Một số bài toán đặc trưng của phân phối hình học hai chiều

✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✣⑨❖ ❆◆❍ ❚❯❻◆ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ✣➄❈ ❚❘×◆● ❈Õ❆ P❍❹◆ P❍➮■ ❍➐◆❍ ❍➴❈ ❍❆■ ❈❍■➋❯ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ❍➔ ◆ë✐✱ ◆➠♠ ✷✵✶✺ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✣⑨❖ ❆◆❍ ❚❯❻◆ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ✣➄❈ ❚❘×◆● ❈Õ❆ P❍❹◆ P❍➮■ ❍➐◆❍ ❍➴❈ ❍❆■ ❈❍■➋❯ ỵ tt st tố t ❤å❝ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✻ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ữớ ữợ P ❍➔ ◆ë✐✱ ◆➠♠ ✷✵✶✺ ✐ ▲❮■ ◆➶■ ✣❺❯ ❑❤✐ ♥❣❤✐➯♥ ❝ù✉ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥➔♦ ✤â✱ t❤æ♥❣ t✐♥ ✤➛② ✤õ ♥❤➜t✱ q✉❛♥ trå♥❣ ♥❤➜t ♠➔ t❛ ♠♦♥❣ ♠✉è♥ ❝â ✤÷đ❝ ❧➔ t❛ ①→❝ ✤à♥❤ ①❡♠ q✉② ❧✉➟t ♣❤➙♥ ♣❤è✐ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤â ❧➔ ♣❤➙♥ ♣❤è✐ ♥➔♦✳ ❈❤➼♥❤ ✈➻ ✈➟② tø ♥❤ú♥❣ t❤➟♣ ♥✐➯♥ ✺✵ ✲ ✻✵ ✲ t trữợ t trữ ♣❤è✐ ①→❝ s✉➜t ✤➣ ♣❤→t tr✐➸♥ r➜t ♠↕♥❤ ♠➩✳ ❚✉②➸♥ t t q t ữợ ữủ ♥❤➔ ❦❤♦❛ ❤å❝ ❧ỵ♥ tr➯♥ t❤➳ ❣✐ỵ✐✿ ▲✐♥♥✐❦ ❨✉✳❱✱ ❑❛❣❛♥ ❆✳▼ ✈➔ ❘❛♦ ❈✳❘✳ tê♥❣ ❦➳t ❧↕✐ tr♦♥❣ ❝✉è♥ ✧❈❤❛r❛❝t❡r✐③❛t✐♦♥ Pr♦❜❧❡♠s ✐♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙t❛t✐st✐❝s✧ ①✉➜t ❜↔♥ ♥➠♠ ✶✾✼✷✳ ▼ët t➼♥❤ ❝❤➜t F = {F (x, θ), θ ∈ O} t➼♥❤ ❝❤➜t S S ✤÷đ❝ ❣å✐ ❧➔ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝❤♦ ❤å ♣❤➙♥ ♣❤è✐ ♥➳✉ t❤➻ t❛ s✉② r❛ X ≈F ∈F X t❤➻ t❛ ❝â t➼♥❤ ❝❤➜t ❝â ♣❤➙♥ ♣❤è✐ t❤✉ë❝ ❤å S ✈➔ ♥❣÷đ❝ ❧↕✐✱ ♥➳✉ ❝â F✳ ❚r♦♥❣ ❝✉è♥ ❝❤✉②➯♥ ❦❤↔♦ tr➯♥ r➜t ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝❤♦ ❝→❝ ❤å ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t q✉❡♥ t❤✉ë❝ ✤➣ ✤÷đ❝ ❝❤➾ r❛✳ ❙♦♥❣ ❦➳t q✉↔ ❝❤õ ②➳✉ t➟♣ tr✉♥❣ ✈➔♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♠ët ❝❤✐➲✉✳ ❚r➯♥ t❤ü❝ t➳ ❝→❝ ♣❤➙♥ ♣❤è✐ ♥❤✐➲✉ ❝❤✐➲✉ q✉❡♥ t❤✉ë❝ ❝ô♥❣ ❝❤➾ ❧➔ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥ ✈➔ ♣❤➙♥ ♣❤è✐ ✤❛ t❤ù❝✳ ❱➻ ✈➟② ①➙② ❞ü♥❣ ❝→❝ ♣❤➙♥ ♣❤è✐ ♥❤✐➲✉ ❝❤✐➲✉ ❦❤→❝ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ❝❤ó♥❣ ✤❛♥❣ ❧➔ ❜➔✐ t♦→♥ ♠ð✱ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ❈→❝ ❦➳t q✉↔ ♥❤➟♥ ✤÷đ❝ t➟♣ tr✉♥❣ ✈➔♦ ♣❤➙♥ ♣❤è✐ ♠ơ ✈➔ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ▲✉➟♥ ✈➠♥ ❝õ❛ ❤å❝ ✈✐➯♥ ❱ô ❚❤à ❚❤↔♦ ❦❤â❛ ✷✵✶✶✲✷✵✶✸ ✈ỵ✐ ✤➲ t➔✐ ✑▼ët sè ❜➔✐ t♦→♥ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ơ ❤❛✐ ❝❤✐➲✉✑ ✤➣ ❝❤♦ ♠ët ❜ù❝ tr❛♥❤ ✤➛② ✤õ ✈➲ ♣❤➙♥ ♣❤è✐ ♠ô ❤❛✐ ❝❤✐➲✉ ✈➔ ❝→❝ ❦➳t q✉↔ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♥➔②✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝õ❛ ❜➔✐ t♦→♥ ♠ð tr➯♥ ✤è✐ ✈ỵ✐ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✳ ▲✉➟♥ ✈➠♥ ✧ ▼ët sè ❜➔✐ t♦→♥ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ỗ t t t ỗ ❤❛✐ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ❈❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ tỵ✐ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ♠ët ❝❤✐➲✉✿ ❦❤→✐ ♥✐➺♠ ❤➔♠ s✐♥❤✱ ❤➔♠ ✤➦❝ tr÷♥❣✱ t➼♥❤ ♠➜t tr➼ ♥❤ỵ✱ ❤➔♠ sè♥❣ sât✱ tè❝ ✤ë t❤➜t ❜↕✐✳ ❙❛✉ ✤â ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✱ ♠ð rë♥❣ t➼♥❤ ♠➜t tr➼ ♥❤ỵ✱ ♣❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥✱ ❝→❝ ổ ợ ữỡ ởt số t ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ r ữỡ ợ t t t trữ ỹ tr t t tr ợ ổ ❜à ❝❤➦t ❝öt✱ ❞ü❛ tr➯♥ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥✱ ❞ü❛ tr➯♥ ❤➔♠ tè❝ ✤ë t❤➜t ❜↕✐ ✈➔ ❤➔♠ sè♥❣ sât tr✉♥❣ ❜➻♥❤✳ ❈→❝ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ✶ ✈➔ ❝❤÷ì♥❣ ✷ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❞ü❛ tr➯♥ ❧✉➙♥ →♥ t✐➳♥ sÿ ❝õ❛ t→❝ ❣✐↔ ▼✉r❛❧❡❡❞❤❛r❛♥ ◆❛✐r ❑✳❘✳ t❤✉ë❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✈➔ ❑ÿ t❤✉➟t ❈♦❝❤✐♥ ✲ ❻♥ ✤ë ✈➔ ❜➔✐ ❜→♦ ✧❇✐✈❛r✐❛t❡ ●❡♦♠❡tr✐❝ ❉✐str✐❜✉t✐♦♥s✧ ❝õ❛ ❊❞✇❛r❞ ❖♠❡② ✈➔ ▲❡❞❛ ❉✳ rữợ tr ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ t ỡ s s tợ P ữớ t t ữợ tổ õ t t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥✱ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ❝ơ♥❣ ♥❤÷ ❝→❝ t❤➛② ❝ỉ ✤➣ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❝❤♦ ❦❤â❛ ❝❛♦ ❤å❝ ❦❤â❛ ✷✵✶✷ ✲ ✷✵✶✹✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ✤➸ tỉ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ♥❤✐➺♠ ✈ö ❝õ❛ ♠➻♥❤✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✹ ✐✈ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ✐ ▲í✐ ❝↔♠ ì♥ ✐✈ ✶ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✶ ✶✳✶ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ♠ët ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❈→❝ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ▼ð rë♥❣ t➼♥❤ ♠➜t tr➼ ♥❤ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✸ P❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✹ ❈→❝ ♠æ ♠❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✺ ❉↕♥❣ ❣✐ỵ✐ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷ ▼ët sè t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✶✽ ✷✳✶ ❈→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❞ü❛ tr➯♥ t➼♥❤ ♠➜t tr➼ ♥❤ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❈→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❞ü❛ tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ỉ ♠❡♥ ❜à ❝❤➦t ❝ưt ✭①❡♠ ❬✺❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ✶✽ ✷✹ ❈→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❞ü❛ tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥ ✭①❡♠ ❬✼❪✮ ✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ❈→❝ ✤➦❝ tr÷♥❣ ❞ü❛ tr➯♥ tè❝ ✤ë t❤➜t ❜↕✐ ✈➔ ❤➔♠ sè♥❣ t❤➯♠ tr✉♥❣ ❜➻♥❤ ✭①❡♠ ❬✺❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❑➳t ❧✉➟♥ ✹✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✹ ✈✐ ❈❤÷ì♥❣ ✶ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ s➩ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✤➣ ✤÷đ❝ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ①➙② ỹ ụ ữ t t õ ữ trữợ ❦❤✐ ✤✐ ✤➳♥ ❝æ♥❣ ✈✐➺❝ ✤â✱ ✤➸ t✐➺♥ t❤❡♦ ❞ã✐✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ♠ët sè ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ♠ët ❝❤✐➲✉✳ ✶✳✶ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ♠ët ❝❤✐➲✉ ❈❤ó♥❣ t❛ ①➨t ❞➣② ♣❤➨♣ t❤û ❧➔ p n ♣❤➨♣ t❤û ❇❡r♥♦✉❧❧✐ ✈ỵ✐ ❦❤↔ ♥➠♥❣ t❤➔♥❤ ❝ỉ♥❣ ❝õ❛ ♠é✐ ✭❦❤↔ ♥➠♥❣ ♣❤➨♣ t❤û t❤➜t ❜↕✐ ❧➔ q = − p)✳ ❈❤ó♥❣ t❛ q✉❛♥ t➙♠ tỵ✐ ❜✐➳♥ ❝è✿ ❝â ❜❛♦ ♥❤✐➯✉ ♣❤➨♣ tỷ trữợ t q t ổ tự ữỡ ố ◆â✐ ❝❤✉♥❣ sè ❝→❝ ❦➳t q✉↔ tr♦♥❣ ①→❝ s✉➜t ✤➸ ❦➳t q✉↔ t❤ù ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ✈✐➳t ν = r + k✳ n r ①↔② r❛ ð ♣❤➨♣ t❤û t❤ù ♠➔ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❳→❝ s✉➜t ✤➸ ❦➳t q✉↔ t❤ù s➩ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ f (k; r, p) trữợ t ổ tự tỷ õ ú k r✳ n r r✱ r ❧➔ sè ♥❣✉②➯♥✱ ν✱ ν, r, p✳ tr♦♥❣ ✤â ❱➻ ν≥r ①↔② r❛ t↕✐ ♣❤➨♣ t❤û ν≤n ♥➯♥ ✤➸ ❝❤♦ t✐➺♥ t❛ (r + k), k = 0, 1, 2, ❇✐➳♥ ❝è ♥❤÷ ✈➟② ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tr♦♥❣ sè t❤➜t ❜↕✐ ✈➔ ♣❤➨♣ t❤û s❛✉ ❝ò♥❣✱ tù❝ ❧➔ ♣❤➨♣ t❤û ❦➳t t❤ó❝ t❤➔♥❤ ❝ỉ♥❣❀ ❝→❝ ①→❝ s✉➜t t÷ì♥❣ ù♥❣ s➩ ❧➔ ✶ k Cn+k−1 pr−1 q k k = 0, 1, 2, ♥❤÷♥❣ ✱ rã r➔♥❣ ❧➔ ❳→❝ s✉➜t ♥➔② ❝❤➼♥❤ ❧➔ ①→❝ s✉➜t ✤➸ ❝â ✤ó♥❣ k f (k; r, p) = Cn+k−1 pr q k ; r✱ ♣❤➨♣ t❤û s➩ ♥❤ä ❤ì♥ k t❤➜t ❜↕✐ (r + k − 1) (r + k) ✈➔ p✱ ✱ ✤÷đ❝ ❞♦ ✤â ❇➙② ❣✐í ❝❤ó♥❣ t❛ ❣✐↔ t❤✐➳t r➡♥❣ ❝→❝ ♣❤➨♣ t❤û ❇❡r♥♦✉❧❧✐ ✈➝♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❦❤✐ t❛ ❝❤÷❛ ♥❤➟♥ ✤÷đ❝ r ❦➳t q✉↔❀ ❤❛② ❝→❝ ♣❤➨♣ t❤û ❇❡r♥♦✉❧❧✐ s➩ ❦➳t t❤ó❝ ♥➳✉ ð ♣❤➨♣ t❤û ❝✉è✐ ♥➔② t❛ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ t❤ù ♠ët ❞➣② ❝❤ù❛ sè ♥❣➝✉ ♥❤✐➯♥ k ✈➔ ❞➣② ✤÷đ❝ ❦➳t t❤ó❝ ❜ð✐ ❝❤ú ❝❤ú A✳ A ✈➔ ✤ó♥❣ r✳ r ❇✐➳♥ ❝è ❝ì ❜↔♥ ✤✐➸♥ ❤➻♥❤ s➩ ❧➔ ❝❤ú A ✭A✿ ♣❤➨♣ t❤û t❤➔♥❤ ❝æ♥❣✮ ❳→❝ s✉➜t ❝õ❛ ❜✐➳♥ ❝è ♥❤÷ t❤➳ s➩ ❧➔ pr q k ▼ët ❝➙✉ ❤ä✐ ✤➦t r❛ ❧➔✿ ❝â t❤➸ ❝→❝ ♣❤➨♣ t❤û ❦❤ỉ♥❣ ❜❛♦ ❣✐í ✤÷đ❝ ❦➳t t❤ó❝✱ tự tỗ t ổ t❤û ✈ỵ✐ sè ❦➳t q✉↔ ♥❤ä ❤ì♥ r✳ f (k; r, p) ❇ð✐ ✈➻ k=0 ❧➔ ①→❝ s✉➜t ✤➸ ❦➳t q✉↔ t❤ù r ①↔② r❛ s❛✉ ♠ët sè ❤ú✉ ❤↕♥ tỷ tỗ t ổ ✈ỵ✐ sè ❦➳t q✉↔ ♥❤ä ❤ì♥ r ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿ ∞ f (k; r, p) = k=0 ❚r♦♥❣ ❜➔✐ t♦→♥ ✈➲ t❤í✐ ❣✐❛♥ ❝❤í ✤đ✐ ♥â✐ tr t ữ tự tr ú ợ ợ ♠å✐ r r r ♣❤↔✐ ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ❞÷ì♥❣✳ ữỡ ố tũ ỵ < p < ❞➣② {f (k; r, p)} ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ ♣❤è✐ tự ợ r ữỡ {f (k; r, p)} ❝â t❤➸ ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ t❤í✐ ❣✐❛♥ ❝❤í ✤đ✐ ❦➳t q✉↔ tự ợ r=1 ú t ữủ X r✳ ❑❤✐ ✤â t❛ ❝â ♣❤➙♥ ♣❤è✐ P❛s❝❛❧✳ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝✳ ✣â ❧➔ ♣❤➙♥ ♣❤è✐ ❝õ❛ ❜✐➸✉ t❤à sè ♣❤➨♣ t❤û ❦❤æ♥❣ t❤➔♥❤ ❝æ♥❣ ❝➛♥ ♣❤↔✐ t❤ü❝ ❤✐➺♥ ❝❤♦ ✤➳♥ ❦❤✐ t❛ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ t❤➔♥❤ ❝ỉ♥❣ t❤➻ ❞ø♥❣ ❧↕✐❀ ♥â✐ ❝→❝❤ ❦❤→❝ X ❧➔ t❤í✐ ❣✐❛♥ ❝❤í ✤đ✐ ❦➳t q✉↔ t❤➔♥❤ ❝ỉ♥❣ ✤➛✉ t✐➯♥✿ f (x) = P (X = x) = q x p ❚r♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❤➔♠ f (x) x = 0, 1, 2, 3, t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❦❤è✐ ❧÷đ♥❣ ①→❝ s✉➜t ✭♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥✮✱ ♥❤÷♥❣ ✤➸ ❝❤♦ t✐➺♥✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ t❛ ❝❤➾ ❣å✐ ❧➔ ❤➔♠ ①→❝ s✉➜t✳ ✷ ✣ó♥❣ r❛ ❤➔♠ f ❝á♥ ♣❤ư t❤✉ë❝ ✈➔♦ t❤❛♠ sè p✱ ♥❤÷♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ t❛ ❝❤➾ ①➨t ❤å ♣❤➙♥ ♣❤è✐ ♥➔②✱ ♥➯♥ t❛ ❜ä q✉❛ ❝❤♦ t✐➺♥✳ EX = 1−p = − 1, p p DX = 1−p p2 ▼✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛② ❣✐→ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♥➔② ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ I1+ ✳ I1+ = {0, 1, 2, } P ố ợ tr ữủ ❣å✐ ❧➔ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ①✉➜t ♣❤→t tø ✵✳ ◆➳✉ t❛ ①➨t ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ Y ❜✐➸✉ t❤à sè ♣❤➨♣ t❤û ❝➛♥ ♣❤↔✐ t❤ü❝ ❤✐➺♥ ❝❤♦ ✤➳♥ ❦❤✐ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ t❤➔♥❤ ❝ỉ♥❣ ✤➛✉ t✐➯♥ t❤➻ ❞ø♥❣ ❧↕✐✱ ❦❤✐ ✤â f (y) = P (Y = y) = q y−1 p y = 1, 2, 3, ●✐→ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♥➔② ❧➔ I1 = {1, 2, 3, } ❚r÷í♥❣ ❤đ♣ ♥➔② ♣❤➙♥ ♣❤è✐ s➩ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ①✉➜t ♣❤→t tø ✶✳ ❘ã r➔♥❣ X = Y − 1✳ ✣➸ ❝❤♦ t✐➺♥✱ t❛ s➩ ❦➼ ❤✐➺✉ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ①✉➜t ♣❤→t tø ✶ ❧➔ EY = , p DY = Ge(p)✳ 1−p p2 ❍➔♠ s✐♥❤ ●✐↔ sû ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ♥❤➟♥ ❝→❝ ❣✐→ trà 0, 1, 2, ✈ỵ✐ ❑❤✐ ✤â P (s) = p0 + p1 s + p2 s2 + p3 s3 + ✸ (−1 ≤ s ≤ 1) P (X = k) = pk ❚❤❛② t1 ❜➡♥❣ t1 + ✈➔ trø ✤✐ ❦➳t q✉↔ ♥❤➟♥ ✤÷đ❝ tø ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✼✮ A1 (t1 , t2 ) R (t1 , t2 ) − A1 (t1 + 1, t2 ) R (t1 + 1, t2 ) ∞ = (x1 + 1) x1 =0 ∞ ∞ − x21 f (t1 + x1 + 1, t2 ) x2 =t2 ∞ = (2x1 + 1) f (t1 + x1 + 1, t2 ) x1 =0 x2 =t2 ∞ ∞ [2 (x1 − t1 ) − 1] f (x1 , x2 ) = x1 =t1 +1 x2 =t2 = 2r1 (t1 + 1, t2 ) R (t1 + 1, t2 ) + R (t1 + 1, t2 ) ✭✷✳✸✽✮ ❑❤✐ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✷✮ t❤ä❛ ♠➣♥ t❤➻ ✭✷✳✸✹✮ ❝â t❤➸ ✤÷đ❝ t ữợ A1 (t1 , t2 ) = V1 (t2 ) + r12 (t1 , t2 ) ❚❤❛② ✈➔♦ ✭✷✳✸✽✮ t❛ ❝â [V1 (t2 ) + r12 (t1 , t2 )]R(t1 , t2 ) = [V1 (t2 ) + r12 (t1 + 1, t2 ) + 2r1 (t1 + 1, t2 ) + 1]R(t1 + 1, t2 ) = [V1 (t2 ) + (r1 (t1 + 1, t2 ) + 1)2 ]R(t1 + 1, t2 ) ✭✷✳✸✾✮ ❚ø ✭✷✳✸✽✮ R (t1 , t2 ) = + r1 (t1 + 1, t2 ) R (t1 + 1, t2 ) r1 (t1 , t2 ) ✭✷✳✹✵✮ ❇➙② ❣✐í ✭✷✳✸✾✮ ❝â t❤➸ ✤÷đ❝ ✈✐➳t ❧↕✐ ❧➔ V1 (t2 ) + r12 (t1 , t2 ) [1 + r1 (t1 + 1, t2 )] = r1 (t1 , t2 ) V1 (t2 ) + (1 + r1 (t1 + 1, t2 ))2 ❤♦➦❝ V1 (t2 ) + r1 (t1 + 1, t2 ) − r1 (t1 , t2 ) r1 (t1 , t2 ) [1 + r1 (t1 + 1, t2 )] = + r1 (t1 + 1, t2 ) − r1 (t1 , t2 ) ✭✷✳✹✶✮ ◆❣❤✐➺♠ ❝õ❛ ✭✷✳✹✶✮ ❧➔ + r1 (t1 + 1, t2 ) = r1 (t1 , t2 ) ✸✶ ✭✷✳✹✷✮ ✈➔ r1 (t1 , t2 ) [1 + r1 (t1 + 1, t2 )] = V1 (t2 ) ✭✷✳✹✸✮ ✭✷✳✹✷✮ ✈➔ ✭✷✳✹✵✮ ❝❤♦ t❛ R (t1 , t2 ) = R (t1 + 1, t2 ) ◆❤÷ ✈➟② R (t1 , t2 ) ✤ë❝ ❧➟♣ ✈ỵ✐ t1 ✱ ♠➔ ✤✐➲✉ ✤â rã r➔♥❣ ❧➔ ❦❤ỉ♥❣ ✤÷đ❝ ❝❤➜♣ ♥❤➟♥✳ ❚ø r1 (t1 + 1, t2 ) = V1 (t2 ) − r1 (t1 , t2 ) = r1 (t1 , t2 ) V1 (t2 ) − r1 (t1 − 1, t2 ) r1 (t1 − 1, t2 ) V1 (t2 ) − r1 (t1 − 1, t2 ) r1 (t1 − 1, t2 ) V1 (t2 ) − = [1 + V1 (t2 )] r1 (t1 − 1, t2 ) − V1 (t2 ) V1 (t2 ) − r1 (t1 − 1, t2 ) = [1 + 2V1 (t2 )] r1 (t1 − 2, t2 ) − V1 (t2 ) [V1 (t2 ) + 1] V1 (t2 ) − (1 + V1 (t2 )) r1 (t1 − 2, t2 ) = [V1 (t2 ) (1 + V1 (t2 ) + 2V1 (t2 )) + 1] r1 (t1 − 3, t2 ) − V1 (t2 ) [1 + 2V1 (t2 )] , [1 + V1 (t2 )] V1 (t2 ) − [1 + 2V1 (t2 )] r1 (t1 − 3, t2 ) ❝❤♦ t❛ ❦➳t q✉↔ ❧➔ r1 (t1 + 1, t2 ) ❧➔ ❤➔♠ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ t2 ✳ ❉♦ ✤â r1 (t1 , t2 ) = r1 (0, t2 ) ❇➙② ❣✐í tø ✭✷✳✹✵✮ R (t1 + 1, t2 ) = r1 (0, t2 ) + r1 (0, t2 ) R (t1 , t2 ) = g1 (t2 ) R (t1 , t2 ) ✭✷✳✹✹✮ tr♦♥❣ ✤â g1 (t2 ) = r2 (0, t2 ) r1 (0, t2 ) = 1 + r1 (0, t2 ) V1 (t2 ) (t❤❡♦ ✭✷✳✹✸✮) ▲➦♣ ❧↕✐ ❧➛♥ ❧÷đt ✈ỵ✐ ❣✐→ trà t1 tr♦♥❣ ✭✷✳✹✹✮✱ ❞➝♥ tỵ✐ R (t1 , t2 ) = [g1 (t2 )]t1 R (0, t2 ) ✸✷ ✭✷✳✹✺✮ ❚r♦♥❣ ✭✷✳✹✺✮ ❝❤♦ t2 ❞➛♥ tỵ✐ ✵ R1 (t1 ) = [g1 (0)]t1 = pt11 , tr♦♥❣ ✤â p1 = r2 (0, 0) r1 (0, 0) = + r1 (0, 0) V1 (0) ❚÷ì♥❣ tü R2 (t2 ) = pt22 ❚❤❛② ❜✐➸✉ t❤ù❝ tr➯♥ ✈➔♦ ✭✷✳✹✺✮ t❤➻ ❝â R (t1 , t2 ) = [g1 (t2 )]t1 pt22 ✭✷✳✹✻✮ R (t1 , t2 ) = [g2 (t1 )]t2 pt11 ✭✷✳✹✼✮ ❚÷ì♥❣ tü ❚ø ✭✷✳✹✻✮ ✈➔ ✭✷✳✹✼✮ ❝❤ó♥❣ t❛ ❝â g1 (t2 ) t2 = p1 g2 (t1 ) t1 p2 ♠➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝❤➾ ❝â t❤➸ ❧➔✿ gi (tj ) = pi θtj ❚ø ✭✷✳✹✽✮ R (t1 , t2 ) = pt11 pt22 θt1 t2 ✣➙② ❧➔ ❤➔♠ sè♥❣ sât ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✳ ❍ì♥ ♥ú❛ V1 (t2 ) = r1 (t1 , t2 ) [1 + r1 (t1 + 1, t2 )] = r1 (0, t2 ) [1 + r1 (0, t2 )] ❤♦➦❝ 1= r1 (0, t2 ) r12 (0, t2 ) + V1 (t2 ) V1 (t2 ) ✸✸ ✭✷✳✹✽✮ ❙✉② r❛ 0≤ r12 (0, t2 ) ≤ V1 (t2 ) ◆❣❤➽❛ ❧➔ ≤ g1 (t2 ) ≤ ❤♦➦❝ ≤ p1 ≤ ●✐→ trà ❝õ❛ θ ✤÷đ❝ rót r❛ tø t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ Vi (tj ) ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ✷✳✸ ❈→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❞ü❛ tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥ ✭①❡♠ ❬✼❪✮ ◆❣♦➔✐ ❝→❝ ❦➳t q✉↔ ♥❤÷ ❧➔ ❝→❝ ♣❤✐➯♥ ❜↔♥ ♠ð rë♥❣ ❝õ❛ tr÷í♥❣ ❤đ♣ ♠ët ❝❤✐➲✉ ❝á♥ ❝â ♠ët ✈➔✐ t➼♥❤ ❝❤➜t ♥↔② s✐♥❤ tø ❝❤➼♥❤ ✈➨❝ tì ❤❛✐ ❝❤✐➲✉✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② s➩ tr➻♥❤ ❜➔② ❜❛ ✤à♥❤ ❧➼✳ ✣➛✉ t✐➯♥ ❝❤ó♥❣ t❛ t❤✐➳t ❧➟♣ ♠ët ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ❜ð✐ ❞↕♥❣ ❝õ❛ ♠➟t ✤ë ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ Xj ≥ tj ; i, j = 1, 2; i = j Xi ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦✳ ✣à♥❤ ❧➼ ✷✳✻ ●✐↔ sû X = (X1, X2) ❧➔ ✈➨❝tì ♥❣➝✉ ♥❤✐➯♥ ợ I2+ X ữủ ố ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✭✷✳✻✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ Xi ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ Xj ≥ tj ; i, j = 1, 2; i = j ✱ ✤➣ ❝❤♦✱ ❧➔ ❤➻♥❤ ❤å❝✱ t❤❛♠ sè pi (tj ) ❧➔ ❤➔♠ ❦❤æ♥❣ t➠♥❣ ❝õ❛ tj ✳ ❈❤ù♥❣ ♠✐♥❤✿ ❑❤✐ ❝→❝ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ Xi ✈ỵ✐ ✸✹ Xj ≥ tj ✤➣ ❝❤♦ ❧➔ ❤➻♥❤ ❤å❝✱ ❤➔♠ sè♥❣ sât ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ X1 ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ X2 ✤➣ ❝❤♦✱ ❧➔ ❜✐➸✉ t❤ù❝ R (t1 |t2 ) = P [X1 ≥ t1 |X2 ≥ t2 ] = [p1 (t2 )]t1 ✭✷✳✹✾✮ < p1 (t2 ) < 1, t1 = 0, 1, 2, ✣➦t t2 = R1 (t1 ) = P [X1 ≥ t1 ] = (p1 )t1 , tr♦♥❣ ✤â p1 = p1 (0) ❚❛ ❝â R (t1 , t2 ) = R (t1 |t2 ) R2 (t2 ) = [p1 (t2 )]t1 pt22 ❚÷ì♥❣ tü✱ t❤❛② ✈❛✐ trá X2 ❝❤♦ X1 ✭✷✳✺✵✮ ❝❤ó♥❣ t❛ ❝â R (t1 , t2 ) = [p2 (t1 )]t2 pt11 ✭✷✳✺✶✮ ❚ø ✭✷✳✺✵✮ ✈➔ ✭✷✳✺✶✮ ❝❤ó♥❣ t❛ ❝â [p1 (t2 )]t1 pt22 = [p2 (t1 )]t2 pt11 Pữỡ tr ú ợ p1 (t2 ) p1 t1 , t2 ✱ 1/t2 = ✭✷✳✺✷✮ ♥❣❤✐➺♠ ❝õ❛ ♥â ❝❤➾ ❝â t❤➸ ❧➔ p2 (t1 ) p1 1/t1 ❧➔ ❤➡♥❣ sè ✤ë❝ ❧➟♣ ✈ỵ✐ t1 ✈➔ t2 ✳ ❚ø ✤â ❝❤♦ pi (tj ) = pi θtj ♠➔ ❦❤✐ t❤❛② ✈➔♦ ✭✷✳✺✶✮ ❤♦➦❝ ✭✷✳✺✷✮ ❞➝♥ ✤➳♥ ✭✷✳✻✮✳ ▼➦t ❦❤→❝✱ ♥➳✉ ❝❤ó♥❣ t❛ ❣✐↔ sû R (t1 , t2 ) = pt11 pt22 θt1 t2 ✸✺ =θ ✭✷✳✺✸✮ ❝❤ó♥❣ t❛ ❝â R (t1 |t2 ) = ❍➔♠ ①→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ X1 R (t1 , t2 ) = pt11 θt1 t2 R2 (t2 ) ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ X2 > t2 f (x1 |X2 ≥ x2 ) = R (t1 |t2 ) − R (t1 + 1|t2 ) = p1 θt2 ✈➔ ♥â ❧↕✐ ❧➔ ❤➻♥❤ ❤å❝ ✈ỵ✐ t❤❛♠ sè ❈ơ♥❣ ✈➻ X1 ✈ỵ✐ ≤ θ ≤ 1, X2 > t2 X1 ≥ t1 ♥➯♥ ✤➣ ❝❤♦✿ t1 − p1 θ t p1 θ t ✳ < p1 θt2 < (∀t2 )✳ ❉♦ ✤â ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤➣ ❝❤♦ ❧➔ ❤➻♥❤ ❤å❝✳ ❚÷ì♥❣ tü ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ X2 ✈ỵ✐ ✤➣ ❝❤♦ ❧➔ ❤➻♥❤ ❤å❝✳ ❑❤✐ ✈➨❝tì ♥❣➝✉ ♥❤✐➯♥ X ❧➔ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✱ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ❤➔♠ ①→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ X1 ❦❤✐ X = x2 ✤➣ ❝❤♦✱ ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✶✳✷✽✮ ✈➔ ❤➔♠ ①→❝ s✉➜t ❜✐➯♥ ❞✉②➯♥ ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✷✳✶✽✮✳ ❉↕♥❣ ❝õ❛ ❤➔♠ ①→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥ ✭✶✳✷✽✮ ❝â t❤➸ ✤÷đ❝ sû ❞ư♥❣ ❞➝♥ ✤➳♥ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ ❤➻♥❤ ❤å❝ ♠ët ❝❤✐➲✉ ❝ơ♥❣ ♥❤÷ ♠ỉ ❤➻♥❤ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ❞↕♥❣ ✭✷✳✻✮✳ ✣à♥❤ ❧➼ ✷✳✼ ◆➳✉ X = (X1, X2) ❧➔ ✈➨❝tì ♥❣➝✉ ♥❤✐➯♥ ✈ỵ✐ ❣✐→ I2+ s❛♦ ❝❤♦ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ X1 ✈ỵ✐ X2 ✤➣ ❝❤♦ ❧➔ ❜✐➸✉ t❤ù❝ ✭✶✳✷✽✮✱ ❦❤✐ ✤â X1 ❧➔ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ♥➳✉ X2 ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝✳ ❈❤ù♥❣ ♠✐♥❤✿ ●✐↔ sû X2 ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝✳ ❑❤✐ ✤â ❤➔♠ ①→❝ s✉➜t ❝õ❛ X2 ❧➔ ❜✐➸✉ t❤ù❝ f2 (x2 ) = px2 (1 − p2 ) , x2 = 0, 1, 2, ❙û ❞ö♥❣ ❜✐➸✉ tự ố ợ st ỗ tớ X f (x1 |x2 ) tø ✭✶✳✷✽✮ ✈➔ f2 (x2 ) ✤÷đ❝ ❝❤♦ ð tr➯♥✱ ❤➔♠ ①→❝ s➩ ❧➔ f (x1 , x2 ) = px1 px2 θx1 x2 −1 − p1 θx2 +1 ✸✻ ✭✷✳✺✹✮ − p2 θx1 +1 + θ − ▲➜② tê♥❣ t❤❡♦ ❣✐→ ❝õ❛ X2 ✱ f1 (x1 ) = px1 (1 − p1 ) , x1 = 0, 1, 2, ❈ị♥❣ t❤í✐ ✤✐➸♠ ♥➔② ♥➳✉ t❤➔♥❤ ♣❤➛♥ ✤➥♥❣ t❤ù❝ X1 ✭✷✳✺✺✮ ✤÷đ❝ ❝❤♦ ❧➔ ❤➻♥❤ ❤å❝ ❞↕♥❣ ✭✷✳✺✺✮✱ t❤➻ ∞ f (x1 |x2 ) f2 (x2 ) f1 (x1 ) = x2 =0 ❝❤♦ t❛ ∞ px1 (1 − p1 ) = x2 =0 px1 θx1 x2 −1 − p2 − p1 θx2 +1 − p1 θx1 +1 + θ − f2 (x2 ) ❤♦➦❝ ∞ θx1 x2 −1 θ − p2 θx1 +1 − p1 θx2 +1 + p1 p2 θx1 +x2 −2 f2 (x2 ) (1 − p1 ) (1 − p2 ) = x2 =0 ∞ θx1 x2 (1 − p1 θx2 ) − p2 θx1 (x2 +1) − p1 θx2 +1 = f2 (x2 ) x2 =0 ❈❤♦ ❝→❝ ❤➺ sè ❝õ❛ θrx1 , r = 0, 1, 2, ✱ ð ❤❛✐ ✈➳ ❜➡♥❣ ♥❤❛✉✿ f2 (x2 ) = px2 (1 − p2 ) , x2 = 0, 1, 2, ✈➔ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ t❛ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔✿ P❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ Xi Xi ✈ỵ✐ X j = xj ✤➣ ❝❤♦ ❧➔ ❜✐➸✉ t❤ù❝ ✭✶✳✷✽✮ ✈➔ ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ (X1 , X2 ) ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✳ ✣à♥❤ ❧➼ ✷✳✽ ●✐↔ sû X = (X1, X2) ❧➔ ✈➨❝tì ♥❣➝✉ ♥❤✐➯♥ ✈ỵ✐ ❣✐→ I2+✳ X ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✭✷✳✻✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✭✐✮ P❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥ ❝õ❛ X1 ❧➔ ❤➻♥❤ ❤å❝❀ ✸✼ ✭✐✐✮ ❑➻ ✈å♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❝õ❛ X2 ✈ỵ✐ X1 ≥ t ✤➣ ❝❤♦✿ E (X2 − t2|X1 ≥ t) ❧➔ p2θt − p2 θ t −1 ❈❤ù♥❣ ♠✐♥❤✿ ✣✐➲✉ ❦✐➺♥ ✭✐✐✮ tr♦♥❣ ✤à♥❤ ❧➼ ❝â t❤➸ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ ∞ (x2 − t2 ) f (x1 , x2 ) = r2 (t1 ) R (t1 , t2 ) = t1 ❚❤❛② t2 ❧➔ t2 + ∞ ∞ t2 R (t1 , t2 + s) ✭✷✳✺✻✮ s=1 ✈➔ trø ✤✐ tø ✭✷✳✺✻✮✿ r2 (t1 ) [R (t1 , t2 ) − R (t1 , t2 + 1)] = R (t1 , t2 + 1) ❤♦➦❝ R (t1 , t2 + 1) = ●✐↔♠ t2 r2 (t1 ) R (t1 , t2 ) + r2 (t1 ) ✭✷✳✺✼✮ ❧✐➯♥ t✐➳♣ t❛ ✤÷đ❝ r2 (t1 ) R (t1 , t2 ) = + r2 (t1 ) t2 R (t1 , 0) ✭✷✳✺✽✮ ❑❤✐ ✭✐✐✮ t❤ä❛ ♠➣♥✿ r2 (t1 ) = p θ t1 + r2 (t1 ) ❚ø ✤✐➲✉ ❦✐➺♥ ✭✐✮ ❝õ❛ ✤à♥❤ ❧➼ R (t1 , 0) = pt11 ❇➙② ❣✐í ✭✷✳✺✽✮ s➩ ❧➔ R (t1 , t2 ) = pt11 pt22 θt1 t2 ◆❣÷đ❝ ❧↕✐ ❦❤✐ X ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✱ ✭✐✮ ✈➔ ✭✐✐✮ ✤÷đ❝ s✉② r❛ tø ✭✷✳✾✮ ✈➔ ✭✶✳✷✼✮ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ✷✳✹ ❈→❝ ✤➦❝ tr÷♥❣ ❞ü❛ tr➯♥ tè❝ ✤ë t❤➜t ❜↕✐ ✈➔ ❤➔♠ sè♥❣ t❤➯♠ tr✉♥❣ ❜➻♥❤ ✭①❡♠ ❬✺❪✮ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ t❛ ✤➲ ❝➟♣ ✤➳♥ ❦❤→✐ ♥✐➺♠ tè❝ ✤ë t❤➜t ❜↕✐ ✭❢❛✐❧✉r❡ r❛t❡✮ ✈➔ ❤➔♠ sè♥❣ t❤➯♠ tr✉♥❣ ❜➻♥❤ ✭♠❡❛♥ r❡s✐❞✉❛❧ ❧✐❢❡ ✭▼❘▲✮✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ rí✐ ✸✽ r↕❝ ✈➔ sû ❞ư♥❣ ♥â ♥❤÷ ❧➔ ❝ỉ♥❣ ❝ư ✤➸ ✤➦❝ tr÷♥❣ ❝❤♦ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✳ ❈❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ tè❝ ✤ë t❤➜t ❜↕✐ ❤❛✐ ❝❤✐➲✉ ♥❤÷ ❧➔ ♠ët ✈➨❝tì ❤❛✐ t❤➔♥❤ ♣❤➛♥ h (t) = (h1 (t) , h2 (t)) hi (t) = tr♦♥❣ ✤â P (Xi = ti , Xj ≥ tj ) ; i, j = 1, 2; i = j P (X1 ≥ t1 , X2 ≥ t2 ) = R2 (t2 ) fi (ti |Xj ≥ tj ) R (t1 , t2 ) ú ỵ r ố õ ❦✐➺♥ tr➯♥ tû sè ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤➙♥ ♣❤è✐ ♠➔ ❝❤ó♥❣ t❛ ✤➣ ❣➦♣ tr♦♥❣ ♣❤➛♥ ✶✳✷✳ ❚r♦♥❣ ❝→❝ ✤à♥❤ ❧➼ t✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ t➼♥❤ ❤➡♥❣ sè ✤à❛ ♣❤÷ì♥❣ ❝õ❛ hi (t1 , t2 ) ❧➔ ♠ët t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✭✷✳✻✮✳ ✣à♥❤ ❧➼ ✷✳✾ ❱➨❝ tì ♥❣➝✉ ♥❤✐➯♥ X = (X1, X2) ✈ỵ✐ ❣✐→ I2+ ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✭✷✳✻✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❤➔♠ tè❝ ✤ë t❤➜t ❜↕✐ ❝õ❛ ♥â ❝â ❞↕♥❣ h (t1, t2) = [h1 (t2) , h2 (t1)] tr♦♥❣ ✤â h1 ✈➔ h2 ❧➔ ❝→❝ ❤➔♠ ❦❤æ♥❣ t➠♥❣ t❤❡♦ ❝→❝ ❜✐➳♥ t÷ì♥❣ ù♥❣✱ s❛♦ ❝❤♦ hi (0) = − pi , i = 1, ❈❤ù♥❣ ♠✐♥❤✿ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ❝➛♥✱ tø ✭✷✳✺✾✮ ❝❤ó♥❣ t❛ ❝â R (t1 , t2 ) h1 (t1 , t2 ) = R (t1 , t2 ) − R (t1 + 1, t2 ) ❉♦ ✤â R (t1 + 1, t2 ) = [1 − h1 (t1 , t2 )] R (t1 , t2 ) ✭✷✳✻✵✮ R (t1 , t2 + 1) = [1 − h2 (t1 , t2 )] R (t1 , t2 ) ✭✷✳✻✶✮ ❚÷ì♥❣ tü ✸✾ ✣➦t t2 = tr♦♥❣ ✭✷✳✻✵✮ R1 (t1 ) = [1 − h1 (t1 − 1, 0)] R (t1 1) ữỡ tr ợ ❣✐→ trà t1 ❣✐↔♠ ❞➛♥ t❛ ✤÷đ❝ t1 [1 − h1 (t1 − r, 0)] R1 (t1 ) = r=1 ❚÷ì♥❣ tü t2 [1 − h2 (0, t2 − r)] R2 (t2 ) = ✭✷✳✻✷✮ r=1 ❍ì♥ ♥ú❛ ❧➦♣ ❧↕✐ ✭✷✳✻✵✮ ✈ỵ✐ ❝→❝ ❣✐→ trà t1 ❣✐↔♠✱ ❝❤♦ t❛ t1 [1 − h1 (t1 − r, t2 )] R (0, t2 ) R (t1 , t2 ) = ✭✷✳✻✸✮ r=1 ❚❤❛② R(0, t2 ) ❜ð✐ ✭✷✳✻✷✮✿ t1 t2 [1 − h1 (t1 − r, t2 )] R (t1 , t2 ) = r=1 [1 − h2 (0, t2 − r)] ✭✷✳✻✹✮ r=1 ◆❤÷ ✈➟② ✭✷✳✻✹✮ ❝❤♦ ❝ỉ♥❣ t❤ù❝ tê♥❣ q✉→t ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ tè❝ ✤ë t❤➜t ❜↕✐ ✈➔ ❤➔♠ sè♥❣ sât✳ ◆â ❜✐➸✉ ❞✐➵♥ r➡♥❣ ❤➔♠ tè❝ ✤ë t❤➜t t ố tữỡ ự ữợ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤à♥❤ ❧➼✱ ✭✷✳✻✸✮ s➩ ❧➔ R (t1 , t2 ) = [1 − h1 (t2 )]t1 R2 (t2 ) ❑❤✐ t2 = ✭✷✳✻✺✮ tr♦♥❣ ✭✷✳✻✺✮ ❝❤ó♥❣ t❛ ❝â R (t1 , t2 ) = pt11 ❙û ❞ư♥❣ ❦ÿ t❤✉➟t t÷ì♥❣ tü✱ ❤➺ t❤ù❝ R (t1 , t2 ) = [1 − h2 (t1 )]t2 R1 (t1 ) ✹✵ ✭✷✳✻✻✮ ❝❤♦ R2 (t2 ) = pt22 ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✻✺✮ ✈➔ ✭✷✳✻✻✮ ❞➝♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✿ [1 − h1 (t2 )]t1 pt22 = [1 − h2 (t1 )]t2 pt11 ◆❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❧➔ − h1 (t2 ) p1 tr♦♥❣ ✤â θ t1 ❧➔ ✤ë❝ ❧➟♣ ✈ỵ✐ ❝↔ t2 − h2 (t1 ) p2 = t1 =θ ✭✷✳✻✼✮ ✈➔ t2 ✳ ✭✷✳✻✼✮ ❝❤♦ ❝❤ó♥❣ t❛ ❝→❝ ❜✐➸✉ t❤ù❝ hi (tj ) = − pi θtj ✭✷✳✻✽✮ ❚❤❛② ✭✷✳✻✽✮ ✈➔♦ ✭✷✳✻✺✮ ❤♦➦❝ ✭✷✳✻✻✮ ❝❤ó♥❣ t❛ ❝â R (t1 , t2 ) = pt11 pt22 θt1 t2 ✈➔ ♥❤÷ ✈➟② (X1 , X2 ) ◆❣÷đ❝ ❧↕✐ ❦❤✐ ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✳ X ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✱ ❝❤ó♥❣ t❛ ❝â Ri (ti ) = ptii ✈➔ f (ti |Xj ≥ tj ) ✤÷đ❝ ❝❤♦ ❜ð✐ ✭✶✳✸✶✮✱ ❞♦ ✤â tø ✭✷✳✺✾✮✿ hi (t1 , t2 ) = − pi θtj ♠➔ rã r➔♥❣ õ ợ ti ự ữủ t❤➔♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ rí✐ r↕❝ ❝❤ó♥❣ t❛ ①→❝ ✤à♥❤ ❤➔♠ sè♥❣ t❤➯♠ tr✉♥❣ ❜➻♥❤ ✭▼❘▲❋✮ ❤❛✐ ❝❤✐➲✉✳ ❑➼ ❤✐➺✉ X = (X1 , X2 ) ✈➔ X>x ♥❣❤➽❛ ❧➔ ❍➔♠ sè♥❣ t❤➯♠ tr✉♥❣ ❜➻♥❤ ❤❛✐ ❝❤✐➲✉ ❝õ❛ X Xi > xi , ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ ❧➔ ♠ët ❤➔♠ ✈➨❝tì r(t) = E (X − t/X > t) , ✹✶ i = 1, tr♦♥❣ ✤â t = (t1 , t2 ) ✈➔ ti ❧➔ ❝→❝ sè t❤ü❝✱ ❦❤æ♥❣ ➙♠✳ ❉♦ ✤â ∞ ∞ t1 t2 (xi − ti ) f (x1 , x2 ) ri (t1 , t2 ) R (t1 , t2 ) = ❙û ❞ö♥❣ ✭✶✳✷✷✮✱ ♥❣❤➽❛ ❧➔ r1 (t1 , t2 ) R (t1 , t2 ) = R (t1 + 1, t2 ) [1 + r1 (t1 + 1, t2 )] ✈➔ sû ❞ư♥❣ ✭✷✳✻✵✮ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ h1 − h1 (t1 , t2 ) = r (t) ✈➔ r1 ✤÷đ❝ ❧✐➯♥ ❤➺ ❜ð✐ ❤➺ t❤ù❝ r1 (t1 , t2 ) + r1 (t1 + 1, t2 ) ✭✷✳✻✾✮ ữ ởt tỡ ợ t ri (t1 , t2 ) = E [Xi − ti |Xi ≥ ti , i = 1, 2] ✭✷✳✼✵✮ ❚❤❡♦ ♥❣æ♥ ♥❣ú ❝õ❛ ▼❘▲❋ ❤❛✐ ❝❤✐➲✉✱ ✤à♥❤ ❧➼ ✷✳✾ ❝â t❤➸ ✤÷đ❝ ♣❤→t ❜✐➸✉ ❧↕✐ ♥❤÷ s❛✉✿ ✣à♥❤ ❧➼ ✷✳✶✵ ❱➨❝tì ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ X tr♦♥❣ ✤à♥❤ ❧➼ ✷✳✾ ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✭✷✳✻✮ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ▼❘▲❋ ❝õ❛ ♥â ❝â ❞↕♥❣ r (t) = (r1 (t2 ) , r2 (t1 )) ✈ỵ✐ ❝↔ ❤❛✐ r1 ✈➔ r2 ❧➔ ❦❤ỉ♥❣ t➠♥❣ t❤❡♦ ❝→❝ ❜✐➳♥ t÷ì♥❣ ù♥❣✱ ✈ỵ✐ ri (0) = pi/1 − pi ❈❤ù♥❣ ♠✐♥❤ ❝õ❛ ✤à♥❤ ❧➼ ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣ tø ✭✷✳✻✾✮ ✈➔ ✤à♥❤ ❧➼ ✷✳✾✳ ✹✷ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✧ ▼ët sè ❜➔✐ t♦→♥ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ✧ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ❞↕♥❣ ❦❤→❝ ♥❤❛✉ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉✱ ❝ơ♥❣ ♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t t÷ì♥❣ ù♥❣ ❝õ❛ ❝❤ó♥❣✳ ❚✐➳♣ ✤➳♥ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ ❞ü❛ tr➯♥ ❝→❝ ✤➦❝ tr÷♥❣ s trữ ỹ tr t t tr ợ ✰ ✣➦❝ tr÷♥❣ ❞ü❛ tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ỉ♠❡♥ ❜à ❝❤➦t ❝ưt❀ ✰ ✣➦❝ tr÷♥❣ ❞ü❛ tr➯♥ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥❀ ✰ ✣➦❝ tr÷♥❣ ❞ü❛ tr➯♥ ❤➔♠ tè❝ ✤ë t❤➜t ❜↕✐ ✈➔ ❤➔♠ sè♥❣ t❤➯♠ tr✉♥❣ ❜➻♥❤✳ ❉♦ ❦❤↔ ♥➠♥❣ ❝â ❤↕♥ ♥➯♥ ♠➦❝ ❞ị ✤➣ ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ ①ât✳ ❚æ✐ r➜t ữủ sỹ õ ỵ qỵ t ổ ✈➔ ❜↕♥ ✤å❝✳ ✹✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❙♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❛ss♦❝✐✲ ❛t❡❞ ✇✐t❤ t❤❡ ❜✐✈❛r✐❛t❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ❞✐str✐❜✉t✐♦♥s ❬✶❪ ❑✳❘✳ ▼✉r❛❧❡❡❞❤❛r❛♥ ◆❛✐r✱ ✭✶✾✾✵✮✳ ❬✷❪ ●✉♣t❛ P✳▲✳✱ ✭✶✾✽✺✮✳ ❙♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❞✐str✐❜✉t✐♦♥s ❜② tr✉♥❝❛t❡❞ ♠♦♠❡♥ts✱ ❙t❛t✐st✐❝s✱ ✶✻✱ ✹✻✺✲✹✼✸✳ ❬✸❪ ▲✉❦❛❝s ❊✳ ❛♥❞ ▲❛❤❛ ❘✳●✳ ✭✶✾✻✹✮✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥s✱ ●r✐❢❢✐♥✱ ▲♦♥❞♦♥✳ ❬✹❪ ◆❛❣❛r❛❥❛ ❍✳◆✳ ✭✶✾✼✺✮✳ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s♦♠❡ ❞✐str✐❜✉t✐♦♥s ❜② ❝♦♥❞✐t✐♦♥❛❧ ♠♦♠❡♥ts✱ ❏✳■♥❞✳❙t❛t✐st✳❆ss♦❝✱ ✶✸✱ ✺✼✲✻✶✳ ❬✺❪ ◆❛✐r ❑✳❘✳▼✳ ❛♥❞ ◆❛✐r ◆✳❯✳ ✭✶✾✽✾✮✳ ❇✐✈❛r✐❛t❡ ♠❡❛♥ r❡s✐❞✉❛❧ ❧✐❢❡✱ ■✳❊✳❊✳❊✳ ❚r❛♥s✳ ❘❡❧✳ ✸✽✱ ♣✳ ✸✻✷✲✸✻✹✳ ❬✻❪ ◆❛✐r ◆✳❯✳ ✭✶✾✽✸✮✳ ❆♠❡❛s✉r❡ ♦❢ ♠❡♠♦r② ❢♦r s♦♠❡ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥s✱ ❏✳■♥❞✳❙t❛t✐st✳❆ss♦❝✳✱ ✷✶✱ ✶✹✶✲✶✹✼ ❬✼❪ ◆❛✐r ◆✳❯✳ ❛♥❞ ◆❛✐r ❑✳❘✳▼✳ ✭✶✾✾✵✮✳ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s♦❢ t❤❡ ●✉♠❜❡❧s ❜✐✈❛r✐✲ ❛t❡ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥✱ ❙t❛t✐st✐❝s ✷✶ ✭t♦ ❛♣♣❡❛r✮✳ ❬✽❪ P❛t❤❛❦ ❆✳●✳ ❛♥❞ ❙r❡❡❤❛r✐ ▼ ✭✶✾✽✶✮✳ ❙♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛ ❜✐✈❛r✐❛t❡ ❣❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥✱ ❏✳■♥❞✳❙t❛t✐st✳❆ss♦❝✳ ✶✾✱ ✶✹✶✲✶✹✻✳ ✹✹ ❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❣❡♦✲ ♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ❛ ❜✐✈❛r✐❛t❡ ❣❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥✱❙❛♥❦❤✉②❛✱ ❆✱ ✸✹✱ ❬✾❪ P❛✉❧s♦♥ ❆✳❙✳ ❛♥❞ ❯♣♣✉❧✉r✐ ❱✳❘✳❘✳ ✭✶✾✼✷✮✳ ✽✽✲✾✶✳ ❬✶✵❪ ❳❡❦❛❧❛❦✐ ❊ ✭✶✾✽✸✮✳ ❍❛③❛r❞ ❢✉♥❝t✐♦♥s ❛♥❞ ❧✐❢❡ ❞✐str✐❜✉t✐♦♥s ✐♥ ❞✐s❝r❡t❡ t✐♠❡✱ ❈♦♠♠✉♥✳ ❙t❛t✐st✳✱ ✶✷✱ ✷✺✵✸✲✷✺✵✾✳ ❬✶✶❪ ❊❞✇❛r❞ ❖♠❡② ❛♥❞ ▲❡❞❛ ❉✳ ▼✐♥❦♦✈❛ ✭✶✾✾✾✮✳ t✐♦♥s✱▲✐r✐❛s✳❍✉❜r✉ss❡❧✳❜❡✱ ✸ ✲ ✼✳ ✹✺ ❇✐✈❛r✐❛t❡ ❣❡♦♠❡tr✐❝ ❞✐str✐❜✉✲ ... ❝❤✐➲✉✱ ♠ð rë♥❣ t➼♥❤ ♠➜t tr➼ ♥❤ỵ✱ ♣❤➙♥ ♣❤è✐ ❜✐➯♥ ❞✉②➯♥ ✈➔ ♣❤➙♥ ♣❤è✐ ❝â ✤✐➲✉ ❦✐➺♥✱ ❝→❝ ổ ợ ữỡ ởt số t ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❤❛✐ ❝❤✐➲✉ r ữỡ ợ t t t trữ ỹ tr t t tr ợ ổ ❜à ❝❤➦t... t|Y ≥ t] = E (Y ) ✹ ✭✶✳✸✮ ✭①❡♠ ❬✻❪✮✳ ❍➔♠ sè♥❣ sât ●✐↔ sû ✤↕✐ ❧÷đ♥❣ ♥❣➝✉ ♥❤✐➯♥ X ❞÷ì♥❣ tũ ỵ t tớ số tớ ❝❤í ✤đ✐✳ R(t) = P (X > t) ✣➦t ❘ã r➔♥❣ R(t) R(t) ❧➔ ①→❝ s✉➜t ✤➸ t❤í✐ ❣✐❛♥ sè♥❣ ✭❦➸ tø ❧ó❝... t❤è♥❣ ❦➯ ù♥❣ ❞ư♥❣✳ P❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ ❝ơ♥❣ ✤÷đ❝ ❜✐➳t ✤➳♥ ♥❤÷ ❧➔ ♠ët ❜↔♥ s❛♦ rí✐ r↕❝ ❝õ❛ ố ụ õ ởt số ỵ tr÷♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ơ ♠ët ❝❤✐➲✉ ✤➣ ✤÷đ❝ ♠ð rë♥❣ ✤➳♥ ❦❤ỉ♥❣ ❣✐❛♥ ♠➝✉ rí✐ r↕❝ ✤➸ ❝❤ù♥❣ ♠✐♥❤