ËÈÊÁÅÇ

Ëåêöèÿ 2: Îñíîâû òåîðèè âåðîÿòíîñòåé

Äìèòðèé Èöûêñîí

ÏÎÌÈ ÐÀÍ

28 ñåíòÿáðÿ 2008

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Ïëàí

• Âåðîÿòíîñòíîå ïðîñòðàíñòâî;

• Ñëó÷àéíûå âåëè÷èíû;

• Averaging argument (íåðàâåíñòâî Ìàðêîâà);

• Ëèíåéíîñòü ìàòåìàòè÷åñêîãî îæèäàíèÿ;

• Óñëîâíûå âåðîÿòíîñòè è íåçàâèñèìîñòü ñëó÷àéíûõ

âåëè÷èí;

• Äèñïåðñèÿ, íåðàâåíñòâî ×åáûøåâà;

• Îöåíêè ×åðíîâà;

• Ìàðêîâñêàÿ öåïü.

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Ëèòåðàòóðà

1 À. À. Áîðîâêîâ. Òåîðèÿ âåðîÿòíîñòåé.

2 Â. Ôåëëåð. Ââåäåíèå â òåîðèþ âåðîÿòíîñòåé è åå

ïðèëîæåíèÿ.

3 Í. Àëîí, Äæ. Ñïåíñåð. Âåðîÿòíîñòíûé ìåòîä.

4 À. Øåíü. Âåðîÿòíîñòü: ïðèìåðû è çàäà÷è.

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σ-àëãåáðà

Îïðåäåëåíèå. Ω íåêîòîðîå ìíîæåñòâî, A ⊆ 2Ω. A íàçûâàåòñÿ

σ-àëãåáðîé, åñëè:

• Ω ∈ A• Äëÿ ïîñëåäîâàòåëüíîñòè Ann∈N, åñëè ∀n ∈ N,An ∈ A, òî⋂

n∈N An ∈ A è⋃

n∈N An ∈ A.• A ∈ A ⇐⇒ A = Ω \ A ∈ A.

Îïðåäåëåíèå. Ω íåêîòîðîå ìíîæåñòâî, A σ-àëãåáðà.Âåðîÿòíîñòíîé ìåðîé íà A íàçûâàåòñÿ îòîáðàæåíèå

p : A → [0, 1]:

• p(Ω) = 1;

• Äëÿ ïîñëåäîâàòåëüíîñòè ïîïàðíî íåïåðåñåêàþùèõñÿ

ìíîæåñòâ Ann∈N âûïîëíÿåòñÿ

p(⋃

n∈N An) =∑

n∈N P(An).

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Âåðîÿòíîñòíîå ïðîñòðàíñòâî

Îïðåäåëåíèå. Âåðîÿòíîñòíûì ïðîñòðàíñòâîì íàçûâàåòñÿ

òðîéêà (Ω,A, p), ãäå

• Ω ïðîñòðàíñòâî ýëåìåíòàðíûõ ñîáûòèé (íåêîòîðîå

ìíîæåñòâî);

• A ⊆ 2Ω ìíîæåñòâî äîïóñòèìûõ ñîáûòèé (σ-àëãåáðà);

• p âåðîÿòíîñòíàÿ ìåðà.

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Ïðèìåðû

Ïðèìåð. Äèñêðåòíîå âåðîÿòíîñòíîå ïðîñòðàíñòâî.

Ω = ω1, ω2, . . . , ωn. A = 2Ω, p1, p2, . . . , pn ≥ 0,∑n

i=1 pi = 1,p(ωi ) = pi .

p(A) =∑

ω∈A p(ω).Çàìå÷àíèå. Ïåðåñå÷åíèå σ-àëãåáð - ýòî σ-àëãåáðà.Ïðèìåð. Ω = R, A ïåðåñå÷åíèå âñåõ σ-àëãåáð, ñîäåðæàùèõâñå îòêðûòûå ìíîæåñòâà íà R (áîðåëåâñêàÿ σ-àëãåáðà B).ρ : R → R+,

∫ +∞−∞ ρ(x)dx = 1.

p(A) =∫A ρ(x)dx .

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Ïðîñòåéøèå ñâîéñòâà âåðîÿòíîñòè

• p(∅) = 0, p(Ω) = 1;

• p(A1 ∪ A2 ∪ · · · ∪ An) ≤∑n

i=1 p(An);

• (ôîðìóëà âêëþ÷åíèé-èñêëþ÷åíèé) p(A1 ∪ A2 ∪ · · · ∪ An) =∑ni=1 p(An)−

∑i<j p(AiAj) + · · ·+ (−1)np(A1A2 . . .An)

• p(A1 ∪ A2 ∪ · · · ∪ An) ≥∑n

i=1 p(An)−∑

i<j p(AiAj)

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Random subsum principle

Îïðåäåëåíèå. Äëÿ x , y ∈ 0, 1n îïðåäåëèì

< x , y >=∑n

i=1 xiyi (mod2).Ëåììà. Ðàññìîòðèì Ω = 0, 1n,A = 2Ω, p(x) = 2−n äëÿ âñåõ

x ∈ Ω. Èçâåñòíî, ÷òî y 6= 0n. Ïóñòü A = x | < x , y >= 1, òîãäàp(A) = 1

2 (èíà÷å Prx∈0,1n< x , y >= 1 = 12).

Äîêàçàòåëüñòâî. Ïóñòü yk = 1, êàæäîìó x ∈ Ω ìîæíî

ñîïîñòàâèòü x (k), ó êîòîðîãî k-é ýëåìåíò èíâåðòèðîâàí,

< y , x >= 1− < y , x (k) >.

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Ñëó÷àéíàÿ âåëè÷èíà

Îïðåäåëåíèå. (Ω,A, p) âåðîÿòíîñòíîå ïðîñòðàíñòâî.

Ñëó÷àéíîé âåëè÷èíîé íàçûâàåòñÿ òàêîå îòîáðàæåíèå

ξ : Ω → R, ÷òî äëÿ âñåõ A ∈ B âûïîëíÿåòñÿ ξ−1(A) ∈ A.

• Åñëè A = 2Ω, òî ëþáîå îòîáðàæåíèå ξ : Ω → R ÿâëÿåòñÿ

ñëó÷àéíîé âåëè÷èíîé.

• Ñëó÷àéíàÿ âåëè÷èíà èíäóöèðóåò âåðîÿòíîñòíóþ ìåðó µ íà

R,B: µ(A) = p(ξ−1(A)).

• Çíàÿ ìåðó µ ìîæíî "çàáûòü"ïðî âåðîÿòíîñòíîå

ïðîñòðàíñòâî. Prξ ∈ A = µ(A).

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Äèñêðåòíîå ðàñïðåäåëåíèå

• Ω = ω1, ω2, . . . , ωn,ξ(ω1) = a1, ξ(ω2) = a2, . . . , ξ(ωn) = an.

• Prξ ∈ A =∑

i :ai∈A µ(ai ).

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Àáñîëþòíî íåïðåðûâíîå ðàñïðåäåëåíèå

Îïðåäåëåíèå. Ðàñïðåäåëåíèå íàçûâàåòñÿ àáñîëþòíî

íåïðåðûâíûì, åñëè ñóùåñòâóåò òàêàÿ ôóíêöèÿ ρ : R → R+,∫ +∞−∞ ρ(x)dx = 1, êîòîðàÿ çàäàåò ìåðó µ ïî ôîðìóëå

µ(A) =∫A ρ(x)dx . Ôóíêöèÿ ρ íàçûâàåòñÿ ïëîòíîñòüþ

ðàñïðåäåëåíèÿ.

Ïðèìåð. U(a; b) ðàâíîìåðíîå ðàñïðåäåëåíèå íà [a; b].

ρ(x) =

1

b − a, x ∈ [a; b]

0, x 6∈ [a; b]

Ïðèìåð. N(µ, σ2) íîðìàëüíîå ðàñïðåäåëåíèå.

ρ(x) = 1σ√

2πe−

(x−µ)2

2σ2 .

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Ìàòåìàòè÷åñêîå îæèäàíèå

Îïðåäåëåíèå. ξ ñëó÷àéíàÿ âåëè÷èíà, µ ìåðà,

èíäóöèðîâàííàÿ ξ. Ìàòåìàòè÷åñêèì îæèäàíèåì ξ íàçûâàåòñÿ

E ξ =∫Ω ξ(ω)dp(ω) =

∫R xdµ.

• Åñëè µ àáñîëþòíî íåïðåðûâíà ñ ïëîòíîñòüþ ρ(x), òîE ξ =

∫ +∞−∞ xρ(x)dx .

• Åñëè µ äèñêðåòíàÿ ìåðà, ïðè êîòîðîé

µ(A1) = p1, . . . , µ(An) = pn, òî

E ξ =∑

ω∈Ω ξ(ω)p(ω) =∑n

i=1 piAi .

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Averaging argument

• a1, a2, . . . , an íåêîòîðûå ÷èñëà, èõ ñðåäíåå

àðèôìåòè÷åñêîå c , òîãäà ñóùåñòâóåò ak ≥ c .

• ξ ñëó÷àéíàÿ âåëè÷èíà, E ξ = m, òîãäà Prξ ≥ m > 0.Äîêàçàòåëüñòâî. Ïóñòü ξ ïðèíèìàåò çíà÷åíèÿ A1 ñ

âåðîÿòíîñòüþ p1, A2 ñ âåðîÿòíîñòüþ p2,..., An ñ

âåðîÿòíîñòüþ pn, ãäå pi > 0,∑

pi = 1.Åñëè âñå Ai < m, òî

m = p1A1 +p2A2 + · · ·+pnAn < p1m+p2m+ · · ·+pnm = m.

Ïðîòèâîðå÷èå!

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Íåðàâåíñòâî Ìàðêîâà

Ëåììà. ξ ýòî íåîòðèöàòåëüíàÿ ñëó÷àéíàÿ âåëè÷èíà. Òîãäà

äëÿ âñåõ k > 0 âûïîëíÿåòñÿ íåðàâåíñòâî Prξ ≥ k E ξ ≤ 1k .

Äîêàçàòåëüñòâî. Îáîçíà÷èì m = E ξ Ïóñòü

A1 ≤ A2 · · · ≤ Ai < mk ≤ Ai+1 ≤ . . .An.

Prξ ≥ km = pi+1 + · · ·+ pn > 1k . Òîãäà

m = p1A1 + p2A2 + · · ·+ pnAn ≥ pi+1Ai+1 + · · ·+ pnAn ≥mk(pi+1 + · · ·+ pn) > m. Ïðîòèâîðå÷èå!

Ïðèìåð. Â ëîòåðåå íà âûèãðûøè óõîäèò 40% ñòîèìîñòè

áèëåòîâ. Áèëåò ñòîèò 100 ðóáëåé. Äîêàæèòå, ÷òî âåðîÿòíîñòü

âûèãðàòü õîòÿ áû 5000 íå áîëåå 1%

• Ìàò. îæèäàíèå âûèãðûøà 40 ðóáëåé.

• 405000 < 1%

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Îöåíêà ñâåðõó

Ëåììà. ξ ∈ [0; 1], m = E ξ. Òîãäà äëÿ âñåõ 0 < c < 1âûïîëíÿåòñÿ íåðàâåíñòâî: Prξ ≤ cm ≤ 1−m

1−cm .Äîêàçàòåëüñòâî. Ïóñòü η = 1− ξ. E η = 1−m.Òîãäà

Prξ ≤ cm = Pr1− η ≤ cm = Prη ≥ 1− cm =

Prη ≥ 1− cm

1−m(1−m)

í-âî Ìàðêîâà≤ 1−m

1− cm.

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Ëèíåéíîñòü ìàòåìàòè÷åñêîãî îæèäàíèÿ

Òåîðåìà. ξ = αξ1 +βξ2, ãäå α, β ∈ R. Òîãäà E ξ = α E ξ1 +β E ξ2.

Äîêàçàòåëüñòâî. E ξ =∑

ω∈Ω ξ(ω)p(ω) =∑ω∈Ω(αξ1(ω) + βξ2(ω))p(ω) = α E ξ1 + β E ξ2.

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Òóðíèð ñ áîëüøèì ÷èñëîì

ãàìèëüòîíîâûõ ïóòåé

• Òóðíèðîì íàçûâàåòñÿ îðèåíòèðîâàííûé ãðàô ìåæäó

ëþáûìè äâóìÿ âåðøèíàìè êîòîðûõ åñòü ðîâíî îäíî

îðèåíòèðîâàííîå ðåáðî.• Ãàìèëüòîíîâ ïóòü ïóòü ïðîõîäÿùèé ïî âñåì âåðøèíàì

ðîâíî 1 ðàç.• Ω = G1,G2, . . . ,G2C2

n ìíîæåñòâî âñåõ òóðíèðîâ íà n

âåðøèíàõ, âñå òóðíèðû ðàâíîâåðîÿòíû.• σ ïåðåñòàíîâêà ÷èñåë îò 1 äî n.

Xσ(G ) =

1, åñëè σ çàäàåò ã.ï. â G

0, èíà÷å

• X =∑

σ Xσ ÷èñëî ãàìèëüòîíîâûõ ïóòåé â ñëó÷àéíîì

ãðàôå.• E X =

∑σ E Xσ = n!

2n−1 .

• Çíà÷èò, ñóùåñòâóåò òóðíèð â êîòîðîì íå ìåíüøå n!2n−1

ãàìèëüòîíîâûõ ïóòåé.17 / 27

Äâóäîëüíûé ïîäãðàô

• Òåîðåìà. Èç ëþáîãî ãðàôà G (V ,E ) ìîæíî âûáðîñèòü íå

áîëåå |E |2 ðåáåð òàê, ÷òîáû îí ñòàë äâóäîëüíûì.

• Äîêàçàòåëüñòâî. Ω = 2V , âñå ïîäìíîæåñòâà

ðàâíîâåðîÿòíû.

• Ïóñòü T ∈ Ω (T ⊆ V ), îïðåäåëèì ñëó÷àéíóþ âåëè÷èíó äëÿ

êàæäîãî ðåáðà (x , y):

Xxy (T ) =

1, åñëè ðîâíî îäíà âåðøèíà èç x , y ñîäåðæèòñÿ â T

0, èíà÷å

• X =∑

(x ,y)∈E Xxy êîëè÷åñòâî ðåáåð, ðîâíî îäíà èç

âåðøèí êîòîðûõ ñîäåðæèòñÿ â T .

• E X =∑

(x ,y)∈E E Xxy = |E |2 .

• Çíà÷èò, ñóùåñòâóåò T ∈ Ω, ÷òî X (T ) ≥ |E |2 .

• Âûêèíåì âñå îñòàëüíûå ðåáðà.

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Óñëîâíûå âåðîÿòíîñòè

• Ω, B ∈ A (B ⊂ Ω), PrB > 0;

• ΩB = A ∩ B|A ∈ A;• PrA|B = Pr AB

PrB ;

• Ïóñòü A1,A2, . . . ,An ïîëíàÿ ñèñòåìà íåñîâìåñòíûõ

ñîáûòèé (AiAj = ∅,⋃

Ai = Ω).

• C = CA1 ∪ CA2 ∪ · · · ∪ CAn.

• (Ôîðìóëà ïîëíîé âåðîÿòíîñòè)

PrC =∑

i PrCAi =∑

i PrC |AiPrAi

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Íåçàâèñèìîñòü

• A,B ⊂ Ω íàçûâàþòñÿ íåçàâèñèìûìè, åñëè

PrAB = PrAPrB;• PrA|B = PrA, PrB|A = PrB;• Ïðèìåð. Ω = 00, 01, 10, 11, âñå èñõîäû ðàâíîâåðîÿòíû.

A ïåðâûé áèò ðàâåí 0, B ñóììà áèòîâ ÷åòíà.

PrA = 12 , PrB = 1

2 , PrAB = 14 ;

• Ñîáûòèÿ Aii∈I íàçûâàþòñÿ âçàèìíî íåçàâèñèìûìè, åñëè

äëÿ âñåõ T ⊆ I âûïîëíÿåòñÿ Pr⋂

i∈T =∏

i∈T PrAi.• (Äèñêðåòíûå) ñëó÷àéíûå âåëè÷èíû ξ è η íàçûâàþòñÿ

íåçàâèñèìûìè, åñëè äëÿ âñåõ a, b ∈ R âûïîëíÿåòñÿ

Prξ = a, η = b = Prξ = aPrη = b.

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Ïðîèçâåäåíèå ìàòîæèäàíèéÒåîðåìà. X1,X2, . . . ,Xn âçàèìíî íåçàâèñèìû. Òîãäà

E[X1X2 . . .Xn] = E[X1] E[X2] . . .E[Xn].Äîêàçàòåëüñòâî.

E[X1X2 . . .Xn] =∑x

x PrX1X2 . . .Xn = X =

∑x1,x2,...,xn

x1x2 . . . xn PrX1 = x1,X2 = x2, . . . ,Xn = xn(íåçàâèñèìîñòü)

=∑x1,x2,...,xn

x1x2 . . . xn PrX1 = x1PrX2 = x2 . . .PrXn = xn =

(∑x1

x1 PrX1 = x1)(∑x2

x2 PrX2 = x2) . . . (∑xn

xn PrXn = xn) =

n∏i=1

E[Xi ]

Îïðåäåëåíèå. Êîâàðèàöèÿ: Cov(X ,Y ) = E[XY ]− E[X ] E[Y ].21 / 27

Äèñïåðñèÿ

Îïðåäåëåíèå. Äèñïåðñèåé ñëó÷àéíîé âåëè÷èíû ξ íàçûâàåòñÿ

âåëè÷èíà D ξ = E(ξ − E ξ)2.D ξ = E(ξ − E ξ)2 = E[ξ2 − 2ξ E ξ + (E ξ)2] = E ξ2 − (E ξ)2 ≥ 0.Ëåììà. (Íåðàâåíñòâî ×åáûøåâà). D ξ = σ2, òîãäà

Pr|ξ − E ξ| ≥ kσ ≤ 1

k2.

Äîêàçàòåëüñòâî. η = (ξ − E ξ)2, E η = σ2.

Pr|ξ − E ξ| ≥ kσ = Prη ≥ k2σ2(íåð-âî Ìàðêîâà)

≤ 1k2 .

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Ëèíåéíîñòü äèñïåðñèè

Òåîðåìà. Åñëè ξ1, ξ2, . . . , ξn ïîïàðíî íåçàâèñèìû, òî

D(ξ1 + ξ2 + · · ·+ ξn) = D ξ1 + D ξ2 + · · ·+ D ξn.

Äîêàçàòåëüñòâî.

D(∑

i

ξi ) = E[(∑

i

ξi −∑

i

E ξi )2] =

E[(∑

i

ξi )2 − 2(

∑i

ξi )(∑

j

E ξj) + (∑

i

E ξi )2] =

∑i

E ξ2i + 2

∑i<j

E ξi E ξj−2∑

i

(E ξi )2 − 4

∑i<j

E ξi E ξj

+∑

i

(E ξi )2 + 2

∑i<j

E ξi E ξj =

=∑

i

E ξ2i −

∑i

(E ξi )2 =

∑i

D[ξi ].

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Îöåíêè ×åðíîâà

• X1,X2, . . . ,Xn âçàèìíî íåçàâèñèìûå ñëó÷àéíûå

âåëè÷èíû, ïðèíèìàþùèå çíà÷åíèÿ èç 0, 1;• X =

∑ni=1 Xi , m = EX ;

• Õî÷åòñÿ ïîëó÷èòü PrX ≥ (1 + δ)m ≤ ÷òî-òî ìàëåíüêîå

• E etXi = (1− pi ) + etpi ;

E etX = E et∑

i Xi = E∏i

etXi =∏i

E etXi =∏i

(1+pi (et−1)) ≤

(1+x≤ex )

≤∏i

epi (et−1) = e

∑i pi (e

t−1) = em(et−1)

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Îöåíêè ×åðíîâà

• E etX = em(et−1)

• PrX ≥ (1 + δ)m = PretX ≥ e(1+δ)mt ≤ E etX

e(1+δ)mt ≤ em(et−1)

e(1+δ)mt

• t = ln(1 + δ)

• PrX ≥ (1 + δ)m ≤ ( eδ

(1+δ)(1+δ) )m ≤ e(δ−(1+δ) ln(1+δ))m ≤

e(δ−(1+δ)(δ−δ2/2))m = e(−δ2/2+δ3/3)m ≤ e−δ2m/6

• PrX ≤ (1− δ)m ≤ ( e−δ

(1−δ)(1−δ) )m ≤ e−δ2m/2

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Îöåíêè ×åðíîâà

• X1,X2, . . . ,Xn îäèíàêîâî ðàñïðåäåëåííûå âçàèìíî

íåçàâèñèìûå ñëó÷àéíûå âåëè÷èíû.

• PrXi = 1 = p,PrXi = 0 = 1− p.E Xi = p,E X = E

∑ni=1 Xi = np;

• Pr|∑

Xi

n − p| ≥ ε = Pr|X − np| ≥ np εp ) ≤ 2e−

ε2n6p

• 10000 ðàç áðîñàëè ìîíåòêó. Îöåíèòü âåðîÿòíîñòü òîãî, ÷òî

âûïàëî áîëüøå 5500 îðëîâ?

• e−( 50010000 )2·10000

3 ≤ 0.00025.

26 / 27

Ìàðêîâñêàÿ öåïü

• Äàí îðèåíòèðîâàííûé ãðàô. Äëÿ êàæäîé âåðøèíû

èçâåñòíû âåðîÿòíîñòè ïåðåõîäîâ ïî ðåáðàì.

"Áëóæäàíèå"ïî òàêîìó ãðàôó ýòî ìàðêîâñêèé ïðîöåññ.

• Ôîðìàëüíî. Âåðøèíû: 1, 2, . . . ,N.• X0,X1,X2, . . . , ïðèíèìàþò çíà÷åíèÿ 1, 2, . . . ,N.• Èçâåñòíû pj ,i = PrXk+1 = j |Xk = i• π(k) = (π

(k)1 , π

(k)2 , . . . , π

(k)n ) ðàñïðåäåëåíèå Xk .

• π(k+1)j =

∑ni=1 PrXk+1 = j |Xk = iPrXk = i =∑n

i=1 pj ,iπ(k)i ;

• π(k+1) = Pπ(k), P = (pj ,i ) ìàòðèöà ïåðåõîäà;

• π(k) = Pkπ(0).

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