Quá trình poisson và các vấn đề liên quan

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲✲✲✲✲✲ ✲✲✲✲✲✲ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❍⑨ ◗❯⑩ ❚❘➐◆❍ P❖■❙❙❖◆ ❱⑨ ❈⑩❈ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✻ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲✲✲✲✲✲ ✲✲✲✲✲✲ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❍⑨ ◗❯⑩ ❚❘➐◆❍ P❖■❙❙❖◆ ❱⑨ ❈⑩❈ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ ▲Þ ❚❍❯❨➌❚ ❳⑩❈ ❙❯❻❚ ❱⑨ ❚❍➮◆● ❑➊ ❚❖⑩◆ ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✵✻ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ữớ ữợ ❍➪❆ ◆❣❤➺ ❆♥ ✲ ✷✵✶✻ ✶ ▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ✶ ▲í✐ ♥â✐ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✹ ✶✳✶✳ ✶✳✷✳ ✶✳✸✳ ✶✳✹✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ✈❡❝tì ♥❣➝✉ ♥❤✐➯♥ P❤➙♥ ♣❤è✐ ♠ô ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ◗✉→ tr➻♥❤ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳ ●✐ỵ✐ t❤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ✷✳✸✳ ❈→❝ q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✵ ✶✸ ✶✺ ✶✼ ✶✼ ✶✾ ✸✺ ✹✵ ✹✷ ✷ ▼Ð ✣❺❯ ◗✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ q✉→ tr q trồ ỵ tt ụ ữ ù♥❣ ❞ư♥❣✳ ✣➙② ❧➔ ❤á♥ ✤→ t↔♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ ♥❣➝✉ ♥❤✐➯♥✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ t✐➳♣ ❝➟♥ ✈➲ q✉→ tr➻♥❤ P♦✐ss♦♥✳ ✣➛✉ t✐➯♥ ữ ởt q tr ợ ợ tớ ❦ý ✤ê✐ ♠ỵ✐ ❧✐➯♥ t✐➳♣ ❝â ♣❤➙♥ ♣❤è✐ ♠ơ✳ ❚❤ù ❤❛✐ ♥❤÷ ♠ët q✉→ tr➻♥❤ ✤➳♠ ❝â sè ❞ø♥❣ ✈➔ sè ❣✐❛ ✤ë❝ ❧➟♣ ✈ỵ✐ ❦❤→❝❤ ✤➳♥ tr➯♥ ♠é✐ ❦❤♦↔♥❣ ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈➔ t❤ù ❜❛ ♥â ♥❤÷ ❧➔ ♠ët ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐✳ ❚❛ t❤➜② r➡♥❣ ♠é✐ ✤à♥❤ ♥❣❤➽❛ ❝✉♥❣ ❝➜♣ ♠ët ❝→❝❤ ♥❤➻♥ r✐➯♥❣ ✈➔♦ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ q✉→ tr➻♥❤✳ ◆❤➜♥ ♠↕♥❤ sü q✉❛♥ trå♥❣ ❝õ❛ t➼♥❤ ❦❤ỉ♥❣ ♥❤ỵ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ơ ð ❝❤é ✈ø❛ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ❤ú✉ ➼❝❤ tr♦♥❣ qt ứ ỵ s❛♦ q✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ ♠ët q✉→ tr➻♥❤ ✤ì♥ ❣✐↔♥✳ ợ õ ữủ tr tr ✷ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❝➛♥ ❞ò♥❣ ❝❤♦ ❝→❝ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ð ❝❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ q✉→ tr➻♥❤ P♦✐ss♦♥ ✈ỵ✐ tữỡ ữỡ ỗ tớ ự q tr➻♥❤ P♦✐ss♦♥ ❧➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❝♦ ❇❡r♥♦✉❧❧✐✳ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ữợ sỹ ữợ ❣✐→♦ ❚❙✳ ◆❣✉②➵♥ ❚r✉♥❣ ❍á❛✳ ❚→❝ ❣✐↔ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t tỵ✐ ❚❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ tờ ỵ tt st tố t ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ t➟♥ t➻♥❤ ❞↕② ❞é✱ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ❚→❝ ❣✐↔ ✸ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ✤➳♥ qỵ ổ rữớ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❜❛♥ ❧➣♥❤ ✤↕♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ỗ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ị t→❝ ❣✐↔ ✤➣ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ♥➠♥❣ ❧ü❝✱ ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❘➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ þ ❦✐➳♥ ✤â♥❣ ❣â♣ q✉þ ❜→✉ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ ♥❣➔② ✷✵ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✻ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚❤✉ ❍➔ ✹ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ✈❡❝tì ♥❣➝✉ ♥❤✐➯♥ ✶✳✶✳✶ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❤➔♠ ♣❤➙♥ ♣❤è✐✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧➔ →♥❤ ①↕ X : Ω → R s❛♦ ❝❤♦ (X x) = {ω ∈ Ω|X(ω) x} ∈ A, ∀x ∈ R, ❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣ X −1 (B) = {ω ∈ Ω|X(ω) ∈ B} ∈ A, ∀B ∈ B ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ F (x) = P {X x}, x ∈ R ❍➔♠ sè ♥➔② ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝➛♥ ✈➔ ✤õ s❛✉✿ ✐✮ ❦❤æ♥❣ ❣✐↔♠❀ ✐✐✮ ❧✐➯♥ tö❝ ❜➯♥ ♣❤↔✐❀ ✐✐✐✮ limx→−∞ F (x) = 0, limx→+∞ F (x) = ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✤÷đ❝ ❣å✐ ❧➔ rí✐ r↕❝ ♥➳✉ t➟♣ t➜t ❝↔ ❝→❝ ❣✐→ trà ❝õ❛ ♥â ❧➔ ❤ú✉ ❤↕♥ ❤❛② ✤➳♠ ✤÷đ❝✳ ❑➼ ❤✐➺✉ (x1, x2, ) ❧➔ ❝→❝ ❣✐→ trà ❝õ❛ X ✳ ❚❛ ✤➦t pn = P (X = xn), (n = 1, 2, ) ✈➔ ❣å✐ (pn) ❧➔ ❞➣② ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ X ✳ ❉➣② sè ♥➔② ❝â ❝→❝ t➼♥❤ ❝❤➜t ✭❝➛♥ ✈➔ ✤õ✮ s❛✉✿ ✐✮ ❦❤æ♥❣ ➙♠✱ tù❝ ❧➔ pn (n = 1, 2, )❀ ✐✐✮ ❝â tê♥❣ ❜➡♥❣ 1✱ tù❝ ❧➔ n p(n) = 1✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ ♥➳✉ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ✺ ♥â ❝â ✤↕♦ ❤➔♠✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ❣å✐ f (x) = F (x), x ∈ R ❧➔ ❤➔♠ ♠➟t ✤ë✳ ❍➔♠ sè ♥➔② ❝â ❝→❝ t➼♥❤ ❝❤➜t ✭❝➛♥ ✈➔ ✤õ s❛✉✮✿ ✐✮ ❦❤æ♥❣ ➙♠✱ tù❝ ❧➔ f (x) ∀x ∈ R❀ +∞ ✐✐✮ ❝â t➼❝❤ ♣❤➙♥ ❜➡♥❣ 1✱ tù❝ ❧➔ −∞ f (x)dx = 1✳ ●✐↔ sû (Ω, A) ✈➔ (E, B) ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ✤♦✳ →♥❤ ①↕ X : Ω → E ✤÷đ❝ ❣å✐ ❧➔ ✤♦ ✤÷đ❝✱ ❤❛② ❝❤➼♥❤ ①→❝ ❤ì♥ ❧➔ (A, B)✲✤♦ ✤÷đ❝ ♥➳✉ X −1 (B) ∈ A, ∀B ∈ B, ❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣ X −1 (C) ∈ A, ∀C ∈ C, tr♦♥❣ ✤â B = σ(C)✳ ◆➳✉ (Ω, A, µ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝â ✤ë ✤♦✱ t❤➻ t❛ ✤➦t µX (B) = µ(X −1 (B)), B ∈ B ❑❤✐ ✤â✱ µX ❧➔ ✤ë ✤♦ ①→❝ ✤à♥❤ tr➯♥ B✳ ❚❛ ❣å✐ µX ❧➔ ✤ë ✤♦ ↔♥❤ ❝õ❛ ✤ë ✤♦ µ q✉❛ →♥❤ ①↕ X r trữớ ủ = P ①→❝ s✉➜t✱ t❤➻ PX ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ ♣❤è✐ ✭①→❝ s✉➜t✮ ❝õ❛ X ✭tr➯♥ ❦❤æ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ E ✮✳ ❑❤✐ E = Rn✱ B = B(Rn)✱ t❤➻ X = (X1, , Xn) ✤÷đ❝ ❣å✐ ❧➔ ✈❡❝tì ♥❣➝✉ ♥❤✐➯♥ ✈➔ PX ữủ ố ỗ tớ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X1 , , Xn ✳ ❈➛♥ ❝❤ó þ r➡♥❣✿ ✲ ▼é✐ ✤ë ✤♦ ①→❝ s✉➜t µ tr➯♥ (R, B) t÷ì♥❣ ù♥❣ ❞✉② ♥❤➜t ✭❝❤➼♥❤ ①→❝ ✤➳♥ ❤➡♥❣ sè ❝ë♥❣✮ ✈ỵ✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t F ✭tù❝ ❧➔ F ❦❤ỉ♥❣ ❣✐↔♠✱ ❧✐➯♥ tư❝ ♣❤↔✐✱ ❣✐ỵ✐ ❤↕♥ ð −∞ ❜➡♥❣ 0✱ ❣✐ỵ✐ ❤↕♥ ð +∞ ❜➡♥❣ 1✮ t❤❡♦ ❝ỉ♥❣ t❤ù❝ µ((a, b]) = F (b) − F (a) ✲ ❚r➯♥ (Rn, Bn) ❝â ✤ë ✤♦ ❞✉② ♥❤➜t λ s❛♦ ❝❤♦ λ✲✤ë ✤♦ ❝õ❛ ❤➻♥❤ ❤ë♣ ❜➡♥❣ t❤➸ t➼❝❤ ❝õ❛ ❤➻♥❤ ❤ë♣✳ ✣ë ✤♦ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❝õ❛ Rn ✳ ▼é✐ t➟♣ t❤✉ë❝ Bn ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❇♦r❡❧ ❀ tr♦♥❣ ❦❤✐ ✤â✱ ♠é✐ t➟♣ ❝õ❛ Bλn ✭σ ✲tr÷í♥❣ ❜ê s✉♥❣ ❝õ❛ B n ✤è✐ ợ ữủ t s t B n ⊂ Bλn ❍➔♠ f : Rn → R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ ♥â ✤♦ ✤÷đ❝ ố ợ Bn ữủ s ữủ õ ữủ ợ Bn tư❝ ❧➔ ❇♦r❡❧✳ ✶✳✶✳✷ ❈→❝ sè ✤➦❝ tr÷♥❣✳ ●✐↔ sû X : (Ω, F, P ) → (R, B(R)) ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â✱ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡ ❝õ❛ X t P tỗ t ữủ ❦ý ✈å♥❣ ❝õ❛ X ✈➔ ❦➼ ❤✐➺✉ ❧➔ EX ✳ EX = XdP tỗ t < (p > 0) t❤➻ t❛ ♥â✐ X ❦❤↔ t➼❝❤ ❜➟❝ p✳ ✣➦❝ ❜✐➺t✱ ♥➳✉ E|X| < ∞ t❤➻ X ✤÷đ❝ ❣å✐ t E|X|p ú ỵ ữủ ỗ ỹ ý ữủ ỗ ❞ü♥❣ t➼❝❤ ♣❤➙♥ ▲❡❜❡s❣✉❡✿ ◆➳✉ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ X = n i=1 IAi t❤➻ n EX = P (Ai ) i=1 ◆➳✉ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤ỉ♥❣ ➙♠ t❤➻ X ❧➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② t➠♥❣ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ {Xn, n 1} n2n Xn = k=1 k−1 I k−1 2n 2n X< 2kn + nI(X n) ❑❤✐ ✤â EX = lim EXn n→∞ ◆➳✉ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý t❤➻ X = X + − X −✱ ✈ỵ✐ X + = max{X; 0} 0, X − = max{−X; 0} ✼ ❑❤✐ ✤â✱ EX = EX + − EX − ✭♥➳✉ ❝â ♥❣❤➽❛✮✳ ❚➼♥❤ ❝❤➜t✳ ✶✮ ✷✮ ✸✮ ✹✮ ✺✮ ✻✮ ◆➳✉ X t❤➻ EX 0✳ ◆➳✉ X = C t❤➻ EX = C tỗ t EX t ợ C ∈ R✱ t❛ ❝â E(CX) = C EX ✳ ◆➳✉ tỗ t EX EY t E(X Y ) = EX ± EY ✳ ◆➳✉ X ✈➔ EX = t❤➻ X = 0✳ ♥➳✉ X rí✐ r↕❝ ✈ỵ✐ P (X = xi) = pi, ♥➳✉ X ❧✐➯♥ tö❝ ❝â ❤➔♠ ♠➟t ✤ë p ❚ê♥❣ q✉→t✿ ◆➳✉ f : R → R ❧➔ ❤➔♠ ✤♦ ✤÷đ❝ ✈➔ Y = f (X) t❤➻ ♥➳✉ X rí✐ r↕❝ ✈ỵ✐ P (X = xi) = pi, i f (xi )pi EY = +∞ −∞ f (x)p(x)dx ♥➳✉ X ❧✐➯♥ tö❝ ❝â ❤➔♠ ♠➟t ✤ë p i xi pi +∞ −∞ xp(x)dx EX = ỵ tử ỡ Xn X (Xn X) tỗ t↕✐ n ✤➸ EXn− < ∞ ✭t÷ì♥❣ ù♥❣✱ EXn+ < ∞✮ t❤➻ EXn ↑ EX ✭t÷ì♥❣ ù♥❣✱ EXn ↓ EX ✮✳ ✽✮ ✭❇ê ✤➲ ❋❛t♦✉✮ ◆➳✉ Xn Y ✈ỵ✐ ♠å✐ n ✈➔ EY > −∞ t❤➻ limEXn ElimXn ◆➳✉ Xn Y ✈ỵ✐ ♠å✐ n ✈➔ EY < +∞ ElimXn ◆➳✉ |Xn| Y ✈ỵ✐ ♠å✐ n ElimXn ✈➔ EY limEXn t❤➻ limEXn t❛ ❝â P (X ε) EX ε ✶✶✮ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ p > 0✱ t❛ ❝â ∞ EX p = p xp−1 P (X > x)dx ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â✱ ❣✐→ trà ✤ë ❧➺❝❤ ❜➻♥❤ ♣❤÷ì♥❣ tr✉♥❣ ❜➻♥❤ DX := E(X EX)2 tỗ t ữủ ữỡ s❛✐ ❝õ❛ X ✳ P❤÷ì♥❣ s❛✐ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ỏ ữủ ỵ V ar(X) t ❚ø ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❦ý ✈å♥❣✱ t❛ s✉② r❛ r➡♥❣ ♣❤÷ì♥❣ s❛✐ ❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X õ t tỗ t ổ tỗ t tỗ t t õ t ữủ t t ổ tự (xi − EX)2 pi ♥➳✉ X rí✐ r↕❝ ✈➔ P (X = xi) = pi, DX = +∞ −∞ (x − EX) p(x)dx ♥➳✉ X ❧✐➯♥ tö❝ ❝â ❤➔♠ ♠➟t ✤ë p P❤÷ì♥❣ s❛✐ ❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ s❛✉ ✤➙②✿ ✶✮ DX = EX − (EX)2✳ ✷✮ DX 0✳ ✸✮ DX = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X = EX = C ❤✳❝✳❝✳ ✹✮ D(CX) = C 2DX ✳ ✺✮ ✭❇➜t ✤➥♥❣ t❤ù❝ ❈❤❡❜②s❤❡✈✮ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý✳ ❑❤✐ ✤â✱ ♥➳✉ tỗ t DX t ợ > t ❝â P (|X − EX| ε) DX , ε2 ✷✽ ♠➟t ✤ë ✤➣ ❜✐➳t ❝õ❛ Sn+1 ✈➔ t➼❝❤ ♣❤➙♥ t+δ P{t < Sn+1 t + δ} = fSn (τ )dτ = fSn (t)(δ + o(δ)) t ❚❤➔♥❤ ♣❤➛♥ o(δ) ❧➔ ✈ỉ ❝ị♥❣ ❜➨ ❜➟❝ ❝❛♦ ❤ì♥ δ✳ ◆❤÷ t❤➳ P{t < Sn+1 t + δ} = fS (t)(δ + o(δ)) ✤ì♥ ❣✐↔♥ ❧➔ ♠ët ❞➣② ❝→❝ ❜✐➳♥ ❝è Sn ❝â ♠➟t ✤ë ①→❝ s✉➜t ❧✐➯♥ tö❝ tr♦♥❣ ❦❤♦↔♥❣ [t, t + δ]✳ ❈→❝❤ ❦❤→❝ ✤➸ t➼♥❤ P{t < Sn+1 t + } trữợ t q st r st ✤➸ ❝â ♥❤✐➲✉ ❤ì♥ ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ (t, t + δ) ❧➔ o(δ)✳ ❑❤✐ ❧í ✤✐ t➼♥❤ ❝❤➜t ♥➔②✱ ❜✐➳♥ ❝è {t < Sn+1 t + δ} ①✉➜t ❤✐➺♥ ♥➳✉ ❝â ✤ó♥❣ n ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ (0, t] ✈➔ ❝❤➾ ♠ët ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ (t, t + δ]✳ ❱➻ t➼♥❤ ❝❤➜t sè ❣✐❛ ✤ë❝ ❧➟♣✱ ✤➙② ❧➔ ♠ët ❜✐➳♥ ❝è ❝â ①→❝ s✉➜t pN (t) (n)(λδ + o(δ))✳ ◆❤÷ ✈➟②✱ n pN (t) (n)(λδ + o(δ)) + o(δ) = fSn+1 (t)(δ + o(δ)) ❈❤✐❛ ❤❛✐ ✈➳ ❝❤♦ δ ✈➔ ❧➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ δ → 0✱ t❛ ♥❤➟♥ ✤÷đ❝ λpN (t) (n) = fSn+1 (t) ❙û ❞ö♥❣ ♠➟t ✤ë ❝õ❛ fS tr♦♥❣ ✭✷✳✶✸✮ t❛ ❝â ✭✷✳✶✻✮✳ ❈❤ù♥❣ ♠✐♥❤ ✷✳ ❈→❝❤ t✐➳♣ ❝➟♥ ð ✤➙② ❧➔ sû ❞ư♥❣ ❤➺ t❤ù❝ ❝ì ❜↔♥ {N (t) n} = {Sn t}✳ ▲➜② ①→❝ s✉➜t ❝õ❛ ❝→❝ ❜✐➳♥ ❝è ♥➔②✱ t❛ ✤÷đ❝ n ∞ t pN (t) (i) = fSn (τ )dτ i=n ✈ỵ✐ ♠å✐ n 1, t > ❇✐➸✉ t❤ù❝ ð ✈➳ ♣❤↔✐ ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ Sn ✈➔ ❜✐➸✉ t❤ù❝ ✈➳ tr→✐ ❧➔ ❤➔♠ ♣❤➙♥ ♣❤è✐ t➼❝❤ ❧ô② ❝õ❛ N (t)✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ t➼❝❤ ❧ô② ✈➔ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ N (t) ①→❝ ✤à♥❤ t ỵ tữỡ ữỡ ợ ❝❤➾ r❛ r➡♥❣ ∞ t (λt)i −λt e = fS (τ )dτ ✭✷✳✶✼✮ i! n i=n ✷✾ ▲➜② ✤↕♦ ❤➔♠ t❤❡♦ t ❝↔ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✶✼✮✱ t❛ t❤➜② r➡♥❣ ❤➛✉ ❤➳t ❝→❝ sè ❤↕♥❣ ð ✈➳ tr→✐ ❜➡♥❣ 0✱ ❞♦ ✤â λn tn−1 e−λt = fSn (t) (n − 1)! ❚❤➳ t❤➻ ✤↕♦ ❤➔♠ t❤❡♦ t ❝õ❛ ♠ët ✈➳ ✭✷✳✶✼✮ ❜➡♥❣ ✤↕♦ ❤➔♠ t❤❡♦ t ❝õ❛ ✈➳ ❦✐❛ ✈ỵ✐ ♠å✐ n ✈➔ t > 0✳ ❈❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ ❦❤✐ t → 0✱ t❛ s➩ s✉② r❛ t ữủ tọ ỡ ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷✳✺ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ❦❤→❝ ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥✳ ✣à♥❤ ♥❣❤➽❛ ✷ ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥✳ ▼ët q✉→ tr➻♥❤ ✤➳♠ P♦✐ss♦♥ ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♠ t❤ä❛ ♠➣♥ ✭✷✳✶✻✮✱ ❝â ❣✐❛ sè ✤ë❝ ❧➟♣ ✈➔ ❣✐❛ sè ❞ø♥❣✳ ❚❛ ✤➣ t❤➜② q✉→ tr➻♥❤ t❤ä❛ ♠➣♥ ✣à♥❤ ♥❣❤➽❛ ✶ ❝â ❝→❝ t➼♥❤ ❝❤➜t ♥â✐ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✷ ✭sû ❞ư♥❣ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ ❧✐➯♥ t✐➳♣ ✤ë❝ ❧➟♣✱ ❝â ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ♠ơ✮✳ ❱➻ ✈➟② tø ✣à♥❤ ♥❣❤➽❛ ✶ s✉② r❛ ✣à♥❤ ♥❣❤➽❛ ✷✳ ❇➔✐ t➟♣ ✷✳✹ ❝❤➾ r❛ r➡♥❣ tø ✣à♥❤ ♥❣❤➽❛ ✷ s✉② r❛ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ ❧✐➯♥ t✐➳♣ ❧➔ ✤ë❝ ❧➟♣✱ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐ ♠ơ✱ ❞♦ ✤â ❤❛✐ ✤à♥❤ ♥❣❤➽❛ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ❈â t❤➸ t❤➟t ❝â ♣❤➛♥ ✤→♥❣ ♥❣↕❝ ♥❤✐➯♥ ✤➸ ✤➛✉ t✐➯♥ ♥❤➟♥ t❤➜② r➡♥❣ ♠ët q✉→ tr➻♥❤ ✤➳♠ ♠➔ ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ t↕✐ ♠é✐ t ❦❤æ♥❣ ♥❤➜t t❤✐➳t ❧➔ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥✱ ✈➔ ♥❤÷ t❤➳ t➼♥❤ ❝❤➜t ❝â ❣✐❛ sè ❞ø♥❣ ✈➔ ✤ë❝ ❧➟♣ ✤â ❝ô♥❣ ❝➛♥ t❤✐➳t✳ ▼ët ❝→❝❤ ✤➸ t❤➜② ✤✐➲✉ ♥➔② ❧➔ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ợ ộ t tr ởt q tr tữỡ ữỡ ợ t ố ợ tớ ❣✐❛♥ ✤➳♥ ✳✳✳✳ ❈ö t❤➸✱ ♠➟t ✤ë ①→❝ s✉➜t ✤è✐ ✈ỵ✐ S1, S2, ❧➔ ✳✳✳✳ ❍➻♥❤ ✷✳✹ ♠✐♥❤ ❤å❛ ữợ t ỗ tớ S1, S2 ✤÷đ❝ ❝❤♦ ❜ð✐ fS S (s1 s2 ) = es ợ s1 s2 ữớ ð ♠ët ❝❤é t➠♥❣ ❞➛♥ ♣r✳ ❱➔ ❜➡♥❣ t↕✐ ♥❤ú♥❣ ♥ì✐ ❦❤→❝✳ ❈→❝ ♠✐♥❤ ❤å❛ ❤➻♥❤ ❤å❝ s➩ ♥❤÷ t t ỗ tớ õ t ✤÷đ❝ t❤❛② ✤ê✐ ♠➔ ❦❤ỉ♥❣ ❝➛♥ t❤❛② ✤ê✐ ❝→❝ {N (t), t > 0} 2 ✸✵ ❝➟♥✳ ❈â q tữỡ tỹ ố ợ q tr r tr ✤â q✉→ tr➻♥❤ ✤➳♠ ❧➔ rí✐ r↕❝✱ tr♦♥❣ ✤â sè ❦❤→❝❤ ✤➳♥ tø ✤➳♥ t✱ ✈ỵ✐ ♠é✐ t ♥❣✉②➯♥ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ♥❤à t❤ù❝✱ ♥❤÷♥❣ q✉→ tr➻♥❤ ❧↕✐ ❦❤æ♥❣ ❧➔ ❇❡r♥♦✉❧❧✐✳ ✣à♥❤ ♥❣❤➽❛ t✐➳♣ t❤❡♦ ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥ ❞ü❛ tr➯♥ t➼♥❤ ❝❤➜t ❣✐❛ sè ❝õ❛ ♥â✳ ❳➨t sè ❦❤→❝❤ ✤➳♥ tr♦♥❣ ♠ët ❦❤♦↔♥❣ r➜t ♥❤ä ♥➔♦ ✤â (t, t + δ]✳ ❱➻ N (t, t + δ) ❝â ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ♥❤÷ N (δ)✱ sû ❞ư♥❣ ✭✷✳✶✻✮ ✤➸ ♥❤➟♥ ✤÷đ❝ P {N (t, t + δ) = 0} = e−λδ ≈ − λδ + o(δ) P {N (t, t + δ) = 1} = λδe−λδ ≈ λδ + o(δ) P {N (t, t + δ) 2} = o(δ) ✭✷✳✶✽✮ ✣à♥❤ ♥❣❤➽❛ ✸ ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥✳ ▼ët q✉→ tr➻♥❤ ✤➳♠ P♦✐ss♦♥ ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♠ t❤ä❛ ♠➣♥ ✭✷✳✶✽✮ ✈➔ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❝â ❣✐❛ sè ❞ø♥❣ ✈➔ ❝â ❣✐❛ sè ✤ë❝ ❧➟♣✳ ❚❛ t❤➜② ✣à♥❤ ♥❣❤➽❛ ✸ ✤÷đ❝ s✉② tø ✣à♥❤ ♥❣❤➽❛ ✶✳ ❇↔♥ ❝❤➜t ❝õ❛ ♣❤→t ❜✐➸✉ t❤❡♦ ❝→❝❤ ❦❤→❝ ❧➔ ð ❝❤é ✈ỵ✐ ♠ët ❦❤♦↔♥❣ ❣✐ú❛ ❤❛✐ ❧➛♥ ✤➳♥ ❧✐➯♥ t✐➳♣ X, FX (x + δ) − FX (x) ❧➔ ①→❝ s✉➜t ❝õ❛ ♠ët ❦❤→❝❤ ✤➳♥ tr♦♥❣ ♠ët ❦❤♦↔♥❣ ❝â ✤ë ❞➔✐ δ✳ ❚❤❡♦ ✭✷✳✶✽✮ ❧➔ λδ + o(δ)✳ ❈❤✉②➸♥ ❧÷đ♥❣ ♥➔② ✈➔♦ tr♦♥❣ ♠ët ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✱ t❛ ♥❤➟♥ ✤÷đ❝ ❝→❝ ❦❤♦↔♥❣ ✤➳♥ t✐➳♣ t❤❡♦ ✭t❤í✐ ❦ý ✤➳♥✮ ❞↕♥❣ ♠ô ♠♦♥❣ ♠✉è♥✳ ✣à♥❤ ♥❣❤➽❛ ✸ ❝â t➼♥❤ trỹ õ ỹ ỵ tữ ❦❤→❝❤ ✤➳♥ ❧➔ ✤ë❝ ❧➟♣ tr♦♥❣ ❦❤✐ ❝→❝ ❦❤♦↔♥❣ t→❝❤ rớ tũ ỵ õ sỹ t ủ ữớ t ❝➛♥ ♣❤↔✐ ❧➔♠ ♠ët sè ✤→♥❣ ❦➸ ❝→❝ ❝æ♥❣ ✈✐➺❝ ✤➸ ❜↔♦ ✤↔♠ ❝❤➢❝ ❝❤➢♥ r➡♥❣ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ➜② ❧➔ t❤ü❝ t➳ ✈➔ ❦❤↔ ♥➠♥❣ ❝❤➢❝ ❝❤➢♥✱ ✈➔ ❝â ❧➩ ❝→❝❤ ❞➵ ❞➔♥❣ ♥❤➜t ❝â t❤➸ ❧➔ ❜➢t ✤➛✉ tø ✣à♥❤ ♥❣❤➽❛ ✶ ✤➸ s✉② r❛ ❝→❝ t➼♥❤ ❝❤➜t ➜②✳ ❱✐➺❝ ❝❤➾ r❛ r➡♥❣ ❝â ♠ët q✉→ tr➻♥❤ ❞✉② ♥❤➜t t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✣à♥❤ ♥❣❤➽❛ ✸ ❧➔ r➜t ❦❤â✳ ◆❤÷♥❣ ♥â ❝❤÷❛ t❤➟t ❝➛♥ t❤✐➳t ✈➔♦ ❧ó❝ ♥➔②✳ ❱➻ t➜t ❝↔ ♥❤ú♥❣ ✤✐➲✉ ❝➛♥ t❤✐➳t ❧ó❝ ♥➔② ❧➔ sû ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t ➜②✳ ▼ö❝ ✷✳✷✳✺ s➩ ♠✐♥❤ ❤å❛ ❝→❝❤ ❞ị♥❣ ✤à♥❤ ♥❣❤➽❛ ♥➔② ♥❤÷ t❤➳ ♥➔♦ ✤➸ tèt ❤ì♥✳ ❈→✐ ♠➔ ✭✷✳✶✽✮ ✤➲ ❝➟♣ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✸✱ ♥❣♦➔✐ sü ❣✐↔ t❤✐➳t ✈➲ ❣✐❛ sè ✤ë❝ ❧➟♣ ✈➔ ❣✐❛ sè ❞ø♥❣✱ ❧➔ sü ❤↕♥ ❝❤➳ sè ✤æ♥❣ ❦❤→❝❤ ✤➳♥✳ ❈❤➥♥❣ ❤↕♥✱ ①➨t ♠ët q✉→ tr➻♥❤ ✤➳♠✱ tr♦♥❣ ✤â ❦❤→❝❤ ✤➳♥ ❧✉æ♥ t❤❡♦ ❝➦♣ ✈➔ ❦❤♦↔♥❣ ❣✐ú❛ ❝→❝ ❝➦♣ ✤➳♥ ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝â ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ♠ơ ✈ỵ✐ t❤❛♠ sè λ ✭①❡♠ ❤➻♥❤ ✷✳✺✮✳ ❱ỵ✐ q✉→ tr➻♥❤ ♥➔②✱ P(N (t, t + δ) = 1) = ✈➔ P(N (t, t + δ) = 2) = λδ + o(δ)✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✽✮✳ ◗✉→ tr➻♥❤ ♥➔② ❝â ❣✐❛ sè ❞ø♥❣ ✈➔ ❝â ❣✐❛ sè ✤ë❝ ❧➟♣✱ t✉② ♥❤✐➯♥ ✈➻ q✉→ tr➻♥❤ ✤÷đ❝ t↕♦ ❜ð✐ ✈✐➺❝ ①❡♠ tø♥❣ ❝➦♣ ❦❤→❝❤ ✤➳♥ ♥❤÷ ♠ët ♣❤➛♥ tû ✭✤ì♥ t❤➸✮ ♥➯♥ ❧➔ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥✳ ❚â♠ ❧↕✐✱ t❛ ❤✐➸✉ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ q✉→ tr➻♥❤ t❤ä❛ ♠➣♥ ♠ët tr♦♥❣ ❜❛ ✤✐➲✉ ❦✐➺♥✿ ✐✮ ▼ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ ♠ët q✉→ tr➻♥❤ ✤ê✐ ♠ỵ✐ tr♦♥❣ ✤â ❝→❝ ❦❤♦↔♥❣ ✤➳♥ ❦➳ t✐➳♣ ❝â ♠ët ❤➔♠ ♣❤➙♥ ♣❤è✐ ♠ơ✱ ♥❣❤➽❛ ❧➔ ✈ỵ✐ ♠ët sè t❤ü❝ λ > 0✱ ✸✷ ♠é✐ Xi ❝â ♠➟t ✤ë fX (x) = ex ợ x số ữủ ❧➔ tè❝ ✤ë ❝õ❛ q✉→ tr➻♥❤✳ ✐✐✮ ▼ët q✉→ tr➻♥❤ ✤➳♠ P♦✐ss♦♥ {N (t), t > 0} ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♠ t❤ä❛ ♠➣♥ ✭✷✳✶✻✮✱ ❝â ❣✐❛ sè ✤ë❝ ❧➟♣ ✈➔ ❣✐❛ sè ❞ø♥❣✳ ✐✐✐✮ ▼ët q✉→ tr➻♥❤ ✤➳♠ P♦✐ss♦♥ ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♠ t❤ä❛ ♠➣♥ ✭✷✳✶✽✮ ✈➔ t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ❝â ❣✐❛ sè ❞ø♥❣ ✈➔ ❝â ❣✐❛ sè ✤ë❝ ❧➟♣✳ ✷✳✷✳✻ ◗✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ ♠ët ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❝♦ ❇❡r♥♦✉❧❧✐✳ ✣à♥❤ ♥❣❤➽❛ ✸ ❝â t❤➸ t❤➜② ➼t trø✉ t÷đ♥❣ ❤ì♥ ❜➡♥❣ ✈✐➺❝ ❜➢t ✤➛✉ ✈ỵ✐ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐✱ ❧➔ q✉→ tr➻♥❤ ❝â ❝→❝ t➼♥❤ ❝❤➜t tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✸ ✈ỵ✐ t❤í✐ ❣✐❛♥ rí✐ r↕❝✳ ❚❛ s➩ ✤✐ ✤➳♥ ♠ët ❣✐ỵ✐ ❤↕♥ t❤➼❝❤ ❤đ♣ ❝õ❛ ♠ët ❞➣② ❝→❝ q✉→ tr➻♥❤ ➜②✱ ✈➔ t❤➜② r➡♥❣ ❞➣② ❝→❝ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ♥➔② ❤ë✐ tö ✤➳♥ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ♥➔♦ ✤â✳ ◆❤ỵ ❧↕✐ r➡♥❣ ♠ët q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ❧➔ ♠ët ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝ò♥❣ ♣❤➙♥ ♣❤è✐ Y1, Y2, ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥❤à ♣❤➙♥ ✈ỵ✐ PY (1) = p ✈➔ PY (0) = − p✳ {Yi = 1} ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ❦❤→❝❤ ✤➳♥ t↕✐ t❤í✐ ✤✐➸♠ i ✈➔ {Yi = 0} ❝â ♥❣❤➽❛ ❧➔ ❦❤→❝❤ ❦❤ỉ♥❣ ✤➳♥✱ ♥❤÷♥❣ t❛ ❝ơ♥❣ ❝â t❤➸ ❝♦ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❝õ❛ q✉→ tr➻♥❤ s❛♦ ❝❤♦ ✈ỵ✐ sè ♥❣✉②➯♥ j > ♥➔♦ ✤â✱ Yi ❧➔ ♠ët ❦❤→❝❤ ✤➳♥ ❤♦➦❝ ❦❤ỉ♥❣ ✤➳♥ t↕✐ t❤í✐ ✤✐➸♠ i2−j ✳ ❚❛ ①➨t ♠ët ❞➣② ✤÷đ❝ ❝❤➾ sè ❤â❛ ❜ð✐ j ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❝♦ ❇❡r♥♦✉❧❧✐ ♥❤÷ t❤➳ ✈➔ ②➯✉ ❝➛✉ ❣✐ú tè❝ ✤ë ✤➳♥ ❧➔ ❤➡♥❣ sè✳ ✣➦t p = λ2−j ✤è✐ ✈ỵ✐ q✉→ tr➻♥❤ t❤ù j ữ ợ ộ ỡ t t j ✱ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ❝♦ ❜ð✐ ✈✐➺❝ ✤➦t ♠é✐ ♠ët ♠è❝ ❜ð✐ ❤❛✐ ♠è❝ ♠➔ ♠é✐ ♠è❝ ✈ỵ✐ ①→❝ st ỷ s ợ trữợ ý sè ❦❤→❝❤ ✤➳♥ tr➯♥ ♠ët ✤ì♥ ✈à t❤í✐ ❣✐❛♥ ❧➔ λ✱ ♣❤ị ❤đ♣ ✈ỵ✐ q✉→ tr➻♥❤ P♦✐ss♦♥ ✤❛♥❣ ①➜♣ ①➾✳ ◆➳✉ t❛ t❤➜② q✉→ tr➻♥❤ t❤ù j ♥➔② ❧✐➯♥ q✉❛♥ ✤➳♥ ✣à♥❤ ♥❣❤➽❛ ✸ ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥✱ t❛ s➩ t❤➜② r➡♥❣ ✈ỵ✐ sü t➠♥❣ ✤➲✉ ✤➦♥ ❝ï δ = 2−j ✱ ①→❝ s✉➜t ✤➸ ♠ët ❦❤→❝❤ ✤➳♥ tr♦♥❣ ♠ët ✤ì♥ ✈à ❧➔ λδ ✈➔ ①→❝ s✉➜t ❦❤ỉ♥❣ ❝â ❦❤→❝❤ ✤➳♥ ❧➔ − λδ✱ ✈➔ ♥❤÷ t❤➳ ✭✷✳✶✽✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➔ ❦❤✐ ✤â o(δ) = 0✳ ❱ỵ✐ ✸✸ ù t tũ ỵ ró r r số ❧✐➯♥ q✉❛♥ ✤ë❝ ❧➟♣ ✈ỵ✐ ❝→❝ ❦❤→❝❤ ✤➳♥✳ ❈→❝ ❣✐❛ sè ❝❤÷❛ ❝❤➢❝ ✤➣ ❞ø♥❣ ✈➻ ♠ët sè ❣✐❛ ❦➼❝❤ tữợ 2j1 õ t ự ởt tớ ý ởt ❜ë✐ ❝õ❛ 2−j ❤♦➦❝ ❦❤ỉ♥❣✱ ♣❤ư t❤✉ë❝ ✈➔♦ ✈à tr➼ ❝õ❛ ♥â✳ ❚✉② ♥❤✐➯♥ ✈ỵ✐ ♠ët sè ❣✐❛ ❝è ✤à♥❤ ❝ï δ✱ sè ❝→❝ ❜ë✐ ❝õ❛ 2−j ✭♥❣❤➽❛ ❧➔ sè ❝→❝ ✤✐➸♠ ✤➳♥ ❞÷ì♥❣✮ ❤♦➦❝ ❧➔ [δ2j ] ✭♣❤➛♥ ♥❣✉②➯♥✮ ❤♦➦❝ ❧➔ + [δ2j ]✳ ◆❤÷ ✈➟②✱ ❦❤✐ j → ∞✱ ❝→❝ ❣✐❛ sè ✈ø❛ ❧➔ ✤ë❝ ❧➟♣ ✈ø❛ ❞ø♥❣✳ ❱ỵ✐ ♠é✐ j ✱ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ t❤ù j ❝â ♠ët q✉→ tr➻♥❤ ✤➳♠ ❇❡r♥♦✉❧❧✐ ] ❧✐➯♥ ❦➳t Nj (t) = [t2 i=1 Yi ✳ ✣➙② ❧➔ sè ❝→❝ ❦❤→❝❤ ✤➳♥ ❝❤♦ tỵ✐ t❤í✐ ✤✐➸♠ t ✈➔ ♥â ❧➔ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ ✈ỵ✐ ♣❤➙♥ ♣❤è✐ ♥❤à t❤ù❝✳ ✣â ❧➔ [t2 ] pN (t) (n) = n pn (1 − p)[t2 ]−n ✱ tr♦♥❣ ✤â p = λ2−j ✳ ❇➙② ❣✐í t❛ s➩ ❝❤➾ r❛ r➡♥❣ ♣❤➙♥ ♣❤è✐ ♥➔② ①➜♣ ①➾ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ❦❤✐ j t➠♥❣✳ j j j j ỵ t q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ❝♦ ❇❡r♥♦✉❧❧✐ ✈ỵ✐ ①→❝ s✉➜t ✤➳♥ λ2−j ✈➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❝ï 2−j ✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ t❤í✐ ✤✐➸♠ t > ❝è ✤à♥❤ ✈➔ ♠ët sè ❦❤→❝❤ ✤➳♥ n ❝è ✤à♥❤✱ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t pN (t) (n) ①➜♣ ①➾ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✭λ ♥➔♦ ✤â✮ ✈ỵ✐ ❣✐❛ sè j ✱ ♥❣❤➽❛ ❧➔ j ✭✷✳✶✾✮ lim pNj (t) (n) = pN (t) (n) j→∞ ự rữợ t t t ố t❤ù❝✱ ✈ỵ✐ [t2j ] ❜✐➳♥ ✈ỵ✐ p = λ2−j ❧➔ lim pNj (t) (n) = lim j→∞ j→∞ λ2−j − λ2−j [t2j ] n n −j j e[t2 ] ln(1−λ2 ) n = lim j→∞ = lim = λ2−j [t2j ] e−λt −j n − λ2 [t2j ].[t2j − 1] [t2j − n + 1] n! j→∞ (λt)n e−λt n! ✭✷✳✷✵✮ λ2−j − λ2−j n e−λt ✭✷✳✷✷✮ ✭✈➻ ln(1 − λ2−j ) = −λ2−j + o(2−j ) ✈➔ limj→∞[t2j − i] ♠å✐ i✮✳ ✭✷✳✷✶✮ λ2−j 1−λ2−j = λt ✈ỵ✐ ✸✹ ❱➻ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ♥❤à t❤ù❝ ❝â ❣✐ỵ✐ ❤↕♥ ❧➔ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ ♠é✐ n✱ ♥➯♥ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ Nj (t) ❝ò♥❣ ❤ë✐ tư ✤➳♥ ❤➔♠ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ ♠é✐ t✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ✈ỵ✐ ♠é✐ t > ❜✐➳♥ ✤➳♠ ♥❣➝✉ ♥❤✐➯♥ Nj (t) ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ❤ë✐ tö✳ ❚❛ ❝â ❤➺ q✉↔ s❛✉✿ ❍➺ q✉↔ ✷✳✷✳✻✳✷✳ ❱ỵ✐ sè ♥❣✉②➯♥ k ❜➜t ❦ý✱ < t1 < t2 < < tk t tũ ỵ tớ õ ố ỗ tớ Nj (t1), Nj (t2), , Nj (tk ) ①➜♣ ①➾ ❤➔♠ ố ỗ tớ N (t1 ), N (t2 ), , N (tk ) ❦❤✐ j → ∞✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝➛♥ ❝❤➾ r❛ ❝→❝ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ỗ tớ tử õ t t ố st ỗ tớ ộ q tr➻♥❤ ❇❡r♥♦✉❧❧✐ ❧➔ > pNj (t1 ), ,Nj (tk ) (n1 , , nk ) = pN (t j ),Nj (t1 ,t2 ), ,Nj (tk−1 ,tk ) (n1 , n2 − n1 , , nk − nk−1 ) k = pNj (t1 ) (n1 ) pN (t ,t j l l−1 ) ✭✷✳✷✸✮ (nl , nl−1 ) l=2 ð ✤➙② t❛ ✤➣ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❝â ❣✐❛ sè ✤ë❝ ❧➟♣ ❝õ❛ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐✳ ❱ỵ✐ q✉→ tr➻♥❤ P♦✐ss♦♥ t❛ ❝â t÷ì♥❣ tü k pN (t1 ), ,N (tk ) (n1 , , nk ) = pN (t1 ) (n1 ) pN (t ,t l l−1 ) (nl , nl−1 ) ✭✷✳✷✹✮ l=2 ❈❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ ❦❤✐ j sỷ ỵ ✤➸ t❤➜② ♠é✐ sè ❤↕♥❣ ❝õ❛ ✭✷✳✷✸✮ ❞➛♥ ✤➳♥ ♠ët sè ❤↕♥❣ ❝õ❛ ✭✷✳✷✹✮✳ ❱ỵ✐ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ N ✱ ✤á✐ ❤ä✐ ♥➔② ✤÷đ❝ s✉② r❛ ❞➵ ❞➔♥❣ t❤❡♦ ỵ tớ tũ ỵ ứ t t r ❝→❝ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ð tr➯♥ ❤ë✐ tö ✈➲ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ t❤❡♦ ❦➳t ❧✉➟♥ ❝õ❛ ❍➺ q✉↔✳ ◆❤ỵ ❧↕✐ r➡♥❣ ♠ö❝ ✶✳✺✳✺ ❝â ♠ët sè ❝→❝❤ ♠➔ ♠ët ❞➣② ❝→❝ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ❤ë✐ tö ✈➔ ❤➺ q✉↔ ✤➣ t❤✐➳t ❧➟♣ ♠ët ❝→❝❤ ✤ì♥ ❣✐↔♥ tr♦♥❣ sè ✤â✳ ❚❛ ❦❤ỉ♥❣ ❝â ♥❤ú♥❣ ❝ỉ♥❣ ❝ư ✸✺ t♦→♥ ❤å❝ ❤❛② ♥❤✉ ❝➛✉ ❧➔♠ s➙✉ s➢❝ ❤ì♥ ✈➜♥ ✤➲ ❤ë✐ tö ♥➔②✳ ❈↔ q✉→ tr➻♥❤ P♦✐ss♦♥ ✈➔ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ❧➔ q✉→ ❞➵ ✤➸ ♣❤➙♥ t➼❝❤ ✤➳♥ ♥é✐ sü ❤ë✐ tö ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❝♦ ❇❡r♥♦✉❧❧✐ ✤➳♥ q✉→ tr➻♥❤ P♦✐ss♦♥ ❤✐➳♠ ❦❤✐ ❧➔ ❝→❝❤ ❞➵ ♥❤➜t ✤➸ t❤✐➳t ❧➟♣ ❝→❝ t➼♥❤ ❝❤➜t ❦❤→❝ ♥ú❛✳ ▼➦t ❦❤→❝✱ sü ❤ë✐ tö ♥➔② ❧➔ ♠ët sü trđ ❣✐ó♣ trü❝ ❣✐→❝ ✤➸ ❤✐➸✉ ♠é✐ q✉→ tr➻♥❤✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ✈➔ q✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ r➜t ❤ú✉ tr ủ ỵ q st ỳ ợ ữ tổ tữớ t ổ tèt ♥❤➜t ✤➸ ♣❤➙♥ t➼❝❤ ♥❤ú♥❣ ✈➜♥ ✤➲ ✤â✳ ✷✳✸ ❈→❝ q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✷✳✸✳✶ ✣à♥❤ ♥❣❤➽❛✳ ◗✉→ tr➻♥❤ P♦✐ss♦♥ ✤÷đ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ♠ët ❤➡♥❣ sè ❝÷í♥❣ ✤ë ✤➳♥ λ ✱ ❝â ➼❝❤ ✤➸ ①➨t ♠ët ❞↕♥❣ q✉→ tr➻♥❤ tê♥❣ q✉→t ❤ì♥ tr♦♥❣ ✤â ❝÷í♥❣ ✤ë ✤➳♥ ❜✐➳♥ t❤✐➯♥ ♥❤÷ ❧➔ ❤➔♠ ❝õ❛ t❤í✐ ❣✐❛♥✳ ▼ët q tr Pss ổ t t ợ ữớ ❜✐➳♥ t❤✐➯♥ t❤❡♦ t❤í✐ ❣✐❛♥ λ(t) ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♠ {N (t); t > 0} ❝â ❣✐❛ sè ✤ë❝ ❧➟♣ ✈➔ ✈ỵ✐ ♠å✐ t 0, δ > ✱ t❤ä❛ ♠➣♥ P (N (t, t + δ) = 0) = − δλ(t) + o(δ) P (N (t, t + δ) = 1) = δλ(t) + o(δ) P (N (t, t + δ) 2) = o(δ) ✭✷✳✷✺✮ tr♦♥❣ ✤â N (t, t + δ) = N (t + δ) N (t) tr Pss ổ ỗ t ổ ❝â ❣✐❛ sè ❞ø♥❣✳ ▼ët ù♥❣ ❞ö♥❣ ❝❤✉♥❣ ①✉➜t ❤✐➺♥ tr♦♥❣ tr✉②➲♥ t❤æ♥❣ q✉❛♥❣ ❤å❝✱ tr♦♥❣ ✤â ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ♠ỉ ❤➻♥❤ ❞á♥❣ ❝❤✉②➸♥ ✤ë♥❣ ❝→❝ ♣❤♦t♦♥ tø ♠ët ❜ë ✤✐➲✉ ❜✐➳♥ q✉❛♥❣❀ sü ✤✐➲✉ ❜✐➳♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❜➡♥❣ ✈✐➺❝ t❤❛② ✤ê✐ ❝÷í♥❣ ✤ë ♣❤♦t♦♥✳ ❚❛ s➩ t❤➜② ù♥❣ ❞ư♥❣ ❦❤→❝ ❦❤ỉ♥❣ ❧➙✉ ♥ú❛ tr♦♥❣ ✈➼ ❞ư t✐➳♣ t❤❡♦✳ ✣ỉ✐ ởt q tr Pss ữ trữợ ✤➙② ✤÷đ❝ ❣å✐ ❧➔ ♠ët q✉→ tr➻♥❤ ✸✻ t❤✉➛♥ ♥❤➜t✳ ❚❛ ❝â t❤➸ sû ❞ö♥❣ ❧↕✐ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ❝♦ ✤➸ ①➜♣ ①➾ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t✳ ❚❤➟t ✈➟②✱ ❣✐↔ t❤✐➳t λ(t) ❜à ❝❤➦♥ ❜ð✐ 0✳ ❈❤✐❛ trư❝ t❤í✐ ❣✐❛♥ t❤➔♥❤ ❝→❝ ❦❤♦↔♥❣ t➠♥❣ s❛♦ ❝❤♦ ❝→❝ ✤ë ❞➔✐ δ ❝õ❛ ❝❤ó♥❣ t➾ ❧➺ ♥❣❤à❝❤ ✈ỵ✐ λ(t)✱ ♥❤÷ ✈➟② ❜↔♦ ✤↔♠ ①→❝ s✉➜t ❝õ❛ ♠ët ❦❤♦↔♥❣ ✤➳♥ ❧➔ ♠ët ❣✐→ trà ❝è ✤à♥❤ p = δλ(t)✳ ◆❤÷ ✈➟② t↕♠ t❤í✐ ❧í ✤✐ sü ❜✐➳♥ t❤✐➯♥ ❝õ❛ λ(t) tr♦♥❣ ♠ët sü t➠♥❣ ❞➛♥✱ P N t, t + P N t, t + P N t, t + p λ(t) p λ(t) p λ(t) =0 = − p + o(p) =1 = p + o(p) ✭✷✳✷✻✮ = o(p) P❤➙♥ ❝❤✐ ♥➔② ✤÷đ❝ ①→❝ ✤à♥❤ ❝❤➼♥❤ ①→❝ ❤ì♥ ✈✐➺❝ ✤à♥❤ ♥❣❤➽❛ m(t) ♥❤÷ t λ(τ )dτ m(t) = ✭✷✳✷✼✮ ❚❤➳ t❤➻ ❣✐❛ sè t❤ù i ❦➳t tú t t s m(t) = ip ữ trữợ ✤➦t {Yi; i 1} ❧➔ ♠ët ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ♥❤à ♣❤➙♥ ✈ỵ✐ P (Yi = 1) = p ✈➔ P (Yi = 0) = − p ✳ ❳➨t q✉→ tr➻♥❤ ✤➳♠ {N (t); t > 0} tr♦♥❣ ✤â Yi✱ ✈ỵ✐ ♠é✐ i 1✱ ❧➔ sè ❝→❝ ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ (ti−1, ti]✱ ♠➔ ti t❤ä❛ ♠➣♥ m(t) = ip✳ ❚❤➳ t❤➻ N (ti ) = Y1 + Y2 + + Yi ✳ ◆➳✉ p ❣✐↔♠ ♥❤÷ 2−j ✱ ♠é✐ sè ❣✐❛ ✤÷đ❝ ❝❤✐❛ ♠ët ❝→❝❤ ❧✐➯♥ t✐➳♣ t❤➔♥❤ ❝➦♣ ❝→❝ sè ❣✐❛✳ ◆❤÷ t❤➳ ❣✐è♥❣ ♥❤÷ ✭✷✳✷✷✮✱ P (N (t) = n) = [1 + o(p)][m(t)]n e−m(t) n! ✭✷✳✷✽✮ ✸✼ ữỡ tỹ ợ t ý (t, ] ❧➜② m(t, τ ) = t = tk ✱ τ = ti ợ k, i õ t ữủ τ t λ(u)du✱ [1 + o(p)][m(t, τ )]n e−m(t,τ ) P (N (t, τ ) = n) = n! ✈➔ ❧➜② ✭✷✳✷✾✮ ▲➜② ❣✐ỵ✐ ❤↕♥ ❦❤✐ p → 0✱ q✉→ tr➻♥❤ ✤➳♠ {N (t); t > 0} ð tr➯♥ t✐➳♣ ❝➟♥ q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✈➔ t❛ ❝â ỵ s ỵ ố ợ ởt q tr Pss ổ t t ợ ữớ tö❝ ♣❤↔✐ λ(t) ❜à ❝❤➦♥ ❜ð✐ 0✱ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ N (t, τ )✱ sè ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ (t, τ ]✱ t❤ä❛ ♠➣♥ [m(t, τ )]n e−m(t,τ ) P (N (t, τ ) = n) = ; n! τ λ(u)du m(t, τ ) = t ✭✷✳✸✵✮ ❱➻ ✈➟② ♥❣÷í✐ t❛ ❝â t❤➸ ①❡♠ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ♥❤÷ ❧➔ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ t❤✉➛♥ ♥❤➜t tr➯♥ ♠ët t❤❛♥❣ t❤í✐ ❣✐❛♥ ♣❤✐ t✉②➳♥✳ ✣â ❧➔ ✤➦t {N ∗(s); s 0} ❧➔ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ t❤✉➛♥ t ợ ữớ tr Pss ổ t t ữủ N (t) = N (m(t)) ợ ♠é✐ t✳ ❱➼ ❞ư ✷✳✸✳✷✳ ✭❍➔♥❣ ✤đ✐ M/G/1✮ ▼ët ❤➔♥❣ ✤đ✐ M/G/∞ ❧➔ ♠ët ❤➔♥❣ ✤đ✐ ✈ỵ✐ ❝→❝ ❦❤→❝❤ ✤➳♥ P♦✐ss♦♥ M ✱ ♠ët ♣❤➙♥ ♣❤è✐ ❞à❝❤ ✈ö tê♥❣ q✉→t G ✈➔ ♠ët sè ❧÷đ♥❣ ✈ỉ ❤↕♥ ❝→❝ ♠→② ❝❤õ✳ ❱➻ ❤➔♥❣ ✤đ✐ M/G/∞ ❝â ✈ỉ ❤↕♥ ♠→② ❝❤õ✱ ❦❤ỉ♥❣ ❜❛♦ ❣✐í ❦❤→❝❤ ✤➳♥ ♣❤↔✐ ①➳♣ ❤➔♥❣✳ ▼é✐ ❦❤→❝❤ ✤➳♥ ✤÷đ❝ ❜➢t ✤➛✉ ♣❤ư❝ ✈ư ♥❣❛② ❜ð✐ ♠ët ♠→② ❝❤õ ♥➔♦ ✤â ✈➔ ❝→❝ t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ư Yi ❝õ❛ ❝→❝ ❦❤→❝❤ ❤➔♥❣ i ❧➔ ✤ë❝ ❧➟♣✱ ❝ò♥❣ ♣❤➙♥ ♣❤è✐ ✈ỵ✐ ❝ị♥❣ ♠ët ❤➔♠ ♣❤➙♥ ♣❤è✐ G(y) ✱ t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ư ❧➔ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ tø ❧ó❝ ❜➢t ✤➛✉ ✤➳♥ ❧ó❝ ❤♦➔♥ t❤➔♥❤ ✈➔ ❝ơ♥❣ ✤ë❝ ❧➟♣ ✈ỵ✐ t❤í✐ ✤✐➸♠ ✤➳♥✳ ❚❛ s➩ t➻♠ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ sè ❝→❝ ❦❤→❝❤ ❤➔♥❣ ✤÷đ❝ ♣❤ư❝ ✈ư t↕✐ t❤í✐ ✤✐➸♠ τ trữợ {N (t); t > 0} q tr Pss ợ ữớ ✳ ❳➨t ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ ❝õ❛ ♥❤ú♥❣ ❦❤→❝❤ ❤➔♥❣ ➜② s➩ ✤÷đ❝ ♣❤ư❝ ✈ư t↕✐ ♠ët t❤í✐ ✤✐➸♠ ❝è õ r ởt ọ tũ ỵ (t, t + τ ] ♥➔♦ ✤â✱ ①→❝ s✉➜t ♠ët ❦❤→❝❤ ✤➳♥ ❧➔ δλ + o(δ) ✈➔ ①→❝ s✉➜t ❝õ❛ ❤❛✐ ❤♦➦❝ ♥❤✐➲✉ ❤ì♥ ❦❤→❝❤ ✤➳♥ ❧➔ ❦❤ỉ♥❣ ✤→♥❣ ❦➸ ✭tù❝ ❧➔ o(δ) ✮✳ ❑❤↔ ♥➠♥❣ ♠ët ♥❣÷í✐ ❦❤→❝❤ ✤➳♥ tr♦♥❣ (t, t + τ ] ✈➔ ✈➝♥ ✤❛♥❣ ✤÷đ❝ ♣❤ư❝ ✈ư t↕✐ ♠ët t❤í✐ ✤✐➸♠ τ > t ❧➔ δλ[1 − G(τ + t)] + o(δ)]✳ ❍➣② ①❡♠ ①➨t ♠ët q✉→ tr➻♥❤ ✤➳♠ {N1 (t); t > 0}✱ tr♦♥❣ ✤â N1 (t) ❧➔ sè ❧÷đ♥❣ ❦❤→❝❤ ✤➳♥ ❣✐ú❛ ✈➔ t ♠➔ ✈➝♥ ❝á♥ ♣❤ö❝ ✈ö t↕✐ τ ✳ ✣➙② ❧➔ q✉→ tr➻♥❤ ❝â ❣✐❛ sè ✤ë❝ ❧➟♣✳ ✣➸ t ữ ỵ tr tờ t {N (t); t > 0} ❝â ❣✐❛ sè ✤ë❝ ❧➟♣✱ ❝ơ♥❣ ♥❤÷ ❝→❝ ❦❤→❝❤ ✤➳♥ tr♦♥❣ {N (t); t > 0} ❝â t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ư ✤ë❝ ❧➟♣✱ ✈➔ t❤➳ t❤➻ t➼♥❤ ✤ë❝ ❧➟♣ ❝â ❤♦➦❝ ❦❤æ♥❣ ð {N1(t); < t τ }✳ ❙✉② r❛ {N1(t); < t τ } ởt q tr Pss ổ t t ợ ữớ ✤ë λ[1 − G(τ − t)] t↕✐ t❤í✐ ❣✐❛♥ t τ ✳ ❑ý ✈å♥❣ ❝→❝ ❦❤→❝❤ ✈➝♥ ✤❛♥❣ ✤÷đ❝ ♣❤ư❝ ✈ư t↕✐ t❤í✐ ✤✐➸♠ τ ❧➔ τ τ [1 − G(τ − t)]dt = λ m(τ ) = λ t=0 [1 − G(t)]dt t=0 ✭✷✳✸✶✮ ❱➔ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ sè ❦❤→❝❤ ✤❛♥❣ ✤÷đ❝ ♣❤ư❝ ✈ư t↕✐ t❤í✐ ✤✐➸♠ τ ❧➔ m(τ )n e−m(τ ) ✭✷✳✸✷✮ P (N1 ( ) = n) = n! ữ ỵ r τ → ∞ t➼❝❤ ♣❤➙♥ tr♦♥❣ ✭✷✳✸✶✮ t✐➺♠ ❝➟♥ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♣❤➙♥ ♣❤è✐ t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ư ✭tù❝ ❧➔✱ ♥â ❧➔ t➼❝❤ ♣❤➙♥ ❝õ❛ ❤➔♠ ♣❤➙♥ ❜è ✤➛② ✤õ − G(t) ❝õ❛ t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ư✮✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ tr♦♥❣ tr↕♥❣ ✸✾ t❤→✐ ê♥ ✤à♥❤ ✭❦❤✐ τ → ∞✮✱ sü ♣❤➙♥ ❜è ❝õ❛ sè ❞à❝❤ ✈ö t↕✐ τ ♣❤ư t❤✉ë❝ ✈➔♦ ♣❤➙♥ ❜è t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ö ❝❤➾ q✉❛ tr✉♥❣ ❜➻♥❤ ❝õ❛ ♥â✳ ❱➼ ❞ö ♥➔② ❝â t❤➸ ✤÷đ❝ sû ❞ư♥❣ ❝❤♦ ❝→❝ ♠ỉ ❤➻♥❤ ❦❤✐ sè ❧÷đ♥❣ ❝→❝ ❝✉ë❝ ❣å✐ ✤✐➺♥ t❤♦↕✐ ✤❛♥❣ ❞✐➵♥ r❛ ð ♠ët t❤í✐ ✤✐➸♠ ✤➣ ❝❤♦✳ ✣✐➲✉ ♥➔② ✤á✐ ❤ä✐ ❝→❝ ❝✉ë❝ ❣å✐ ✤➳♥ ✤÷đ❝ ♠ỉ ❤➻♥❤ ❤â❛ ♥❤÷ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ✈➔ t❤í✐ ❣✐❛♥ ❝õ❛ ♠é✐ ❝✉ë❝ ❣å✐ ữủ ữ ởt ợ t❤í✐ ❣✐❛♥ ✈➔ t❤í✐ ✤✐➸♠ ❣å✐ ❝õ❛ ❝✉ë❝ ❣å✐ ❦❤→❝✳ ❈✉è✐ ❝ị♥❣ ♥❤÷ ✤➣ ✤÷đ❝ ❝❤➾ r❛ tr♦♥❣ ❍➻♥❤ ✷✳✽✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝♦✐ {N1(t); < t τ } ♥❤÷ ♠ët ♣❤➛♥ t→❝❤ r❛ tø q✉→ tr➻♥❤ ✤➳♥ {N (t); t > 0}✳ ❇ð✐ ✈➻ ❝ò♥❣ ♠ët ❧♦↕✐ t❤❛♠ sè tr♦♥❣ ▼ư❝ ✷✳✸✱ sè ❦❤→❝❤ ✤➣ ✤÷đ❝ ♣❤ư❝ ✈ư ①♦♥❣ t↕✐ t❤í✐ ✤✐➸♠ τ ❧➔ ✤ë❝ ❧➟♣ ✈ỵ✐ sè ❦❤→❝❤ ✤❛♥❣ ❝á♥ ✤÷đ❝ ♣❤ư❝ ✈ư✳ ✹✵ ❑➌❚ ▲❯❾◆ ❈❤ó♥❣ t❛ ✤➣ ❜➢t ✤➛✉ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ q tr Pss ợ tữỡ ữỡ t ữ ởt q tr ợ ợ t❤í✐ ❦ý ✤ê✐ ♠ỵ✐ ❧✐➯♥ t✐➳♣ ❝â ♣❤➙♥ ♣❤è✐ ♠ơ✳ ❚❤ù ❤❛✐✱ ♥❤÷ ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♠ ❝â ❣✐❛ sè ❞ø♥❣ ✈➔ ❣✐❛ sè ✤ë❝ ❧➟♣ ✈ỵ✐ ❦❤→❝❤ ✤➳♥ tr➯♥ ♠é✐ ❦❤♦↔♥❣ ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈➔ t❤ù ❜❛✱ õ ữ ởt ợ q tr ❇❡r♥♦✉❧❧✐ ❝♦✳ ❚❛ ✤➣ t❤➜② r➡♥❣ ♠é✐ ✤à♥❤ ♥❣❤➽❛ ❝✉♥❣ ❝➜♣ ♠ët ❝→❝❤ ♥❤➻♥ r✐➯♥❣ ✈➔♦ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ q✉→ tr➻♥❤✳ ◆❤➜♥ ♠↕♥❤ sü q✉❛♥ trå♥❣ ❝õ❛ t➼♥❤ ❦❤æ♥❣ ♥❤ỵ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ơ ð ❝❤é ✈ø❛ ❧➔ ♠ët ❝ỉ♥❣ ❝ư ❤ú✉ ➼❝❤ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ✈➜♥ ✤➲ ứ ỵ s q tr Pss ♠ët q✉→ tr➻♥❤ ✤ì♥ ❣✐↔♥✳ ❚✐➳♣ t❤❡♦ ✤➣ ❝❤➾ r❛ r➡♥❣ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ❝â t❤➸ ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ t❤✉➛♥ ♥❤➜t tr➯♥ ♠ët t❤❛♥❣ t❤í✐ ❣✐❛♥ ♣❤✐ t✉②➳♥✳ ✣✐➲✉ ♥➔② ❝❤♦ ♣❤➨♣ ♠å✐ t➼♥❤ ❝❤➜t ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥ t❤✉➛♥ ♥❤➜t ✤÷đ❝ ❝❤✉②➸♥ ♠ët ❝→❝❤ trü❝ t✐➳♣ s❛♥❣ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t✳ ❑➳t q✉↔ ✤ì♥ ❣✐↔♥ ♥❤➜t ✈➔ ❝â ➼❝❤ ♥❤➜t tø ✤✐➲✉ ♥➔② ❧➔ ✭✷✳✸✵✮✱ ❝❤➾ r❛ r➡♥❣ sè ❧÷đ♥❣ ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ ✭✤♦↕♥✮ ❜➜t ❦ý ❝â ♠ët ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥✳ ❑➳t q✉↔ ♥➔② ❝ơ♥❣ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤➾ r❛ r➡♥❣ sè ❝→❝ ❦❤→❝❤ ✤÷đ❝ ♣❤ư❝ ✈ư t↕✐ ♠ët t❤í✐ ✤✐➸♠ τ ❜➜t ❦ý tr♦♥❣ ♠ët ❤➔♥❣ ✤đ✐ M/G/∞ ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ ❦ý ✈å♥❣ ①➜♣ ①➾ λ ❧➛♥ ❦ý ✈å♥❣ ❝õ❛ t❤í✐ ❣✐❛♥ ♣❤ư❝ ✈ư ❦❤✐ τ → ∞✳ ❈✉è✐ ❝ị♥❣✱ t❛ t➻♠ ✤÷đ❝ ♣❤➙♥ ♣❤è✐ ❝õ❛ t❤í✐ ✤✐➸♠ ✤➳♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ n ❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ (0, t]✳ ❈→❝ t❤í✐ ✤✐➸♠ ✤➳♥ ♥❤÷ ✈➟② ❝â ũ ởt ố ỗ tớ ữ tố t❤ù tü ❝õ❛ n ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ✹✶ ❧➟♣ ❝â ❝ò♥❣ ♣❤➙♥ ♣❤è✐ ✤➲✉ tr➯♥ (0, t]✳ ❇➡♥❣ ❝→❝❤ sû ❞ư♥❣ t➼♥❤ ✤è✐ ①ù♥❣✱ ❝→❝❤ ♥❤➻♥ ❧ị✐ ✈➔ t✐➳♥ ❣✐ú❛ ♥❤ú♥❣ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ ✤➲✉ ✈➔ q✉→ tr➻♥❤ ✤➳♥ P♦✐ss♦♥✱ ❝❤ó♥❣ t❛ t➻♠ t❤➜② ♣❤➙♥ ♣❤è✐ ❝õ❛ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥✱ t❤í✐ ❦ý ✤➳♥✱ ✈➔ ❝→❝ ố õ ỳ ữợ t tr ❙➩ t➻♠ ❤✐➸✉ ✈➲ ✈✐➺❝ ❦➳t ❤ñ♣ ✈➔ ♣❤➙♥ ❝❤✐❛ q tr Pss ỗ tớ ự ởt số ự ❞ö♥❣ tr♦♥❣ t❤ü❝ t➳✳ ✹✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ❉✉② ❚✐➳♥ ✭✷✵✵✵✮✱ ❈→❝ ♠æ ❤➻♥❤ ①→❝ s✉➜t ✈➔ ù♥❣ ❞ö♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ụ t ỵ t❤✉②➳t ①→❝ s✉➜t✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ◗✉↔♥❣ ✭✷✵✵✽✮✱ ❳→❝ s✉➜t ♥➙♥❣ ❝❛♦✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❬✹❪ ❤tt♣✿✴✴♦❝✇✳♠✐t✳❡❞✉✴❝♦✉rs❡s✴❡❧❡❝tr✐❝❛❧✲❡♥❣✐♥❡❡r✐♥❣✲❛♥❞✲❝♦♠♣✉t❡r✲ s❝✐❡♥❝❡✴✻✲✷✻✷✲❞✐s❝r❡t❡✲st♦❝❤❛st✐❝✲♣r♦❝❡ss❡s✲s♣r✐♥❣✲✷✵✶✶✳