➔ ✤÷đ❝ ❜✐➸✉ t❤à ♥❤÷ ❍➻♥❤ 1.3

❍➻♥❤ ✶✳✸✿ ❍➔♠ t✐♥ ❝➟② R(t)

✽


✶✳✹ ❍➔♠ t➾ ❧➺ rõ✐ r♦
❳→❝ s✉➜t ✤➸ ♠ët ♠ö❝ s➩ t❤➜t ❜↕✐ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ (t, t + t] ợ

ố tữủ số tớ ✤✐➸♠ t ❧➔✿
Pr (t < T ≤ t + ∆t | T > t) =

Pr (t < T ≤ t + ∆t)
F (t + ∆t) − F (t)
=
Pr (T > t)
R(t)
✭✶✳✺✮

❇➡♥❣ ❝→❝❤ ❝❤✐❛ ①→❝ s✉➜t tr➯♥ ❝❤♦ ❣✐❛ sè t❤í✐ ❣✐❛♥ ∆t✱ ✈➔ ❝❤♦ ∆ → 0✱ ✤÷đ❝
❤➔♠ t➾ ❧➺ rõ✐ r♦ λ(t) ❝õ❛ ♠ët ♠ö❝✿

λ(t) = lim

Pr (t < T ≤ t + ∆t | T > t)

∆t
F (t + ∆t) − F (t) 1
f (t)
= lim
=
∆t→0
∆t
R(t) R(t)
∆t→0

✭✶✳✻✮

❱ỵ✐ ∆t ọ t ổ tự (1.6) õ t t ữợ

Pr (t < T ≤ t + ∆t | T > t) ≈ λ(t) · ∆t

◆❤➟♥ ①➨t✿ ❙ü ❣✐è♥❣ ✈➔ ❦❤→❝ ♥❤❛✉ ❣✐ú❛ ❤➔♠ ♠➟t ✤ë ①→❝ s✉➜t f (t) ✈➔ ❤➔♠
t➾ ❧➺ rõ✐ r♦ λ(t)✳
Pr (t < T ≤ t + ∆t) ≈ f (t) · ∆t

✭✶✳✼✮

Pr (t < T ≤ t + ∆t | T > t) ≈ λ(t) · ∆t

✭✶✳✽✮

❚ø ❝→❝ ❝æ♥❣ t❤ù❝ (1.7) ✈➔ (1.8) ♥❤➟♥ t❤➜② r➡♥❣✿

• ❚↕✐ t❤í✐ ✤✐➸♠ t = 0✱ ①→❝ s✉➜t ✤➸ ♠ët ♠ư❝ s➩ t❤➜t ❜↕✐ tr♦♥❣
❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ (t, t + ∆t] ❜➡♥❣ t➼❝❤ ❝õ❛ ❤➔♠ ♠➟t ✤ë ①→❝
s✉➜t f (t) t↕✐ t❤í✐ ✤✐➸♠ t ✈ỵ✐ sè ❣✐❛ t❤í✐ ❣✐❛♥ ∆t✳
• ❳→❝ s✉➜t ✤➸ ♠ët ♠ư❝ s➩ t❤➜t ❜↕✐ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥
(t, t + ∆t] ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤è✐ t÷đ♥❣ ✈➝♥ sè♥❣ ❝❤♦ ✤➳♥ t❤í✐
✤✐➸♠ t ❜➡♥❣ t➼❝❤ ❝õ❛ ❤➔♠ t➾ ❧➺ rõ✐ r♦ λ(t) t↕✐ t❤í✐ ✤✐➸♠ t ợ
số tớ t
ú t ữ ởt số ữủ ợ ử ố t t ở
t t❤í✐ ✤✐➸♠ t t❤➻ t➼❝❤ λ(t) · ∆t s➩ ✤↕✐ ❞✐➺♥ ❝❤♦ t✛ ❧➺ t÷ì♥❣ ✤è✐ ❝→❝ ✤è✐

✾


t÷đ♥❣ ✈➝♥ ❤♦↕t ✤ë♥❣ t↕✐ t❤í✐ ✤✐➸♠ t✱ ♥❤÷♥❣ t❤➜t ❜↕✐ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐
❣✐❛♥ (t, t + ∆t] t✐➳♣ t❤❡♦✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝✿

f (t) =

d
d
F (t) = (1 − R(t)) = −R (t)
dt
dt

✭✶✳✾✮

f (t)
R (t)
d
=−
= − ❧♥ R(t)
R(t)
R(t)
dt

✭✶✳✶✵✮

t❤✉ ✤÷đ❝✿

λ(t) =
❱➻ R(0) = 1 ♥➯♥✿
t

t

−

λ(t) dt =
0

0

d
❧♥ R(t) dt = −❧♥ R(t)
dt

✭✶✳✶✶✮

❚ø ✤â s✉② r❛✿
t

R(t) = ❡①♣ −

✭✶✳✶✷✮

λ(u) du
0

❚ø ❝→❝ ❝æ♥❣ t❤ù❝ (1.6) ✈➔ (1.12) s✉② r❛✿
t

f (t) = λ(t) Ã

ợ t > 0

(u) du

0

õ t ữủ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝→❝ ❤➔♠ F (t), f (t), R(t) ✈➔ λ(t) ♥❤÷ s❛✉✿
t

t

f (u)du = 1 − R(t) = 1 − ❡①♣ −

F (t) =

λ(u) du

0

d
d
f (t) = F (t) = − R(t) = λ(t) · ❡①♣ −
dt
dt

t

λ(u) du
t

f (u)du = ❡①♣ −

R(t) = 1 − F (t) = −

✭✶✳✶✺✮

0

∞
t

λ(t) =

✭✶✳✶✹✮

0

dF (t)
1
·
=
dt
1 − F (t)

λ(u) du

✭✶✳✶✻✮

d
❧♥ R(t)
dt

✭✶✳✶✼✮

0

f (t)
∞

−

f (u)du

=−

t

❚ø ❝æ♥❣ t❤ù❝ (1.12) t❤➜② r➡♥❣ ❤➔♠ t✐♥ ❝➟② ✭❤➔♠ sè♥❣ sât✮ R(t) ✤÷đ❝ ①→❝
✤à♥❤ ❞✉② ♥❤➜t t❤ỉ♥❣ q✉❛ ❤➔♠ t➾ ❧➺ rõ✐ r♦ λ(t)✳ ✣➸ ①→❝ ✤à♥❤ ❞↕♥❣ ❝õ❛ λ(t)

✶✵


❝❤♦ ♠ët ♠ö❝ ❝ö t❤➸✱ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ❝→❝ t❤➼ ♥❣❤✐➺♠ s❛✉✿
❈❤✐❛ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ (0, t) t❤➔♥❤ ❝→❝ ❦❤♦↔♥❣ rí✐ r↕❝ ❝â ✤ë ❞➔✐ ❜➡♥❣ ∆t✳
❙❛✉ ✤â✱ ❝❤♦ n ♠ư❝ ❣✐è♥❣ ❤➺t ♥❤❛✉ ✈➔♦ ❤♦↕t ✤ë♥❣ t↕✐ t❤í✐ ✤✐➸♠ t = 0✳ ❑❤✐
♠ët ♠ư❝ t❤➜t ❜↕✐✱ ❝❤ó♥❣ t❛ ❣❤✐ ❧↕✐ t❤í✐ ❣✐❛♥ ❝ư t❤➸ ✈➔ ❧♦↕✐ ❜ä ♠ư❝ ✤â✳ ❱ỵ✐
♠é✐ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ∆t✱ ❣❤✐ ❝❤➨♣ ❝→❝ ✤✐➲✉ s❛✉✿

• ❙è ♠ư❝ n(i) t❤➜t ❜↕✐ tr♦♥❣ ❦❤♦↔♥❣ i.
• ❚❤í✐ ❣✐❛♥ ❤♦↕t ✤ë♥❣ ❝❤♦ ❝→❝ ♠ö❝ ❝õ❛ ❝→ ♥❤➙♥ tr♦♥❣ ❦❤♦↔♥❣

t❤í✐ ❣✐❛♥ i ❧➔ (T1i , ..., Tni )✱ tr♦♥❣ ✤â Tji ❧➔ ♠ö❝ t❤ù j ✤➣ ❤♦↕t
✤ë♥❣ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ i✳ Tji = 0 ♥➳✉ ♠ư❝ j t❤➜t
trữợ i ợ j = 2, ..., n
n



Tji ❧➔ tê♥❣ t❤í✐ ❣✐❛♥ ❤♦↕t ✤ë♥❣ ❝õ❛ ❝→❝ ♠ư❝ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐
j=1

❣✐❛♥ i✳ ❚➾ sè

n(i)
n

Tji
j=1

❜✐➸✉ t❤à sè ❧÷đ♥❣ ❧é✐ tr➯♥ ♠é✐ ✤ì♥ ✈à t❤í✐ ❣✐↕♥ ❤♦↕t ✤ë♥❣ tr♦♥❣ ❦❤♦↔♥❣ i
✈➔ ữợ t tỹ ừ t rừ r tr♦♥❣ ❦❤♦↔♥❣ i✳
✣à♥❤ ♥❣❤➽❛ m(i) ❧➔ sè ❝→❝ ♠ö❝ ✤❛♥❣ ❤♦↕t ✤ë♥❣ t↕✐ t❤í✐ ✤✐➸♠ ❜➢t ✤➛✉ ❝õ❛
❦❤♦↔♥❣ i✳ ❑❤✐ õ

(i)

n(i)
m(i)t

õ

(i)t

n(i)
m(i)

ởt ỗ ổ t (i) ♠ët ❤➔♠ ❝õ❛ i ❝â ❞↕♥❣ ♥❤÷ ❍➻♥❤ 1.4

✶✶


❍➻♥❤ ✶✳✹✿ ❍➔♠ ❜➟❝ t❤❛♥❣ λ(t)
❑❤✐ n r➜t ❧ỵ♥✱ ❝â t❤➸ sû ❞ư♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ r➜t ♥❤ä✳ ◆➳✉ ∆t → 0✱s➩ ❦➻
✈å♥❣ r➡♥❣ ❤➔♠ ❜➟❝ t❤❛♥❣ λ(t) t✐➺♠ ❝➟♥ ✤➳♥ ♠ët ✤÷í♥❣ ❝♦♥❣ ❧✐➯♥ tư❝✱ ♥❤÷
❍➻♥❤ 1.5

❍➻♥❤ ✶✳✺✿ ❍➔♠ ❜➟❝ t❤❛♥❣ λ(t) t✐➺♠ ❝➟♥ ✤➳♥ ♠ët ✤÷í♥❣ ❝♦♥❣ ❧✐➯♥ tư❝
✶✳✺ ❚✉ê✐ t❤å tr✉♥❣ ❜➻♥❤
❚✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ✭▼❚❚❋✮ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿
∞

▼❚❚❋ = E(T ) =

tf (t) dt

✭✶✳✶✽✮

0

❑❤✐ t❤í✐ ❣✐❛♥ ❝➛♥ t❤✐➳t ✤➸ sû❛ ❝❤ú❛ ❤♦➦❝ t❤❛② t❤➳ ♠ët ♠ư❝ ❜à ❤ä♥❣ r➜t
♥❣➢♥ s♦ ✈ỵ✐ t✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ✭▼❚❚❋✮ t❤➻ ▼❚❚❋ ❝ơ♥❣ t❤➸ ❤✐➺♥ t❤í✐ ❣✐❛♥

✶✷


tr✉♥❣ ❜➻♥❤ ❣✐ú❛ ❝→❝ ❧➛♥ t❤➜t ❜↕✐ ✭▼❚❇❋✮✳ ◆➳✉ t❤í✐ sỷ ỳ ổ
t ọ q t ỗ ❧✉ỉ♥ ❝↔ t❤í✐ ❣✐❛♥ sû❛ ❝❤ú❛ ✭▼❚❚❘✮✳
❚ø f (t) = −R (t)✱ t❤✉ ✤÷đ❝✿
∞

▼❚❚❋ = −

tR (t) dt
0

❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✱ t❤✉ ✤÷đ❝✿
▼❚❚❋ =

−[tR(t)]|∞
0

∞

+

R(t) dt
0

◆➳✉ ▼❚❚❋ < ∞✱ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ [tR(t)]|∞
0 = 0✳ ❚ø ✤â s✉② r❛✿
∞

▼❚❚❋ =

R(t) dt

✭✶✳✶✾✮

0

❈ô♥❣ ❝â t❤➸ ①→❝ ✤à♥❤ t✉ê✐ t❤å tr✉♥❣ ❜➻♥❤ ❜➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❜✐➳♥ ✤ê✐
▲❛♣❧❛❝❡✳ ❇✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡ ❝❤♦ ❤➔♠ t✐♥ ❝➟② ✭❤➔♠ sè♥❣ sât✮ R(t) ✤÷đ❝ ①→❝
✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿
∞

∗

R(t) e−st dt

✭✶✳✷✵✮

R(t) dt = ▼❚❚❋

✭✶✳✷✶✮

R (s) =
0

❑❤✐ s = 0 t❤✉ ✤÷đ❝✿
∞

∗

R (0) =
0

❚r✉♥❣ ✈à ✭▼❡❞✐❛♥✮✿ ▼❚❚❋ ❝❤➾ ❧➔ ♠ët tr♦♥❣ ♥❤✐➲✉ ❜✐➺♥ ♣❤→♣ ✤♦ tr✉♥❣
t➙♠ ❝õ❛ ♣❤➙♥ ♣❤è✐✳ ▼ët ❜✐➺♥ ♣❤→♣ t❤❛② t❤➳ ❦❤→❝ ❧➔ tr✉♥❣ ✈à✱ ✤÷đ❝ ✤à♥❤
♥❣❤➽❛ ❜ð✐✿

R(tm ) = 0.5

✭✶✳✷✷✮

❚r✉♥❣ ✈à ❝❤✐❛ ♣❤➙♥ ♣❤è✐ ❧➔♠ ✷ ♥û❛✳ ▼ët ♥û❛ s➩ tt trữợ tớ
tm ợ st 50% ♥û❛ ❝á♥ ❧↕✐ s➩ t❤➜t ❜↕✐ s❛✉ t❤í✐ ❣✐❛♥ tm ❝ơ♥❣ ✈ỵ✐
①→❝ s✉➜t 50%✳

▼♦❞❡✿ ▼♦❞❡ ❝õ❛ ♣❤➙♥ ♣❤è✐✱ ❦➼ ❤✐➺✉ tmode ❧➔ t❤í✐ ❣✐❛♥ ❧➔♠ ❝ü❝ ✤↕✐ ❤➔♠
♠➟t ✤ë ①→❝ s✉➜t f (t)✿

f (tmode ) = ♠❛① 0≤t<∞ f (t)

✭✶✳✷✸✮

❍➻♥❤ 1.6 ❜✐➸✉ t❤à ✈à tr➼ ❝õ❛ ▼❚❚❋✱ tr✉♥❣ ✈à tm ✈➔ tmode ❝õ❛ ♠ët ♣❤➙♥
♣❤è✐✳

✶✸

❍➻♥❤ ✶✳✻✿ ❱à tr➼ ❝õ❛ ▼❚❚❋✱ tm ✈➔ tmode ❝õ❛ ♠ët ♣❤➙♥ ♣❤è✐
✶✳✻ ❑✐➸♠ ❞✉②➺t ✈➔ ❝❤➦t ❝öt ❞ú ❧✐➺✉
❑✐➸♠ ❞✉②➺t
❳➨t T ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t✉ê✐ t❤å✳
◆➳✉ ❝❤➾ ❜✐➳t T > t tø ♠ët q✉❛♥ s→t t✱ t❤➻ t
✤÷đ❝ ❣å✐ ❧➔ ✧❦✐➸♠ ❞✉②➺t ♣❤↔✐ ✧✳
• ❑✐➸♠ ❞✉②➺t tr→✐ ✿ ◆➳✉ ❝❤➾ ❜✐➳t T < t tø ♠ët q✉❛♥ s→t t t t
ữủ ồ t tr
ã ❞✉②➺t ❦❤♦↔♥❣ ✿ ❚r♦♥❣ ✧❦✐➸♠ ❞✉②➺t ❦❤♦↔♥❣ ✧ ❜✐➳t a < T < b
♥❤÷♥❣ ❦❤ỉ♥❣ ❜✐➳t ❝❤➼♥❤ ①→❝ ❣✐→ trà ❝õ❛ T ✳
• ❚❤í✐ ❣✐❛♥ ❦✐➸♠ ❞✉②➺t ✿ ◆➳✉ c ❧➔ t❤í✐ ❣✐❛♥ ♠➔ ❝❤ó♥❣ t❛ ❝❤➾
q✉❛♥ s→t c ❦❤✐ T > c t❤➻ c ✤÷đ❝ ❣å✐ ❧➔ t❤í✐ ❣✐❛♥ t
ớ số sõt t ữủ t

ã

❞✉②➺t ♣❤↔✐ ✿

t = T ∧ c = min(T, c)
•

❍➺ sè ❦✐➸♠ ❞✉②➺t ✿

δ = I{T ≤c} =

❍➺ sè ❦✐➸♠ ❞✉②➺t δ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿

1 ♥➳✉ T ≤ c (❦❤ỉ♥❣ ❦✐➸♠ ❞✉②➺t);
0 ♥➳✉ T > c (❦✐➸♠ ❞✉②➺t),

✈ỵ✐ IA ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝✿

✶✹


IA =

1 ♥➳✉ A ①↔② r❛;
0 ♥➳✉ A ❦❤æ♥❣ ①↔② r❛.

❈→❝ ❦✐➸✉ ❦✐➸♠ ❞✉②➺t
❚❤í✐ ❣✐❛♥ ❦✐➸♠ ❞✉②➺t ❧➔ ♠ët ❤➡♥❣ sè ✭✤➣
❜✐➳t✮✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ t❤í✐ ❣✐❛♥ ❧➔ ✺ ♥➠♠ tr♦♥❣ tr÷í♥❣ ❤đ♣
❝❤➼♥❤ s→❝❤ ❜↔♦ ❤✐➸♠ ♥❤➙♥ t❤å ✺ ♥➠♠❀ ❤♦➦❝ ♠ët ♥❣❤✐➯♥ ❝ù✉
✷ ♥➠♠ ✈➲ ❝→❝ ❝❤õ ✤➲ ✤➣ ❝â s➤♥✳
• ❑✐➸♠ ❞✉②➺t ▲♦↕✐ ■■ ✿ ❈❤➜♠ ❞ùt ♥❣❛② s❛✉ ❦❤✐ q✉❛♥ s→t t❤➜t
❜↕✐✳ ❑✐➸♠ ❞✉②➺t ♥➔② t❤÷í♥❣ ✤÷đ❝ sû ❞ö♥❣ tr♦♥❣ ❝→❝ ❜➔✐ ❦✐➸♠
tr❛ ✤ë t✐♥ ❝➟②✱ tr♦♥❣ ✤â ♠ët ❜➔✐ ❦✐➸♠ tr❛ ❦➳t t❤ó❝ ❦❤✐ ❝→❝ s↔♥
♣❤➞♠ t❤➜t ❜↕✐✳
• ❑✐➸♠ ❞✉②➺t ♥❣➝✉ ♥❤✐➯♥ ✿ ✣✐➸♠ ❜➢t ✤➛✉ ✈➔ ✤✐➸♠ ❦➳t t❤ó❝ q✉❛♥
s→t ❧➔ ♥❣➝✉ ♥❤✐➯♥✳ ❑✐➸♠ ❞✉②➺t ♥❣➝✉ ♥❤✐➯♥ ❧➔ ❦✐➸♠ ❞✉②➺t ♣❤ê
❜✐➳♥ ♥❤➜t tr♦♥❣ ❝→❝ ♠æ ❤➻♥❤ sè♥❣ sât✳

•

❑✐➸♠ ❞✉②➺t ▲♦↕✐ ■ ✿

❈❤➦t ❝ưt
❈❤➾ q✉❛♥ s→t T = t ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ T > a ✈ỵ✐
a tr trữợ r trữớ ủ ữ q st t

ữủ ồ t ửt tr t a
ã ❈❤➦t ❝öt ♣❤↔✐ ✿ ❈❤➾ q✉❛♥ s→t T = t ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ T < a ✈ỵ✐ a
❧➔ ❣✐→ trà trữợ

ã

t ửt tr

ỹ ỳ t ✈➔ ✧❝❤➦t ❝ưt✧
• ▼ët ✧❦✐➸♠ ❞✉②➺t✧ t ❧➔ q✉❛♥ s→t tr tt ố tữủ t
r T > t
ã ▼ët ✧❝❤➦t ❝öt✧ t ❧➔ q✉❛♥ s→t ❝❤➾ tr➯♥ ♠ët ❝→ ♥❤➙♥ ✈ỵ✐ T > a
❜✐➳t r➡♥❣ T = t ♥➳✉ ✤è✐ t÷đ♥❣ ✤÷đ❝ q✉❛♥ s→t✳

✶✺


✷ ✣ë t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣
❈❤ó♥❣ t❛ s➩ t➻♠ ❤✐➸✉ ❝→❝❤ t❤ù❝ t➼♥❤ ❤➔♠ t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣✱ ♥❤÷ ❧➔ ♠ët
❤➔♠ t✐♥ ❝➟② ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❝õ❛ õ t õ ởt tố
ỗ k ❤➺ t❤è♥❣ ❝♦♥ ✭t❤➔♥❤ ♣❤➛♥✮✱ ❝â ❝→❝ ❤➔♠ ✤ë t✐♥ ❝➟② R1 (t), ..., Rk (t)✱
t❤➻ ✤ë t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣ ❧➔✿

Rsys (t) = ψ(R1 (t), ..., Rk (t));

t≥0

❍➔♠ ψ(.) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❤➔♠ ❝➜✉ tró❝ ✭str✉❝t✉r❡ ❢✉♥❝t✐♦♥✮✳ ❙❛✉ ✤➙② s➩
tr➻♥❤ ❜➔② ✈➲ ♠ët sè ❤➔♠ ❝➜✉ tró❝ ❝õ❛ ❝→❝ ❤➺ t❤è♥❣ ✤ì♥ ❣✐↔♥✳
●✐↔ sû r➡♥❣ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ T1 , T2 , ..., Tk ❧➔ ✤ë❝ ❧➟♣ ✈ỵ✐ ♥❤❛✉ ✈➔ ✤↕✐

❞✐➺♥ ❝❤♦ t❤í✐ ❣✐❛♥ sè♥❣ ❝õ❛ ❝→❝ ❤➺ t❤è♥❣ ❝♦♥✳ ❳➨t ✈ỵ✐ ♠ët ❤➺ t❤è♥❣ ❝â ❤❛✐
❤➺ t❤è♥❣ ❝♦♥ C1 ✈➔ C2 ✳
◆➳✉ ❧é✐ ❝õ❛ ♠ët tr♦♥❣ ❤❛✐ ❤➺ t❤è♥❣ ❝♦♥ ❧➟♣ tù❝ ❣➙② r❛ t❤➜t ❜↕✐ ❝❤♦ ❝↔ ❤➺
t❤è♥❣✱ t❛ ♥â✐ r➡♥❣ ❤➺ t❤è♥❣ ❝♦♥ ✤÷đ❝ ❦➳t ♥è✐ t✉➛♥ tü ✭❝♦♥♥❡❝t❡❞ ✐♥ s❡r✐❡s✮✳
❚r→✐ ❧↕✐✱ ởt tố ỗ k tố ữủ ❣å✐ ❧➔ ❦➳t ♥è✐ s♦♥❣
s♦♥❣✱ ♥➳✉ ❤➺ t❤è♥❣ ❜à ❧é✐ ❝❤➾ ❦❤✐ t➜t ❝↔ ❝→❝ ❤➺ t❤è♥❣ ❝♦♥ ❜à ❧é✐✳ ❚r♦♥❣ ♠ët
❤➺ t❤è♥❣ ❦➳t ♥è✐ s♦♥❣ s♦♥❣ ❝❤➾ ❝➛♥ ➼t ♥❤➜t ♠ët ❤➺ t❤è♥❣ ❝♦♥ ❤♦↕t ✤ë♥❣ ❧➔
✤õ ❝❤♦ t♦➔♥ ở tố t ở
ữợ t t ố ố t s s ởt sỡ ỗ ố ữ
tr 1.7

ỡ ỗ ố tố ✤÷đ❝ ❦➳t ♥è✐ ♥è✐ t✐➳♣ ✈➔ s♦♥❣ s♦♥❣

✶✻


✷✳✶ ❍➺ t❤è♥❣ ♥è✐ t✐➳♣ ✈➔ s♦♥❣ s♦♥❣
❚❛ ❣å✐ Ii , (i = 1, 2, ..., k) ❧➔ ❜✐➳♥ ❝❤➾ ✤à♥❤✳ Ii ♥❤➟♥ ❣✐→ trà ✶ ♥➳✉ t❤➔♥❤ ♣❤➛♥
Ci ✱ ❦❤ỉ♥❣ ❜à ❧é✐ tr♦♥❣ ♠ët ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ q✉② ✤à♥❤ (0, t0 ) ✈➔ ♥❤➟♥ ❣✐→
trà ✵ ♥➳✉ t❤➔♥❤ ♣❤➛♥ Ci ❜à ❤ä♥❣ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ (0, t0 )✳ ❑ý ✈å♥❣
❝õ❛ ❜✐➳♥ Ii ❧➔✿

E[Ii ] = P [Ii = 1] = Ri (t0 )

✭✶✳✷✹✮

❍➺ t❤è♥❣ ✤÷đ❝ ❦➳t ♥è✐ ♥è✐ t
trú srs strtr t ừ ỗ k t❤➔♥❤ ♣❤➛♥ ✤÷đ❝
❦➳t ♥è✐ ♥è✐ t✐➳♣ ❧➔✿
k

ψ(I1 , I2 , ..., Ik ) =

Ii

✭✶✳✷✺✮

i=1

❍➔♠ t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣ ✤÷đ❝ ❦➳t ♥è✐ ♥è✐ t✐➳♣✱ tr÷í♥❣ ❤đ♣ ❝→❝ t❤➔♥❤
♣❤➛♥ ✤ë❝ ❧➟♣ ✤÷đ❝ ❝❤♦ ❜ð✐✿
(s)
Rsys
(t0 ) = E[ψs (I1 , I2 , ..., Ik )]
k

=

Ri (t0 )
i=1

= ψs (R1 (t0 ), ..., Rk (t0 ))

✭✶✳✷✻✮

❱➻ ✈➟②✱ ❤➔♠ t✐♥ ❝➟② ❝❤♦ ❝→❝ ❤➺ t❤è♥❣ ❝♦♥ ♠➔ ❦➳t ♥è✐ ♥è✐ t✐➳♣ ✤÷đ❝ ❝❤♦ ❜ð✐
ψs (R1 , ..., Rk )✱ ✈ỵ✐ R1 , ..., Rk ❧➔ ❝→❝ ❣✐→ trà t✐♥ ❝➟② ❝õ❛ ❝→❝ t❤➔♥❤ ♣❤➛♥✳

❍➺ t❤è♥❣ ✤÷đ❝ ❦➳t ♥è✐ s♦♥❣ s♦♥❣
❍➔♠ ❝➜✉ tró❝ ❝õ❛ ❤➺ ỗ k t ữủ t ố s s

k

p (I1 , I2 , ..., Ik ) = 1 −

(1 − Ii )− i)

✭✶✳✷✼✮

i=1

❍➔♠ t✐♥ ❝➟② ❝õ❛ ❤➺ t❤è♥❣ ✤÷đ❝ ❦➳t ♥è✐ s♦♥❣ s♦♥❣✱ tr÷í♥❣ ❤đ♣ ❝→❝ t❤➔♥❤
♣❤➛♥ ✤ë❝ ❧➟♣ ✤÷đ❝ ❝❤♦ ❜ð✐✿
(s)
Rsys
(t0 ) = E[ψp (I1 , I2 , ..., Ik )]

✶✼


k

=1−

(1 − Ri (t0 ))

✭✶✳✷✽✮

i=1

▼ët t❤➫ ♠→② t➼♥❤ ❝â ✷✵✵ t❤➔♥❤ ♣❤➛♥ ✭♠♦❞✉❧♦✮✱ ✈➔ ❝➛♥ ❤♦↕t ✤ë♥❣

❝❤➼♥❤ ①→❝✳ ✣ë t✐♥ ❝➟②✱ ❝õ❛ ♠é✐ t❤➔♥❤ ♣❤➛♥✱ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ✷✵✵ ❣✐í
❤♦↕t ✤ë♥❣ ❧➔ R = 0.9999✳ ❈→❝ t❤➔♥❤ ♣❤➛♥ ❧➔ ✤ë❝ ❧➟♣ ✈ỵ✐ ♥❤❛✉✳ ✣ë t✐♥ ❝➟②
❝õ❛ t❤➫ ❝❤♦ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ♥➔② ❧➔ ❜❛♦ ♥❤✐➯✉❄
❱➼ ❞ư✿

❱➻ ❝➛♥ ❝→❝ t❤➔♥❤ ♣❤➛♥ ✈➟♥ ❤➔♥❤ t❤❡♦ ✤ó♥❣ t❤✐➳t ❦➳ ♥➯♥ ①❡♠ ①➨t ♠ët ❤➔♠
❝➜✉ tró❝ ♥è✐ t✐➳♣✳ ❉♦ ✤â✱ ✤ë t✐♥ ❝➟② ❤➺ t❤è♥❣ ❝❤♦ t0 = 200 ✭❣✐í✮ ❧➔✿
(s)
Rsys
(t0 ) = (0.9999)200 = 0.9802

◆❤÷ ✈➟②✱ ❞ị t❤ü❝ t➳ ❧➔ ♠é✐ t❤➔♥❤ ♣❤➛♥ ❧➔ ❤➛✉ ♥❤÷ ❦❤ỉ♥❣ t❤➸ ❜à ❧é✐✱ ❝â ♠ët
①→❝ s✉➜t ✵✳✵✷ r➡♥❣ t❤➫ s➩ ❤ä♥❣ tr♦♥❣ ✈á♥❣ ✷✵✵ ❣✐í✳ ◆➳✉ ♠é✐ t❤➔♥❤ ♣❤➛♥
❝❤➾ ❝â ♠ët ✤ë t✐♥ ❝➟② ✵✳✾✾ t❤➻ ✤ë t✐♥ ❝➟② t❤➫ ❧➔✿
(s)
Rsys
(t0 ) = (0.99)2 00 = 0.134

●✐↔ sû r➡♥❣ t❤➫ ❝â ❝❤é trè♥❣ ❝❤♦ ♠ët sè ❧✐♥❤ ❦✐➺♥ ❞ü ♣❤á♥❣✳ ❉♦ ✤â✱ t❛
q✉②➳t ✤à♥❤ sû ❞ö♥❣ ❝→❝ ❧✐♥❤ ❦✐➺♥ ❝â ✤ë t✐♥ ❝➟② R = 0.99 ✈➔ ❧➦♣ ❧↕✐ ♠é✐
❧✐♥❤ ❦✐➺♥ tr♦♥❣ ♠ët ❝➜✉ tró❝ s♦♥❣ s♦♥❣✳ ❈➜✉ tró❝ s♦♥❣ s♦♥❣ ❝õ❛ ❝→❝ ❧✐♥❤
❦✐➺♥ trị♥❣ ❧➦♣ ✤÷đ❝ ❝♦✐ ♥❤÷ ❧➔ ♠ët ♠♦❞✉❧♦✳ ✣ë t✐♥ ❝➟② ❝õ❛ ♠é✐ ♠♦❞✉❧♦
❧➔ RM = 1 − (1 − 0.99)2 = 0.9999✳ ❑❤✐ ✤â✱ ✤ë t✐♥ ❝➟② ❝õ❛ ✭t♦➔♥ ❜ë✮ ❤➺
t❤è♥❣ ❧➔✿
(s)
200
Rsys
= RM
= 0.9802

❉♦ ✤â✱ ❜➡♥❣ ❝→❝❤ t❤❛② ✤ê✐ ❝➜✉ tró❝ ❝õ❛ t❤➫ t❛ ❝â t❤➸ ✤↕t ✤÷đ❝ ✤ë t✐♥
ợ t ộ t ỗ ởt ❝➦♣ ❝→❝ ❧✐♥❤ ❦✐➺♥✱ ♠é✐
❧✐♥❤ ❦✐➺♥ ❝â ✤ë t✐♥ ❝➟② ✵✳✾✾✳
❈→❝ ❤➺ t❤è♥❣ ❝â t❤➸ ❝â ❝➜✉ tró❝ ♣❤ù❝ t↕♣ ỡ 1.8 t sỡ ỗ ố
ừ ởt tố ỗ t
ồ R1 , R2 , ..., R5 ❜✐➸✉ t❤à ❝❤♦ ❝→❝ ❣✐→ trà ✤ë t✐♥ ❝➟② ❝õ❛ ♥➠♠ t❤➔♥❤ ♣❤➛♥
C1 , C2 , ..., C5 tữỡ ự ồ M1 ỗ t C1 , C2
ồ M2 ỗ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❦❤→❝ ð ♥❤→♥❤ t❤ù ❤❛✐✳ ✣ë
t✐♥ ❝➟② ❝õ❛ M1 ❝❤♦ ♠ët ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ q✉② ✤à♥❤ ❧➔✿

RM1 = R1 R2

✶✽