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❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍
✲✲✲✲✲✲ ✲✲✲✲✲✲

◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❍⑨

◗❯⑩ ❚❘➐◆❍ P❖■❙❙❖◆ ❱⑨ ❈⑩❈
❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆

❈❤✉②➯♥ ♥❣➔♥❤✿ ▲Þ ❚❍❯❨➌❚ ❳⑩❈ ❙❯❻❚ ❱⑨ ❚❍➮◆● ❑➊ ❚❖⑩◆
▼➣ sè✿

ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿ ❚❙✳ ◆●❯❨➍◆ ❚❘❯◆● ❍➪❆

◆❣❤➺ ❆♥ ✲ ✷✵✶✻


✶

▼Ư❈ ▲Ư❈
▼ư❝ ❧ư❝
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▲í✐ ♥â✐ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✹
✶✳✶✳
✶✳✷✳
✶✳✸✳
✶✳✹✳

❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ✈❡❝tì ♥❣➝✉ ♥❤✐➯♥
P❤➙♥ ♣❤è✐ ♠ơ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
P❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷ ◗✉→ tr➻♥❤ P♦✐ss♦♥

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✷✳✶✳ ●✐ỵ✐ t❤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥
✷✳✸✳ ❈→❝ q✉→ tr➻♥❤ P♦✐ss♦♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳
❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✹
✶✵
✶✸
✶✺

✶✼

✶✼
✶✾
✸✺
✹✵
✹✷


✷

▼Ð ✣❺❯
◗✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ q✉→ tr➻♥❤ q trồ
ỵ tt ụ ữ ự ử ✣➙② ❧➔ ❤á♥ ✤→ t↔♥❣ ❝õ❛ ♠æ ❤➻♥❤ ♥❣➝✉
♥❤✐➯♥✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ t✐➳♣ ❝➟♥ ✈➲ q✉→ tr➻♥❤
P♦✐ss♦♥✳ ✣➛✉ t✐➯♥ ♥❤÷ ❧➔ ♠ët q✉→ tr➻♥❤ ✤ê✐ ♠ỵ✐ ✈ỵ✐ ❝→❝ 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 ❜❛ ♥â ữ ởt ợ ừ 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♥ ❣✐↔♥✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤
✤â ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✷ ❝❤÷ì♥❣✳
❈❤÷ì♥❣ ✶✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ q✉→
tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ ❝→❝ ❦✐➳♥ t❤ù❝ ❝➛♥ ❞ị♥❣ ❝❤♦ ❝→❝ ♥ë✐
❞✉♥❣ ❝❤➼♥❤ ð ❝❤÷ì♥❣ ✷✳
❈❤÷ì♥❣ ✷✿ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ q✉→ tr Pss
ợ tữỡ ữỡ ỗ tớ ❝ù✉ q✉→ tr➻♥❤ P♦✐ss♦♥
❧➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❝→❝ q✉→ tr➻♥❤ ❝♦ ❇❡r♥♦✉❧❧✐✳

▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ồ ữợ sỹ ữợ
ồ ừ ❚❙✳ ◆❣✉②➵♥ ❚r✉♥❣ ❍á❛✳ ❚→❝ ❣✐↔ ❜➔② tä ❧á♥❣
❜✐➳t ì♥ s s t tợ ổ tr tờ ỵ t❤✉②➳t ①→❝ s✉➜t ✈➔
t❤è♥❣ ❦➯ t♦→♥ ❝õ❛ ❑❤♦❛ ❚♦→♥ ✲ ❚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
ồ ❚r÷í♥❣ ✣↕✐ ❤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❤✐➳✉ sõt t ữủ ỵ õ õ qỵ ❜→✉ ✤➸ ❧✉➟♥ ✈➠♥
✤÷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❛✉✿
✐✮ ❦❤ỉ♥❣ ❣✐↔♠❀
✐✐✮ ❧✐➯♥ tư❝ ❜➯♥ ♣❤↔✐❀
✐✐✐✮ limx→−∞ F (x) = 0, limx→+∞ F (x) = 1.
❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ 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 0 (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) 0 ∀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 à = P ở st 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 ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ ố ỗ tớ ừ
X1 , ..., Xn
ú ỵ r
ộ ở st 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✉♥❣ ừ B n ố ợ ữủ ồ t ▲❡❜❡s❣✉❡✳ ❚➜t


✻

♥❤✐➯♥
B n ⊂ Bλn .

❍➔♠ f : Rn → R ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❇♦r❡❧ ♥➳✉ ♥â ✤♦ ✤÷đ❝ ✤è✐ ợ Bn
ữủ ồ s ữủ õ ữủ ợ Bn
tử r

❈→❝ sè ✤➦❝ tr÷♥❣✳

●✐↔ sû X : (Ω, F, P ) → (R, B(R)) ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â✱ t
s ừ X t ở P tỗ t↕✐✮ ✤÷đ❝ ❣å✐ ❧➔ ❦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 ồ ữủ ỗ ỹ t

s
X ♥❣➝✉ ♥❤✐➯♥ ✤ì♥ ❣✐↔♥ X =

n
i=1 ai IAi

t❤➻

n

EX =

ai P (Ai ).
i=1

◆➳✉ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ➙♠ 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}

0.


✼

❑❤✐ ✤â✱ EX = EX + − EX − ✭♥➳✉ ❝â ♥❣❤➽❛✮✳

❚➼♥❤ ❝❤➜t✳
✶✮
✷✮
✸✮
✹✮

✺✮
✻✮

◆➳✉ X 0 t❤➻ EX 0✳
◆➳✉ X = C t EX = C
tỗ t EX t ✈ỵ✐ ♠å✐ C ∈ R✱ t❛ ❝â E(CX) = C EX
tỗ t EX EY t E(X Y ) = EX ± EY ✳
◆➳✉ X 0 ✈➔ EX = 0 t❤➻ X = 0✳

♥➳✉ X rí✐ r↕❝ ✈ỵ✐ P (X = xi) = pi,
♥➳✉ X ❧✐➯♥ tư❝ ❝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 ở 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 1 ✈➔ EY > −∞ t❤➻
limEXn .

ElimXn

◆➳✉ Xn

Y

✈ỵ✐ ♠å✐ n

1

✈➔ EY

< +∞

ElimXn

◆➳✉ |Xn|

Y

✈ỵ✐ ♠å✐ n
ElimXn

1

EY

limEXn

t

limEXn .
<

t

limEXn

ElimXn .

ỵ s ở tử ❜à ❝❤➦♥✮ ◆➳✉ |Xn| Y ✈ỵ✐ ♠å✐ n 1✱
EY < ∞ ✈➔ Xn → X t❤➻ X ❦❤↔ t➼❝❤✱ E|Xn − X| → 0 ✈➔ EXn → EX
❦❤✐ n → ∞✳


✽

✶✵✮ ✭❇➜t ✤➥♥❣ t❤ù❝ ▼❛r❦♦✈✮ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤ỉ♥❣ ➙♠✳
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ ε > 0 t❛ ❝â
P (X

ε)

EX
.
ε

✶✶✮ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤ỉ♥❣ ➙♠✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ p > 0✱
t❛ ❝â
∞
EX p = p

xp−1 P (X > x)dx.
0

●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥✳ ❑❤✐ ✤â✱ ❣✐→ trà ✤ë ❧➺❝❤ ❜➻♥❤ ữỡ
tr
DX := E(X EX)2 tỗ t.
ữủ ồ ❧➔ ♣❤÷ì♥❣ 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 ❝â t tỗ t ổ tỗ t
tỗ t t❤➻ ❝â t❤➸ ✤÷đ❝ t➼♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝
(xi − EX)2 pi
♥➳✉ X rí✐ r↕❝ ✈➔ P (X = xi) = pi,
DX =
+∞
2
−∞ (x − EX) p(x)dx ♥➳✉ X ❧✐➯♥ tö❝ ❝â ❤➔♠ ♠➟t ✤ë p.
P❤÷ì♥❣ s❛✐ ❝â ♥❤ú♥❣ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ s❛✉ ✤➙②✿
✶✮ DX = EX 2 − (EX)2✳
✷✮ DX 0✳
✸✮ DX = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X = EX = C ❤✳❝✳❝✳
✹✮ D(CX) = C 2DX ✳
✺✮ ✭❇➜t ✤➥♥❣ t❤ù❝ ❈❤❡❜②s❤❡✈✮ ●✐↔ sû X ❧➔ ❜✐➳♥ ♥❣➝✉ t ý
õ tỗ t DX t ợ ♠å✐ ε > 0✱ t❛ ❝â
P (|X − EX|

ε)

DX
,
ε2


✾

tr♦♥❣ ✤â C ❧➔ ❤➡♥❣ sè✱ ❤➛✉ ❝❤➢❝ ❝❤➢♥ ✤÷đ❝ ✈✐➳t ❣å♥ ❜ð✐ ❤✳❝✳❝✳

◆➳✉ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ rí✐ r↕❝ t❤➻ ❣✐→ trà x0 ✤÷đ❝ ❣å✐ ❧➔ ♠♦❞❡ ❝õ❛
X ♥➳✉ X ❝â ①→❝ s✉➜t ❧ỵ♥ ♥❤➜t t↕✐ x0 ✳
◆➳✉ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧✐➯♥ tö❝ ❝â ❤➔♠ ♠➟t ✤ë p t❤➻ ❣✐→ trà x0 ✤÷đ❝
❣å✐ ❧➔ ♠♦❞❡ ❝õ❛ X ♥➳✉ p(x) ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t t↕✐ x0✳
◆➳✉ x0 ❧➔ ♠♦❞❡ ❝õ❛ X t❤➻ t❛ ❦➼ ❤✐➺✉ x0 = modX ✳
❙è xp, (0 < p < 1) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ ✈à ❝➜♣ p ❝õ❛ ❤➔♠ ♣❤➙♥ ♣❤è✐ F
❝õ❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ♥➳✉
F (xp ) p ✈➔ F (xp + 0) p,
tr♦♥❣ ✤â F (xp + 0) = limx→x F (x)✳
❘ã r➔♥❣✱ ♥➳✉ F ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t❤➻ F (xp) = 0✳
◆➳✉ p = 21 t❤➻ xp = x1/2 ✤÷đ❝ ❣å✐ ❧➔ tr✉♥❣ ✈à ❤❛② ♠❡❞✐❛♥ ừ X
ữủ ỵ m(X)
+
p

tỡ ✈➔ ♣❤➙♥ ♣❤è✐ ♥❤✐➲✉ ❝❤✐➲✉✳

▼ët ✈❡❝tì ♥❣➝✉ ♥❤✐➯♥ n ❝❤✐➲✉ ❧➔ ♠ët ❜ë ❝â t❤ù tü (X1, X2, ..., Xn) ✈ỵ✐
X1 , X2 , ..., Xn ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉
tỡ 2 ỵ (X, Y ) ✈ỵ✐ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥

t❤ù ♥❤➜t✱ Y ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ t❤ù ❤❛✐✳
❱❡❝tì ♥❣➝✉ ♥❤✐➯♥ n ❝❤✐➲✉ ❧✐➯♥ tư❝ ❤❛② rí✐ r↕❝ ♥➳✉ t➜t ❝↔ ❝→❝ ❜✐➳♥ ♥❣➝✉
♥❤✐➯♥ t❤➔♥❤ ♣❤➛♥ ❧➔ ❧✐➯♥ tư❝ ❤❛② rí✐ r↕❝✳
◆➳✉ ♠ët ✈❡❝tì ♥❣➝✉ ♥❤✐➯♥ ❝❤ù❛ ❝↔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
rí✐ r↕❝ ✈➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❧✐➯♥ tư❝ t❤➻ ♥â ✤÷đ❝ ❣å✐ ❧➔
✈❡❝tì ♥❣➝✉ ♥❤✐➯♥ ❤é♥ ❤đ♣✳ ✣➸ ✤ì♥ ❣✐↔♥✱ t❛ s➩ ❦❤ỉ♥❣ ①➨t tr÷í♥❣ ❤đ♣ ♥➔②✳
❱❡❝tì ♥❣➝✉ ♥❤✐➯♥ 2 ❝❤✐➲✉ ✤÷đ❝ ❣å✐ ✤ì♥ ❣✐↔♥ ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ 2 ❝❤✐➲✉✳
❈❤♦ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ (X, Y )✳ ❍➔♠ ♣❤➙♥ ♣❤è✐ ①→❝ s✉➜t ❝õ❛ ❜✐➳♥ 2 ❝❤✐➲✉
(X, Y ) ❧➔ F (x, y) ✈ỵ✐
F (x, y) = P (X

x, Y

y),

∀x, y ∈ R.


✶✵

✶✳✷ P❤➙♥ ♣❤è✐ ♠ô
P❤➙♥ ♣❤è✐ ♠ô ✤â♥❣ ✈❛✐ trá r➜t q trồ tr ự ử ừ
st ữợ t❛ ❝❤➾ tr➻♥❤ ❜➔② ✈➔✐ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ♣❤è✐
♠ô✳

✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t✳
❚❛ ♥â✐ r➡♥❣ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝â ♣❤➙♥ ♣❤è✐ ♠ơ ✈ỵ✐ t❤❛♠ sè λ > 0
♥➳✉ ♠➟t ✤ë ①→❝ s✉➜t ❝õ❛ ♥â ❝â ❞↕♥❣
λe−λx ♥➳✉ x 0,
f (x) =

0
♥➳✉ x < 0.
❚❛ ❝â t q s
ã X õ ố ụ ợ t❤❛♠ sè λ > 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❤➔♠ ♣❤➙♥ ♣❤è✐
①→❝ s✉➜t ❝õ❛ ♥â ❝â ❞↕♥❣
1 − e−λx ♥➳✉ x 0,
f (x) =
0
♥➳✉ x < 0.
• X ❝â ♣❤➙♥ ♣❤è✐ ♠ơ ✈ỵ✐ t❤❛♠ sè λ > 0 t❤➻
2
1
1
E(X) = , E(X 2 ) = 2 , V ar(X) = 2 .



ã

õ r X ổ ợ ♥➳✉
P (X > s + t|X > t) = P (X > s),

ợ ồ s, t

0.



ó r r tữỡ ữỡ ợ ởt tr s
P (X>s+t,X>t)

P (X>t)

= P (X > s).
P (X > s + t) = P (X > t)P (X > s).

✭✶✳✷✮

❚❛ ❤➣② ❤➻♥❤ ❞✉♥❣ X ❧➔ t❤í✐ ❣✐❛♥ sè♥❣ ❝õ❛ ❜â♥❣ ✤➧♥ ✤✐➺♥ ❝❤➥♥❣ ❤↕♥✳
❑❤✐ ✤â ✭✶✳✶✮ ❝â ♥❣❤➽❛ ❧➔ ①→❝ s✉➜t ✤➸ ❜â♥❣ ✤➧♥ sè♥❣ q✉→ s + t ✈ỵ✐ ✤✐➲✉
❦✐➺♥ ❜â♥❣ ✤➣ sè♥❣ q✉→ s ❜➡♥❣ ①→❝ s✉➜t ✭❦❤æ♥❣ ✤✐➲✉ ❦✐➺♥✮ ✤➸ ❜â♥❣ ✤➧♥


✶✶

sè♥❣ q✉→ s✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❜â♥❣ ✤➧♥ ✧❦❤ỉ♥❣ ♥❤ỵ õ ữủ ũ q
tớ t rỗ
t❤➜② ✧❳ ❝â ♣❤➙♥ ♣❤è✐ ♠ô ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ X ❦❤ỉ♥❣ ♥❤ỵ ✧✳
❱ỵ✐ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝â ♣❤➙♥ ♣❤è✐ ♠ơ ❝â ❝÷í♥❣ ✤ë λ > 0✱
P {X > x} = e−λx ✤è✐ ✈ỵ✐ x
0✳ ✣✐➲✉ ✤â t❤ä❛ ♠➣♥ ✭✷✳✹✮ ✤è✐ ✈ỵ✐ ♠å✐
x 0✱ t 0✱ ♥➯♥ X ❧➔ q✉➯♥✳ ◆❣÷đ❝ ❧↕✐✱ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❜➜t ❦ý X
❧➔ q✉➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥➳✉ ♥â ❝â ♣❤➙♥ ♣❤è✐ ♠ô✳ ✣➸ t❤➜② ✤✐➲✉ ♥➔②✱ ❣✐↔ sû
h(x) = ln[P {X > x}] ỵ r P {X > x} ❧➔ ❦❤ỉ♥❣ t➠♥❣ t❤❡♦ x✱
h(x) ❝ơ♥❣ t❤➳✳ ❚❤➯♠ ✈➔♦ ✤â✱ ✭✷✳✹✮ ♥â✐ r➡♥❣ h(t + x) = h(x) + t(x) ✈ỵ✐
♠å✐ t, x 0✳ ❍❛✐ ♣❤→t ❜✐➸✉ ✤â ử ỵ r h(x) t t t x
P {X > x} ❧➔ ❤➔♠ ♠ơ t❤❡♦ x✳

✶✳✷✳✷ ❱➼ ❞ư✳ ●✐↔ sû t❤í✐ ❣✐❛♥ sè♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❜â♥❣ ✤➧♥ ✤✐➺♥ tr♦♥❣

♣❤á♥❣ ❧➔ 10 ❣✐í ✈➔ ❝â ♣❤➙♥ ♣❤è✐ ♠ơ✳ ◆❛♠ ✈➔♦ ♣❤á♥❣ t❤➜② ✤➧♥ ✤❛♥❣ s→♥❣✳

❚➼♥❤ ①→❝ s✉➜t ✤➸ ◆❛♠ ❝â t❤➸ ❧➔♠ ✈✐➺❝ 5 ❣✐í ❧✐➲♥ ❦❤✐ sû ❞ư♥❣ ❜â♥❣ ✤➧♥
♥➔②✳
●✐↔✐✳ ●å✐ X ❧➔ t❤í✐ ❣✐❛♥ sè♥❣ ❝õ❛ ❜â♥❣ ✤➧♥✳ ❑❤✐ ✤â t❛ ❝â
1
E(X) = 10 = ,
λ

s✉② r❛ λ = 0, 1.

❱➻ X ❦❤ỉ♥❣ ♥❤ỵ ✭✤➣ t❤➢♣ s rỗ st t
P (X > 5) = 1 − P (X

5) = 1 − F (5) = 1 − e−5λ = e−0,5 .

❈❤ó þ r➡♥❣ ♥➳✉ X ❜✐➳t ♥❤ỵ t❤➻ ①→❝ s✉➜t ♣❤↔✐ t➻♠ ❧➔
P (X > 5 + t, X > t)
P (X > t)
1 − P (X 5 + t)
=
1 − P (X t)

P (X > 5 + t|X > t) =

P (X > 5 + t)
P (X > t)
1 − F (5 + t)
=
,
1 − F (t)

=

tr♦♥❣ ✤â t ❧➔ tớ õ ữủ sỷ ử trữợ ữợ
ỏ


✶✷

✶✳✷✳✸ ❱➼ ❞ö✳ ●✐↔ sû X1, X2 ❧➔ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝â ♣❤➙♥

♣❤è✐ ♠ơ ✈ỵ✐ t❤❛♠ sè t÷ì♥❣ ù♥❣ ❧➔ λ1, λ2✳ ❍➣② t➼♥❤ ①→❝ s✉➜t P (X1 < X2)✳
●✐↔✐✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ①→❝ s✉➜t ✤➛② ✤õ✱ t❛ ❝â
+∞

P (X1 < X2 ) =

P (X1 < X2 |X2 = x)λ2 e−λ2 x dx

0
+∞

=

P (X1 < x)λ2 e−λ2 x dx

0
+∞

=

(1 − λ1 e−λ1 x )λ2 e−λ2 x dx =

0

• ●✐↔ sû X1 , X2 , ..., Xn

λ1
.
λ1 + λ2

❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝ị♥❣ ♣❤➙♥ ♣❤è✐
♠ơ ✈ỵ✐ t❤❛♠ sè λ✳ ❑❤✐ ✤â X1 + X2 + ... + Xn ❝â ♣❤➙♥ ♣❤è✐ ❣❛♠♠❛ ✈ỵ✐
❝→❝ t❤❛♠ sè n ✈➔ λ✱ tù❝ ❧➔✱ ❤➔♠ ♠➟t ✤ë ❝õ❛ ♥â ❝â ❞↕♥❣
(λx)
λe−λx ♥➳✉ x 0,
(n−1)!
fn (x) =
0
♥➳✉ x < 0.
n−1

❚❤➟t ✈➟②✱ ❤✐➸♥ ♥❤✐➯♥ ❦➳t ❧✉➟♥ tr➯♥ ✤ó♥❣ ✈ỵ✐ n = 1✳ ●✐↔ sû ❦➳t ❧✉➟♥
tr➯♥ ✤ó♥❣ ✈ỵ✐ n − 1✳ ❑❤✐ ✤â t❛ ❝â
+∞

f (x − y)fn−1 (y)dy

fn (x) =
0
+∞

λe

=
0

n−2
−λ(x−y) (λy)

(λx)n−1 −λx
λe dy =
λe
,
(n − 2)!
(n − 1)!
λy

tr♦♥❣ ✤â f (x) t ở ừ X1

ú ỵ X õ ♣❤➙♥ ♣❤è✐ ❣❛♠♠❛ ✈ỵ✐ ❝→❝ t❤❛♠ sè n ✈➔ λ t❤➻
✶✳✷✳✹ ❱➼ ❞ö✳ ✭❍➔♠

n
n
E(X) = , V ar(X) = 2 .
λ
λ
tè❝ ✤ë ❤ä♥❣✮ ●✐↔ sû X ❧➔ t❤í✐

❣✐❛♥ sè♥❣ ❝õ❛ t❤✐➳t

❜à ♥➔♦ ✤â✳ ❚❛ ①❡♠ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ❤➔♠ ♠➟t ✤ë ✈➔ ❤➔♠ ♣❤➙♥
♣❤è✐ ❧➔ f (t) ✈➔ F (t) t÷ì♥❣ ù♥❣✳ ❍➔♠ tè❝ ✤ë ❤ä♥❣ r(t) ❝õ❛ X ✤÷đ❝ t➼♥❤
t❤❡♦ ❝ỉ♥❣ t❤ù❝
r(t) =

f (t)
.
1 − F (t)




ỵ ừ r(t) t sỷ r X ✤➣ ✈÷đt q✉→ t ✭t❤✐➳t ❜à ✤➣ ❧➔♠
✈✐➺❝ ✤÷đ❝ t tớ rỗ ố t st X ❦❤ỉ♥❣ ✈÷đt q✉→
t + ∆t ✭t❤✐➳t ❜à s➩ ❜à ọ trữợ t + t ợ t > 0 s✉➜t ✤â ❧➔
P (X ∈ (t, t + ∆t), X > t)
P (X > t)
f (t)∆t
P (X ∈ (t, t + ∆t))
=
= r(t)∆t,
=
P (X > t)
1 − F (t)

P (X ∈ (t, t + ∆t)|X > t) =

tù❝ ❧➔✱ r(t) ❜✐➸✉ t❤à ♠➟t ✤ë ①→❝ s✉➜t ❝â ✤✐➲✉ ❦✐➺♥ ✤➸ t❤✐➳t ❜à ❝â t✉ê✐ t s➩
❦❤ỉ♥❣ ❧➔♠ ✈✐➺❝ ✤÷đ❝ ♥ú❛✳
❉➵ t❤➜② r➡♥❣✱ ♥➳✉ X ❝â ♣❤➙♥ ♣❤è✐ ♠ơ ✈ỵ✐ t❤❛♠ sè λ t❤➻

f (t)
λe−λt
r(t) =
= −λt = λ.
1 − F (t)
e

◆❤÷ ✈➟②✱ ❤➔♠ tè❝ ✤ë ❤ä♥❣ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ơ ❧➔ ♠ët ❤➡♥❣ sè ✭❜➡♥❣
t❤❛♠ sè ❝õ❛ ♣❤➙♥ ♣❤è✐ ➜②✮✳

❈❤ó þ✳ ❍➔♠ tè❝ ✤ë ❤ä♥❣ r(t) ①→❝ ✤à♥❤ ❞✉② ♥❤➜t ❤➔♠ ♣❤➙♥ ♣❤è✐

F (t)✳

❚❤➟t ✈➟②✱ t❛ ❝â
r(t) =

s✉② r❛

F (t)
1 − F (t)
t

F (t) = 1 − exp{−

r(u)du}.
0

❉♦ ✤â✱ ♣❤➙♥ ♣❤è✐ ♠ô ❧➔ ♣❤➙♥ ♣❤è✐ ❞✉② ♥❤➜t ❝â ❤➔♠ tè❝ ✤ë ❤ä♥❣ ❧➔ ❤➡♥❣
sè✳

✶✳✸ P❤➙♥ ♣❤è✐ P♦✐ss♦♥
❚❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ♥❤➜t ❝õ❛ ♣❤➙♥ ♣❤è✐
P♦✐ss♦♥


✶✹

✶✳✸✳✶ ✣à♥❤ ♥❣❤➽❛✳
❚❛ ♥â✐ r➡♥❣ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ t❤❛♠ sè
λ > 0 ♥➳✉
k
P (X = k) =

λ −λ
e (k = 0, 1, 2, ...).
k!

❚❛ ✤➣ ❜✐➳t r➡♥❣✱ ♥➳✉ X ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ t❤❛♠ sè λ > 0 t❤➻
E(X) = V ar(X) = λ.

✶✳✸✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥✳
◆➳✉ X, Y ❧➔ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣✱ ❝â ♣❤➙♥ ♣❤è✐ Pss ợ
t số 1, 2 tữỡ ự t X + Y ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ t❤❛♠ sè
λ1 + λ2 ✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ①→❝ s✉➜t ✤➛② ✤õ✱ t❛ ❝â
•

n

n

P (X = k, Y = n − k) =

P (X + Y = n) =
k=0
n

=
k=0

=

P (X = k)P (Y = n − k)
k=0

λk1 −λ1 λn−k
1
2
e
e−λ2 = e−(λ1 +λ2 )
k!
(n − k)!
n!

n

k=0

n!

λk1 λn−k
2
k!(n − k)!

)n

(λ1 + λ2 −(λ1 +λ2 )
e
(n = 0, 1, 2, ...).
n!

❑➳t q✉↔ s❛✉ ✤➙② ❝ô♥❣ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ❝ỉ♥❣ t❤ù❝ ①→❝ s✉➜t ✤➛②
✤õ✳
• ●✐↔ sû N ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ t❤❛♠ sè λ > 0
✈➔ X ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ s❛♦ ❝❤♦
P (X = k|N = n) = Cnk pk (1 − p)n−k (k = 0, 1, 2, ..., n).

❑❤✐ ✤â X ❝â ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥ ✈ỵ✐ t❤❛♠ sè λp✳


✶✺

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ t❤➜②
∞

P (X = k) =

P (X = k|N = n)P (N = n)
n=0
∞

=
n=0

n!
pk (1 − p)n−k
k!(n − k)!

(λp)k −λ
=
e
k!
=

(λp)k
k!

∞
n=0

[λ(1 − p)]n−k
(n − k)!

−λ λ(1−p)

e

λn −λ
e
n!

e

(λp)k −λp
=
e ,
k!

✈ỵ✐ ♠å✐ k = 0, 1, 2, ...✳
• ▲✉➟t ❜✐➳♥ ❝è ❤✐➳♠✳ ●✐↔ sû A ❧➔ ❜✐➳♥ ❝è ♥➔♦ ✤â ①↔② r❛ ✈ỵ✐ ①→❝ s✉➜t
p✳ ❑➼ ❤✐➺✉ X ❧➔ sè ❧➛♥ ①✉➜t ❤✐➺♥ A tr♦♥❣ n ❧➛♥ q✉❛♥ s→t✳ ❑❤✐ ✤â X ❝â
♣❤➙♥ ♣❤è✐ ♥❤à t❤ù❝
P (X = k) = Cnk pk (1 − p)n−k , k = 0, 1, 2, ..., n.

❚❛ ✤➣ ❜✐➳t✱ ▲✉➟t ❜✐➳♥ ❝è ❤✐➳♠ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ❦❤✐ p ❦❤→ ❜➨ ✈➔ n ❦❤→
❧ỵ♥ t❤➻ ❝â t❤➸ ①➜♣ ①➾ ♣❤➙♥ ♣❤è✐ ♥❤à t❤ù❝ ❜➡♥❣ ♣❤➙♥ ♣❤è✐ P♦✐ss♦♥✳ ❈ö t❤➸
❧➔
k
P (X = k) ≈

tr♦♥❣ ✤â λ = np✳

λ −λ
e ,
k!

✶✳✹ ◗✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥
●✐↔ sû T ❧➔ t➟♣ ✈ỉ ❤↕♥ ♥➔♦ ✤â✳ ◆➳✉ ✈ỵ✐ ♠é✐ t ∈ T ✱ Xt ❧➔ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
t❤➻ t❛ ❦➼ ❤✐➺✉ X = {Xt, t ∈ T }✱ ✈➔ ❣å✐ X ❧➔ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ❤❛②
❤➔♠ ♥❣➝✉ ♥❤✐➯♥ ✭✈ỵ✐ t❤❛♠ ❜✐➳♥ t ∈ T ✮✳

• ◆➳✉ T ❧➔ t➟♣ ✤➳♠ ✤÷đ❝ t❤➻ t❛ ❣å✐ X = {Xt , t ∈ T } ❧➔ q✉→ tr➻♥❤
♥❣➝✉ ♥❤✐➯♥ ✈ỵ✐ t❤❛♠ sè rí✐ r↕❝✳


✶✻

◆➳✉ T = N t❤➻ t❛ ❣å✐ X = {Xn, n ∈ N} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
✭♠ët ♣❤➼❛✮✳
• ◆➳✉ T = Z t❤➻ t❛ ❣å✐ X = {Xn , n ∈ Z} ❧➔ ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
❤❛✐
ã T ởt ừ ữớ t tỹ tù❝ ❧➔✱ T t❤✉ë❝ ♠ët
tr♦♥❣ ❝→❝ t➟♣ s❛✉✿
•

(−∞, ∞), [a, ∞), (−∞, b], [a, b), [a, b], (a, b], (a, b),

t❤➻ t❛ ❣å✐ X = {Xt, t ∈ T } ❧➔ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ ✈ỵ✐ t❤❛♠ sè ❧✐➯♥ tư❝✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥❤÷ t❤➳✱ t❤❛♠ sè T ✤â♥❣ ✈❛✐ trá t❤í✐ ❣✐❛♥✳
• ◆➳✉ T ❧➔ t➟♣ ❝♦♥ ❝õ❛ Rd ✱ t❤➻ t❛ ❣å✐ X = {Xt , t ∈ T } ❧➔ tr÷í♥❣ ♥❣➝✉
♥❤✐➯♥✳




ì

P
ợ t
q tr
ởt q✉→ tr➻♥❤ ✤➳♥ ❧➔ ♠ët ❞➣② t➠♥❣ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ 0 < S1 <

S2 < ...✱ tr♦♥❣ ✤â Si+1 − Si > 0✳ ◆❣❤➽❛ ❧➔ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ♥❤÷
t❤➳ t❤➻ FX (0) = 0✳ ❈→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ S1, S2, ... ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ t❤í✐
✤✐➸♠ ✤➳♥ ✭t❤✉➟t ♥❣ú ✧t❤í✐ ❣✐❛♥✧ ❧➔ ❜à ❧↕♠ ❞ư♥❣ ✤è✐ ✈ỵ✐ ❝❤õ ✤➲ ♥➔②✮ ✈➔
❜✐➸✉ ❞✐➵♥ ❝→❝ t❤í✐ ✤✐➸♠ ♠➔ t↕✐ ✤â ♠ët ❤✐➺♥ t÷đ♥❣ ♥➔♦ ✤â ❧↕✐ ❞✐➵♥ r❛
✭❝❤➥♥❣ ❤↕♥ ởt ỵ r q tr➻♥❤ ❜➢t ✤➛✉
t↕✐ t❤í✐ ✤✐➸♠ 0 ✈➔ ♥❤✐➲✉ ❦❤→❝❤ ❤➔♥❣ ổ t t ỗ tớ
tữủ õ t ữủ ỷ ỵ rở ỡ ❜➡♥❣
✈✐➺❝ ❣➢♥ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❞÷ì♥❣ ❝❤♦ ♠é✐ ❦❤→❝❤✮✳ ổ t s t
ữủ ỗ tớ ❦❤→❝❤ ✤➳♥ t↕✐ t❤í✐ ✤✐➸♠ 0 ♥❤÷ ❧➔ ❜✐➳♥
❝è ❝â ①→❝ s✉➜t 0✱ ♥❤÷♥❣ ✤✐➲✉ ➜② ❝â t❤➸ ✤÷đ❝ ❜ä q✉❛✳ ✣➸ ①❡♠ ①➨t ✤➛② ✤õ
q✉→ tr➻♥❤ S1, S2, ..., t ố ỗ tớ ừ ❝♦♥
S1 , ..., Sn ✈ỵ✐ ♠å✐ n > 1✳
▼➦❝ ❞ị t❛ t❤÷í♥❣ ①❡♠ ❝→❝ q✉→ tr➻♥❤ ♥❤÷ ✈➟② ♥❤÷ ❧➔ q✉→ tr➻♥❤ ✤➳♥✱
♥❤÷♥❣ ❝❤ó♥❣ ❝â t❤➸ ❧➔ ♠ỉ ❤➻♥❤ ①✉➜t ♣❤→t ❝õ❛ ♠ët ❤➺✱ ❤♦➦❝ ❞➣② ❜➜t ❦ý
❝→❝ sü ❝è ❦❤→❝✱ ✤➦❝ ❜✐➺t tr♦♥❣ ❧➽♥❤ ✈ü❝ ♠æ ♣❤ä♥❣ ✤➸ ❝❤➾ ❝→❝ sü ❝è ❤♦➦❝
❝→❝ sü ❦✐➺♥ ✤➳♥ ✈➔ t❛ s➩ ổ t ú ỵ ớ t❤ù
n✱ Sn ❧➔ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔ {Sn t}✱ ❧➔ ♠ët ❜✐➳♥ ❝è✳ ✣✐➲✉ ✤â ✤æ✐
❦❤✐ s➩ ❧➔♠ q✉➯♥ ✤✐ ✈✐➺❝ ❤✐➸✉ r➡♥❣ ❦❤→❝❤ ✤➳♥ t❤ù n ❝ô♥❣ ❧➔ ♠ët ❜✐➳♥ ❝è✳


✶✽

◆❤÷ ♠✐♥❤ ❤å❛ tr♦♥❣ ❍➻♥❤ ✷✳✶✱ q✉→ tr➻♥❤ ✤➳♥ ❜➜t ❦ý ❝â t❤➸ ❝ơ♥❣ ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐ ❤❛✐ q✉→ tr➻♥❤ ♥❣➝✉ ♥❤✐➯♥ t❤➔♥❤ ♣❤➛♥✳ ❚❤➔♥❤ ♣❤➛♥ t❤ù ♥❤➜t
❧➔ ❞➣② ❝→❝ t❤í✐ ❦ý ✤➳♥ X1, X2, ...✳ ❈❤ó♥❣ ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❞÷ì♥❣
✤÷đ❝ ①→❝ ✤à♥❤ t❤ỉ♥❣ q✉❛ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ ❜ð✐ X1 = S1 ✈➔ Xi = Si −Si−1
✈ỵ✐ i > 1✳ ❚÷ì♥❣ tü✱ ♥➳✉ ❝❤♦ Xi✱ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ Si ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷
n

Sn =

Xi .

✭✷✳✶✮

i=1

◆❤÷ ✈➟②✱ ♣❤➙♥ ♣❤è✐ ❦➳t ❤đ♣ ❝õ❛ X1, ..., Xn ✈ỵ✐ ♠å✐ n > 1 ❧➔ ✤õ ✭✈➲ ♥❣✉②➯♥
t➢❝✮ ✤➸ ①→❝ ✤à♥❤ q✉→ tr➻♥❤ ✤➳♥✳ ❱➻ ❝→❝ t❤í✐ ❦ý ✤➳♥ ❧✐➯♥ t✐➳♣ ❧➔ ❝ị♥❣ ♣❤➙♥
♣❤è✐ tr♦♥❣ ❤➛✉ ❤➳t ❝→❝ tr÷í♥❣ ❤đ♣ ✤÷đ❝ q✉❛♥ t➙♠✱ ♥â ❝ơ♥❣ t❤÷í♥❣ ❞➵
❞➔♥❣ ❤ì♥ ✤➸ ①→❝ ✤à♥❤ ♣❤➙♥ ♣❤è✐ ❦➳t ❤đ♣ ❝õ❛ Xi ❤ì♥ ❧➔ ❝õ❛ Si✳
❑❤➼❛ ❝↕♥❤ t❤ù ❤❛✐ ✤➸ ①→❝ ✤à♥❤ ♠ët q✉→ tr➻♥❤ ✤➳♥ ❧➔ q✉→ tr➻♥❤ ✤➳♠ N (t)✱
tr♦♥❣ ✤â ✈ỵ✐ ♠é✐ t > 0✱ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ N (t) ❧➔ sè ❝→❝ ❦❤→❝❤ ❤➔♥❣ tỵ✐
❝❤♦ tỵ✐ t❤í✐ ✤✐➸♠ t✳
❈→❝ q✉→ tr➻♥❤ ✤➳♠ {N (t), t > 0} ✤÷đ❝ ♠✐♥❤ ❤å❛ tr♦♥❣ ❤➻♥❤ ✷✳✶✱ ❧➔ ♠ët
❤å ✈ỉ ❤↕♥ ❦❤ỉ♥❣ ✤➳♠ ✤÷đ❝ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {N (t), t > 0} tr♦♥❣ ✤â
N (t)✱ ✈ỵ✐ ♠é✐ t > 0✱ ❧➔ sè ❦❤→❝❤ ❤➔♥❣ tỵ✐ tr♦♥❣ ❦❤♦↔♥❣ (0, t]✳ ❈→❝ ✤➛✉
♠ót ❝õ❛ ❝→❝ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ➜② ✤ỉ✐ ❦❤✐ ❧➔ q✉❛♥ trå♥❣✳ N (0) ✤÷đ❝ ✤à♥❤
♥❣❤➽❛ ❧➔ 0 ✈ỵ✐ ①→❝ s✉➜t 1✱ t❤❡♦ ♥❣❤➽❛ t❛ ✤❛♥❣ ①❡♠ ①➨t ❝❤➾ ❝→❝ ❦❤→❝❤ ❤➔♥❣
tỵ✐ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❞÷ì♥❣✳
◗✉→ tr➻♥❤ ✤➳♠ {N (t), t > 0} ✤è✐ ✈ỵ✐ q✉→ tr➻♥❤ ✤➳♥ ❜➜t ❦ý ❝â t➼♥❤ ❝❤➜t ❧➔
N (τ ) N (t) ✈ỵ✐ ♠å✐ τ t > 0 ✭♥❣❤➽❛ ❧➔ N (τ ) − N (t) ❧➔ ♠ët ❜✐➳♥ ♥❣➝✉


✶✾

♥❤✐➯♥ ❦❤ỉ♥❣ ➙♠✮✳
❱ỵ✐ ♠å✐ sè ♥❣✉②➯♥ n

✈➔ t❤í✐ ✤✐➸♠ t > 0✱ Sn ✈➔ N (t) ❧✐➯♥ ❤➺ ❜ð✐

{Sn t} = {N (t) n}.

t ữủ ỵ {Sn t} ❧➔ ❜✐➳♥ ❝è ❦❤→❝❤ ❤➔♥❣ t❤ù n
①✉➜t ❤✐➺♥ trữợ tớ t ỹ N (t)✱ sè ❝→❝ ❦❤→❝❤ ❤➔♥❣
ð t❤í✐ ✤✐➸♠ t✱ ➼t ♥❤➜t ♣❤↔✐ ❧➔ n❀ ❝â ♥❣❤➽❛ ❧➔ ❜✐➳♥ ❝è {N (t) n}✳ ❚÷ì♥❣
tü✱ {N (t) n} ❝â ♥❣❤➽❛ ❧➔ {Sn t}✱ tù❝ ❧➔ ❝â ✭✷✳✷✮✳ ❍➺ t❤ù❝ ♥➔② ❧➔ rã
r➔♥❣ tø ❤➻♥❤ ✷✳✶✳ ❱➔ ♥❤÷ t❤➳ t❛ ❧↕✐ ❝â
{Sn > t} = {N (t) < n}
✭✷✳✸✮
❈❤➥♥❣ ❤↕♥✱ ❜✐➳♥ ❝è {S1 > t} ❝â ♥❣❤➽❛ ❧➔ ❦❤→❝❤ ✤➛✉ t✐➯♥ ✤➳♥ s❛✉ t✱ ♥â
❝â ♥❣❤➽❛ ❧➔ {N (t) < 1} ✭❝â ♥❣❤➽❛ ❧➔ {N (t) = 0}✮✳ ▲✐➯♥ ❤➺ ➜② s➩ ✤÷đ❝ sû
❞ư♥❣ t❤÷í♥❣ ①✉②➯♥ tr♦♥❣ ♠è✐ ❧✐➯♥ ❤➺ q✉❛ ❧↕✐ ❣✐ú❛ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ ✈➔
❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤➳♠✳ ❱➲ ♥❣✉②➯♥ t➢❝✱ ✭✷✳✷✮ ❤♦➦❝ ✭✷✳✸✮ ❝â t❤➸ ✤÷đ❝ sû
❞ư♥❣ ✤➸ ①→❝ ố ỗ tớ ừ tớ tổ q
ố ỗ tớ ừ ♥❣÷đ❝ ❧↕✐✱ ✈➻ ✈➟② ♠ët ❧➛♥ ♥ú❛
✤➦❝ tr÷♥❣ ❝â t❤➸ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ①→❝ ✤à♥❤ ♠ët q✉→ tr➻♥❤ ✤➳♥✳
❚â♠ ❧↕✐✱ ♠ët q✉→ tr➻♥❤ ✤➳♥ ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♣❤➙♥ ♣❤è✐ ❦➳t ❤đ♣
❝õ❛ ❝→❝ t❤í✐ ✤✐➸♠ ✤➳♥ Sn✱ ❝→❝ t❤í✐ ❦ý ✤➳♥ ✭❦❤♦↔♥❣ ✤➳♥ ❧✐➯♥ t✐➳♣✮ Xn✱
❤♦➦❝ ❝→❝ ❜✐➳♥ ✤➳♠ ✭♥❣➝✉ ♥❤✐➯♥✮ N (t)✳ ❱➲ ♥❣✉②➯♥ t➢❝ ❦❤✐ ①→❝ ✤à♥❤ ❜➜t
❦ý ♠ët tr♦♥❣ ❝❤ó♥❣ t❤➻ ♥❤ú♥❣ ❝→✐ ❦❤→❝ ❝ơ♥❣ ✤÷đ❝ ①→❝ ✤à♥❤✳
1

✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥
▼ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ ♠ët ✈➼ ❞ư ✈➲ ♠ët q✉→ tr➻♥❤ ✤➳♥✱ ✈➔ ❝→❝ t❤í✐
❦ý ✤➳♥ ❦➳ t✐➳♣ ❝✉♥❣ ❝➜♣ ♠ët ♠ỉ t↔ t✐➺♥ ❧đ✐ ♥❤➜t ✈➻ ❝→❝ t❤í✐ ❦ý ✤➳♥ ❦➳
t✐➳♣ ✤÷đ❝ ①→❝ ✤à♥❤ ❧➔ ❝ị♥❣ ♣❤➙♥ ♣❤è✐✳ ❈→❝ q✉→ tr➻♥❤ ✈ỵ✐ ❝→❝ t❤í✐ ❦ý ❦➳
t✐➳♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐ ❧➔ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ✈➔ ❤➻♥❤ t❤➔♥❤ ♥➯♥ ♥❤✐➲✉ ❜➔✐
t♦→♥ ✤÷đ❝ ①❡♠ ①➨t ✈➲ s❛✉✳

✷✵

✷✳✷✳✶ ❍➔♠ ♠➟t ✤ë ①→❝ s✉➜t ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥✳
◗✉→ tr➻♥❤ ✤ê✐ ♠ỵ✐ ❧➔ ♠ët q✉→ tr➻♥❤ ✤➳♥ tr♦♥❣ ✤â ❞➣② ❝→❝ t❤í✐ ❦ý ✤➳♥
❧➔ ♠ët ❞➣② ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✤ë❝ ❧➟♣ ❝ò♥❣ ♣❤➙♥ ♣❤è✐✳
▼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) = λe−λx ✈ỵ✐ x 0✳ ❚❤❛♠ sè λ ✤÷đ❝ ❣å✐ ❧➔ tè❝
✤ë ❝õ❛ q✉→ tr➻♥❤✳
❚❛ s➩ t❤➜② s❛✉ ♥➔② ❧➔ ✈ỵ✐ ♠ët ✤ë ❞➔✐ t ❜➜t ❦ý✱ λt ❧➔ ❦ý ✈å♥❣ ❝õ❛ sè
❦❤→❝❤ ✤➳♥ tr♦♥❣ ❦❤♦↔♥❣ ✤â✳ ◆❤÷ t❤➳ λ ✤÷đ❝ ❣å✐ ❧➔ ❝÷í♥❣ ✤ë ✤➳♥ ❝õ❛ q✉→
tr➻♥❤✳

✷✳✷✳✷ ❚➼♥❤ ❝❤➜t q✉➯♥ ✭❦❤ỉ♥❣ ♥❤ỵ✮✳
✣✐➲✉ ❣➻ ❦❤✐➳♥ q✉→ tr➻♥❤ P♦✐ss♦♥ ❧➔ q✉→ tr➻♥❤ ❞✉② ♥❤➜t tr♦♥❣ sè ❝→❝
q✉→ tr➻♥❤ ✤ê✐ ♠ỵ✐ ❝â t➼♥❤ q✉➯♥ ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ơ✳

✣à♥❤ ♥❣❤➽❛✳ ✭❈→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ q✉➯♥ ✮✳ ▼ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X

✤÷đ❝ ❣å✐ ❧➔ ❝â t➼♥❤ ❝❤➜t ❦❤ỉ♥❣ ♥❤ỵ ♥➳✉ X ❧➔ ởt ữỡ
ợ ồ x 0 t 0✱
P {X > t + x} = P {X > x}P {X > t}.



ỵ r ởt ♣❤→t ❜✐➸✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➔♠ ♣❤➙♥ ♣❤è✐ ❜ò ❝õ❛
X ✳ ❑❤æ♥❣ ❝â r➔♥❣ ❜✉ë❝ ❣➻ ❣✐ú❛ ❜✐➳♥ ❝è {X > t + x} tr ữỡ tr
ợ ỳ ố {X > t} ❤♦➦❝ {X > x}✳
❱➻ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤ỉ♥❣ ♥❤ỵ X ❧➔ ❝â ♣❤➙♥ ♣❤è✐ ♠ơ✱ P {X >
t} > 0 ✈ỵ✐ ♠å✐ t 0✳ ❉♦ ✤â ❝â t❤➸ ✈✐➳t ✭✷✳✹✮ ♥❤÷

P {X > t + x|X > t} = P {X > x}.

✭✷✳✺✮

◆➳✉ X ✤÷đ❝ ❤✐➸✉ ❧➔ t❤í✐ ❣✐❛♥ ❝❤í ❝❤♦ tỵ✐ ❦❤✐ ♠ët ❦❤→❝❤ ♥ú❛ tợ t
ữủ ổ trữợ t t❤➻ ♣❤➙♥ ♣❤è✐ ❝õ❛ t❤í✐
❣✐❛♥ ❝❤í ❝á♥ ❧↕✐ ✭✤÷đ❝ ❝❤♦ ❜ð✐ x tr➯♥ ✈➳ tr→✐ ❝õ❛ ✭✷✳✺✮✮ ❝ô♥❣ ❧➔ ♣❤➙♥ ♣❤è✐


✷✶

❝õ❛ t❤í✐ ❣✐❛♥ ❝❤í ❜❛♥ ✤➛✉ ✭✤÷đ❝ ❝❤♦ ❜ð✐ ✈➳ ♣❤↔✐ ❝õ❛ ✭✷✳✺✮✮✱ ♥❣❤➽❛ ❧➔ t❤í✐
❣✐❛♥ ❝❤í ❝á♥ ❧↕✐ ❦❤ỉ♥❣ ợ ớ trữợ õ

ử sỷ X t❤í✐ ❣✐❛♥ ❝❤í ✤➸ ♠ët ①❡ ❜✉s ✤➳♥✱ ❜➢t ✤➛✉ tø 0✱

✈➔ ❣✐↔ t❤✐➳t ❧➔ X ❦❤ỉ♥❣ ♥❤ỵ✳ ❙❛✉ ❦❤✐ ❝❤í tø 0 ✤➳♥ t✱ ♣❤➙♥ ♣❤è✐ ❝õ❛ t❤í✐
❣✐❛♥ ❝❤í ❝á♥ ❧↕✐ tø t ❝ơ♥❣ ♥❤÷ ♣❤➙♥ ♣❤è✐ ❝õ❛ t❤í✐ ❣✐❛♥ ❝❤í tø 0✳ ❍➔♥❤
❦❤→❝❤ ✈➝♥ ❝á♥ ♣❤↔✐ ❝❤í✱ t❤❡♦ ♥❣❤➽❛ ð t❤í✐ ✤✐➸♠ t ❦❤ỉ♥❣ tèt ❤ì♥ s♦ ✈ỵ✐
t❤í✐ ✤✐➸♠ 0✳ ▼➦t ❦❤→❝✱ ♥➳✉ ①❡ ❜✉s ❜➻♥❤ t❤÷í♥❣ s➩ tỵ✐ ❝→❝❤ ♥❤❛✉ ✤➲✉ 16
♣❤ót ✈➔ t = 15✱ t❤➻ ❝❤➢❝ ❝❤➢♥ ♥â s➩ tỵ✐ tr♦♥❣ 1 ♣❤ót✱ ✈➔ X õ ợ
ữủ s tữớ tr t❤➻ ♠➦❝ ❞➛✉ ✤➣ ❝❤í 15 ♣❤ót✱ sü
❝❤í ✤đ✐ t✐➳♣ t❤❡♦ ❝❤➢❝ s➩ ❝á♥ r➜t ❞➔✐✱ ✈➻ t❤➳ ♠ët ❧➛♥ ♥ú❛✱ X ❧➔ ❦❤ỉ♥❣
♥❤ỵ✳
▼➦❝ ❞➛✉ ❝→❝ ♣❤➙♥ ♣❤è✐ ❦❤ỉ♥❣ ♥❤ỵ ❝➛♥ ♣❤↔✐ ❧➔ ♣❤➙♥ ♣❤è✐ ♠ô✱ ❝â t❤➸
t❤➜② r➡♥❣ ♥➳✉ t ổ ợ ữủ ử ố ợ t❤í✐ ✤✐➸♠
♥❣✉②➯♥✱ t❤➻ ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝ s➩ trð ♥➯♥ ❦❤ỉ♥❣ ❝â ❦➼ ù❝✳ ❚❤❡♦ ♥❣❤➽❛
♥➔②✱ q✉→ tr➻♥❤ ❇❡r♥♦✉❧❧✐ ✭t❤í✐ ❦ý ✤➳♥ ❧✐➯♥ t✐➳♣ ❝â ♣❤➙♥ ♣❤è✐ ❤➻♥❤ ❤å❝✮
❣✐è♥❣ ♥❤÷ ♠ët ♣❤✐➯♥ ❜↔♥ ❝õ❛ q✉→ tr➻♥❤ P♦✐ss♦♥ ✭❝â ❝→❝ t❤í✐ ❦ý ✤➳♥ ❧✐➯♥
t✐➳♣ ❝â ♣❤➙♥ ♣❤è✐ ❞↕♥❣ ♠ơ✮✳

❇➙② ❣✐í t❛ sû ❞ư♥❣ t➼♥❤ ❦❤ỉ♥❣ ♥❤ỵ ❝õ❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♠ô ✤➸
t➻♠ ♣❤➙♥ ♣❤è✐ ❝õ❛ ❧➛♥ ✤➳♥ ✤➛✉ t✐➯♥ tr♦♥❣ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ s❛✉ ♠ët
t❤í✐ ✤✐➸♠ ❜➜t ❦ý t > 0✳ ❚❛ ❦❤æ♥❣ ❝❤➾ t➻♠ ♣❤➙♥ ♣❤è✐ ♥➔② ♠➔ ❝á♥ ❝❤➾ r❛
r➡♥❣ ❧➛♥ ✤➳♥ ✤➛✉ t✐➯♥ s❛✉ t ❧➔ ✤ë❝ ❧➟♣ ✈ỵ✐ t➜t ❝↔ ❝→❝ ❧➛♥ ✤➳♥ ❝❤♦ tỵ✐ tớ
t tt ỡ t ự ỵ s

ỵ ợ ởt q tr Pss tố ở λ✱ ✈➔ ♠ët t❤í✐

✤✐➸♠ ✤➣ ❝❤♦ ❜➜t ❦ý t > 0✱ ✤ë ❞➔✐ ❝õ❛ ❦❤♦↔♥❣ tø t ✤➳♥ ❧➛♥ ✤➳♥ ✤➛✉ t✐➯♥
s❛✉ t ❧➔ ♠ët ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❦❤æ♥❣ ➙♠ Z ✈ỵ✐ ❤➔♠ ♣❤➙♥ ♣❤è✐ 1 − e−λz ✈ỵ✐
z
0✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ♥➔② ✤ë❝ ❧➟♣ ✈ỵ✐ t➜t ❝↔ ❝→❝ t❤í✐ trữợ
tớ t ở ợ t ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ {N (τ ), τ t}✳
ý t÷ð♥❣ ỡ ừ ỵ r Z ✭✈ỵ✐ ✤✐➲✉ ❦✐➺♥ τ




tớ ừ ố ũ trữợ t ❧➔ t❤í✐ ❣✐❛♥ ❝á♥ ❧↕✐ ❝❤♦ tỵ✐
t❤í✐ ✤✐➸♠ ✤➳♥ ❦➳ t✐➳♣✳ ❱➻ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❜➢t ✤➛✉ t↕✐ τ ❧➔ ❝â ♣❤➙♥ ♣❤è✐
♠ơ✱ ✈➔ ❞♦ ✤â ❦❤ỉ♥❣ ♥❤ỵ✱ Z ✤ë❝ ợ t ợ ồ trữợ
õ ự s tt õ ỵ tữ ♥➔②✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû Z ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø t tớ
t s t rữợ t t ①➨t N (t) = 0 ✭①❡♠ ❍➻♥❤ ✷✳✷✮✳ ❈❤♦ N (t) = 0✱ t❛
t❤➜② r➡♥❣ X1 > t ✈➔ Z = X1 − t✳ ❉♦ ✤â
P {Z > z|N (t) = 0} = P {X1 > z + t|N (t) = 0}

✭✷✳✻✮
= P {X1 > z} = e−λz .

✭✷✳✼✮
❉➜✉ ✤➥♥❣ t❤ù❝ ✭✷✳✻✮ ✤ó♥❣ ✈➻ {N (t) = 0} = {X1 > t} ✤÷đ❝ t❤➸ ❤✐➺♥ tr♦♥❣
❍➻♥❤ ✷✳✶ ✈➔ tø ✭✷✳✸✮✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ð ✭✷✳✼✮ ✤ó♥❣ ✈➻ sû ❞ư♥❣ ✤✐➲✉ ❦✐➺♥
❦❤ỉ♥❣ ♥❤ỵ tr♦♥❣ ✭✷✳✺✮ ✈➔ X1 ❧➔ ♠ơ✳
❚✐➳♣ t❤❡♦ t❛ ①➨t N (t) = n ✭✈ỵ✐ n > 1 tị② þ✮ ✈➔ Sn = τ ✭✈ỵ✐ τ t tị② þ✮✳
❈ì trữớ ủ ụ ữ ợ N (t) = 0✱ ❝❤➾ ❝â ✈➔✐ ❝❤✐ t✐➳t ❦❤→❝
❜✐➺t ✭①❡♠ ❍➻♥❤ ✷✳✸✮✳
= P {X1 > z + t|X1 > t}


✷✸

❚ø ✤✐➲✉ ❦✐➺♥ N (t) = n ✈➔ Sn = τ ✱ ❦❤→❝❤ ✤➳♥ ✤➛✉ t✐➯♥ s❛✉ t ❧➔
❦❤→❝❤ ✤➳♥ ✤➛✉ t✐➯♥ s❛✉ ❦❤→❝❤ ✤➳♥ t↕✐ Sn✱ ♥❣❤➽❛ ❧➔ Z = z tữỡ ự ợ
Xn+1 = z + (t τ )✳
P {Z > z|N (t) = n, Sn = τ }
= P {Xn+1 > z + t − τ |N (t) = n, Sn = τ }
= P {Xn+1 > z + t − τ |Xn+1 > t − τ, Sn = τ }
= P {Xn+1 > z + t − τ |Xn+1 > t − τ }
= P {Xn+1 > z} = e−λz .

✭✷✳✽✮
✭✷✳✾✮
✭✷✳✶✵✮
✭✷✳✶✶✮

❉➜✉ ✤➥♥❣ t❤ù❝ ð ✭✷✳✾✮ ❝â ✤÷đ❝ ✈➻ ❦❤✐ ❝❤♦ Sn = τ t✱ t❛ ❝â {N (t) =
n} = {Xn+1 > t − τ } ✭①❡♠ ❤➻♥❤ ✷✳✸✮✳
❉➜✉ ✤➥♥❣ t❤ù❝ ð ✭✷✳✶✵✮ ❝â ✤÷đ❝ ✈➻ Xn+1 ✤ë❝ ❧➟♣ ✈ỵ✐ Sn✳
❉➜✉ ✤➥♥❣ t❤ù❝ ð ✭✷✳✶✶✮ õ ữủ ổ ợ tr

Xn+1 ❧➔ ♠ơ✳
✣✐➲✉ ❦✐➺♥ ✭✷✳✽✮ ❦❤ỉ♥❣ ❝❤➾ tr➯♥ Sn ♠➔ ❝ơ♥❣ ❝á♥ ❝↔ tr➯♥ S1, ..., Sn−1✳ ❱➻
✤✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈➲ N (τ ) ✈ỵ✐ ♠å✐ τ tr♦♥❣ (0, t]✱ t❛ ❝â
P {Z > z|N (τ ), 0

τ

t}} = e−λz .

✭✷✳✶✷✮

❚✐➳♣ t❤❡♦✱ ①➨t ❞➣② ❝♦♥ ❝→❝ ❦❤♦↔♥❣ t s ởt tớ t
trữợ ợ m 2✱ ❣✐↔ sû Zm ❧➔ ❦❤♦↔♥❣ tỵ✐ ❦➳ t✐➳♣ ❦➸ tø ❦❤♦↔♥❣ tỵ✐ t❤ù
m − 1 s❛✉ t ✤➳♥ ❦❤♦↔♥❣ tỵ✐ t❤ù m s❛✉ t✳ ●✐↔ sû Z tr♦♥❣ ✭✷✳✶✷✮ ❧➔ Z1 ✳
❑❤✐ ❝❤♦ N (t) = n✱ Sn = τ ✱ t❛ t❤➜② r➡♥❣ Zm = Xm+n ✈ỵ✐ m 2✱ ✈➔ ❞♦
✤â Z1, Z2, ... ❧➔ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ❝ị♥❣ ♣❤➙♥ ♣❤è✐ ♠ơ✱ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥
N (t) = n ✈➔ Sn = τ ✳ ❚ø ✤â ✤ë❝ ❧➟♣ ✈ỵ✐ N (t) ✈➔ Sn ✱ ♥➯♥ Z1 , Z2 , ... ❧➔ ❝ị♥❣
♣❤➙♥ ♣❤è✐ ❦❤ỉ♥❣ ✤✐➲✉ ❦✐➺♥ ✈➔ ❝ơ♥❣ ✤ë❝ ❧➟♣ ✈ỵ✐ N (t) ✈➔ Sn✳ ✣✐➲✉ ✤â rã
r➔♥❣ ❧➔ Z1, Z2, ... ✤ë❝ ❧➟♣ ✈ỵ✐ {N (t); 0 < τ t}✳
❱ø❛ q✉❛ ❝❤➾ r❛ r➡♥❣ t❤➸ ❤✐➺♥ ❝õ❛ ♠ët q✉→ tr➻♥❤ P♦✐ss♦♥ ❜➢t ✤➛✉ tø ♠ët
t❤í✐ ✤✐➸♠ ❜➜t ❦ý t > 0 ❧➔ ♠ët ♠æ ❤➻♥❤ ①→❝ s✉➜t ❝õ❛ ♠ët q✉→ tr➻♥❤ ❜➢t