Теория вероятностей и математическая статистика. Билялов Р.Ф. - 9 стр.

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P (Ω) = 1
A B
AB = ∅ P (A + B) = P (A) + P (B)
A
1
⊃ A
2
⊃ ···A
n
⊃ ··· ,
∞
∩
n=1
A
n
= ∅ lim
n→∞
P (A
n
) = 0
P (
¯
A) = 1 − P (A).
A+
¯
A = Ω, A ∩
¯
A = ∅, P (Ω) = 1 = P (A+
¯
A) = P (A)+ P (
¯
A),
P (
¯
A) = 1 − P (A).
P (A + B) = P (A) + P (B) − P (AB).
A + B = A + BΩ = A + B(A +
¯
A) = A + B
¯
A, A(B
¯
A) =
A
¯
AB = ∅, P (A + B
¯
A) = P (A) + P (B
¯
A).
BA + B
¯
A = B, (BA)(B
¯
A) = ∅, P (BA) + P (B
¯
A) = P (B).
P (A + B) = P (A) + P (B) − P (AB).
Ω
|Ω| s Ω = {ω
1
, ω
2
, ..., ω
s
}.
P (ω
1
) = ... = P (ω
s
) ⇒
P (ω
i
) =
1
s
, i = 1, ..., s. P (A) =
|A|
s
.
Ω.
(Ω, , P )
ℵ = {1, 2, ..., N} ω = (i
1
, i
2
, ..., i
n
)
n i
k
∈ ℵ, k = 1, 2, ..., n.
i
k
Ω = {ω}.
N
i
1
i
2
|Ω| = N
n
.
i
1
6= ß
2
6= ··· 6= i
n
,
n ≤ N, |Ω| = N (N − 1) ···(N − (n − 1)).
ℵ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
1) P (Ω) = 1,
2) Äëÿ ëþáîé ïàðû íåñîâìåñòíûõ ìåæäó ñîáîé ñîáûòèé A è B
     ( AB = ∅), âûïîëíÿåòñÿ P (A + B) = P (A) + P (B) (àääèòèâíîå
     ñâîéñòâî),

3) äëÿ ëþáîé ïîñëåäîâàòåëüíîñòè A1 ⊃ A2 ⊃ · · · An ⊃ · · · , òàêîé,
           ∞
     ÷òî ∩ An = ∅, èìååò ìåñòî ðàâåíñòâî lim P (An ) = 0 (íåïðå-
          n=1                                   n→∞
     ðûâíîñòü).

   Ñëåäñòâèå 1. P (Ā) = 1 − P (A).
    A+Ā = Ω, A∩Ā = ∅, ïîýòîìó P (Ω) = 1 = P (A+Ā) = P (A)+P (Ā),
îòêóäà P (Ā) = 1 − P (A).
    Ñëåäñòâèå 2. P (A + B) = P (A) + P (B) − P (AB).
    A + B = A + BΩ = A + B(A + Ā) = A + B Ā, êðîìå òîãî A(B Ā) =
AĀB = ∅, ïîýòîìó P (A + B Ā) = P (A) + P (B Ā). Ñ äðóãîé ñòîðîíû,
BA + B Ā = B, (BA)(B Ā) = ∅, ïîýòîìó P (BA) + P (B Ā) = P (B). Â
èòîãå P (A + B) = P (A) + P (B) − P (AB).
    Êëàññè÷åñêîå îïðåäåëåíèå âåðîÿòíîñòè. Ïóñòü Ω  êîíå÷-
íîå ìíîæåñòâî è ÷èñëî åãî ýëåìåíòîâ |Ω| ðàâíî s: Ω = {ω1 , ω2 , ..., ωs }.
Ïóñòü ýëåìåíòàðíûå ñîáûòèÿ ðàâíîâåðîÿòíû: P (ω1 ) = ... = P (ωs ) ⇒
P (ωi ) = 1s , i = 1, ..., s. Ïîëîæèì P (A) = |A|s .  êà÷åñòâå ýëåìåí-
òîâ äëÿ A âîçüì¼ì âñå âîçìîæíûå ïîäìíîæåñòâà èç Ω. Ó íàñ åñòü
(Ω, A, P )  âåðîÿòíîñòíîå ïðîñòðàíñòâî.
    Óðíîâûå ñõåìû. Ïóñòü ℵ = {1, 2, ..., N }, ω = (i1 , i2 , ..., in ) 
óïîðÿäî÷åííûé íàáîð n ÷èñåë ñ ik ∈ ℵ, k = 1, 2, ..., n.
    1) Ñõåìà ñëó÷àéíîãî âûáîðà ñ âîçâðàùåíèåì. ik ïðîèçâîëüíû,
è Ω = {ω}. Óðíîâàÿ èíòåðïðåòàöèÿ äàííîé ñõåìû òàêîâà. Â óðíå
N ïðîíóìåðîâàííûõ øàðîâ. Øàðû òùàòåëüíî ïåðåìåøèâàåì, çàòåì
âûòàñêèâàåì øàð, çàïèñûâàåì åãî íîìåð i1 , çàòåì øàð êëàäåì îáðàò-
íî, âñå ïåðåìåøèâàåì, âûòàñêèâàåì øàð ñ íîìåðîì i2 , è ò.ä. |Ω| = N n .
    2) Ñõåìà ñëó÷àéíîãî âûáîðà áåç âîçâðàùåíèÿ. i1 6= ß2 6= · · · 6= in ,
(øàðû íå âîçâðàùàåì). n ≤ N, |Ω| = N (N − 1) · · · (N − (n − 1)).
    3) Ñëó÷àéíûå ÷èñëà. Ê íèì ïðèäåì, åñëè ïðèìåíèì óðíîâóþ ñõå-
ìó ñ âîçâðàùåíèåì äëÿ ℵ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
    Ãåîìåòðè÷åñêèå âåðîÿòíîñòè.

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