Кудрявцев А.А. Пособие по теории вероятностей
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(a, b] [a, b] (−∞, b] a, b ∈ IR
σ σ
Ω = [0, 1] σ σ(B)
Ω B
B = {Ω}
B = Ω
B = {∅}
B = {[0, 2/3], [1/3, 1]}
B = {[0, 1/2], [1/2, 1]}
B = {{0}, {1}}
B = {[1/3, 1/2]}
{B ∩[0, 1], B ∈ B}
σ B
[0, 1]
[0, 1]
5
130 129 128
σ
σ
n
F σ Ω
F
F
A
1
, A
2
, . . .
Ω σ
Ω σ
(Ω, F, P)
ξ : Ω −→ IR
ξ
−1
(B) ≡ {ω|ξ(ω) ∈ B} ∈ F (7.1)
B ∈ B
ξ
−1
(B) B
ω
Ω
ω
ξ(ω)
1
1
1
2
0
Ω = [0, 1] F = {∅, Ω}
ξ(ω)
P(ξ < 1/2)
P
ξ <
1
2
=P{ω|ξ(ω) = 0}=P
ω ∈
1
2
, 1
.
1/2 (1/2, 1]
[0, 1]
(Ω, F) P P(∅) = 0
P(Ω) = 1
ξ(ω)
ξ
−1
((−∞, 1/2)) = (1/2, 1] σ
F
σ F [0, 1/2] (1/2, 1]
P([0, 1/2]) = 0.1 P((1/2, 1]) = 0.9
0.9 F
σ [0, 1]
(Ω, F)
([0, 1], B
[0, 1]
) ξ(ω)
ξ
B
{ξ ∈ B}
σ F
σ
σ B B
IR
σ B
B
IR σ
σ
σ
(S, H) (IR, B)
ξ
ξ
P
ξ
: B −→ IR B ∈ B
P
ξ
(B) = P(ξ ∈ B). (7.2)
P
ξ
P
ξ
P
ξ
(Z) = 1 Z
η(ω)
ζ(ω)
ω
1
1
1
2
1
4
0
([0, 1], B
[0, 1]
, λ) λ
η(ω) ζ(ω)
η ζ
σ B B
1
0 1 B
2
0 1 B
3
0 1 B
4
0 1
B = B
1
∪B
2
∪B
3
∪B
4
B
i
B
j
= ∅ i 6= j
B
1
∈ B
1
P{ω|ζ(ω) ∈ B
1
} = λ([0, 1]\{1/4}) = λ([0, 1]) = P(η ∈ B
1
) = 1.
B
i
∈ B
i
i = 2, 3, 4
P(ζ ∈ B
2
) = λ({1/4}) = λ(∅) = P(η ∈ B
2
) = 0
P(ζ ∈ B
3
) = P(ζ ∈ B
4
) = P(η ∈ B
3
) = P(η ∈ B
4
) =
1
2
.
B
P
η
(B) P
ζ
(B)
ξ
η ξ(ω) = η(ω)
ω ∈ Ω ξ ≡ η
ξ
η
P{ω|ξ(ω) = η(ω)} = 1 ξ = η
ξ
η P
ξ
(B) =
P
η
(B) B ξ
d
= η
ξ
η ζ ξ
ζ
σ
B
ξ
F
ξ
: IR −→ IR
F
ξ
(x) = P(ξ < x) F
ξ
(x) = P
ξ
((−∞, x))
x ∈ IR
x
1
≤ x
2
F (x
1
) ≤ F (x
2
)
lim
x→x
0
−0
F (x) = F (x
0
)
lim
x→−∞
F (x) = 0 lim
x→+∞
F (x) = 1
F (x)
(Ω, F, P)
ξ F (x) = P(ξ < x) x ∈ IR
1
F (x)
x
0
1
F (x)
x
0
1
F (x)
x
0
ξ(ω)
(Ω, F)
(Ω, F) ξ(ω)
g(x) ξ(ω) g(x)
ξ(ω)
g(ξ(ω))
g(x) ξ(ω)
g(ξ(ω))
g(x) ξ(ω) g(x)
ξ(ω)
g(ξ(ω))
g(x)
ξ(ω)
(Ω, F, P) g(ξ(ω))
(Ω, F, P)
ξ
F (x) aξ ξ+b a, b ∈ IR
1/2(ξ + |ξ|)
ξ
ξ
2
|ξ| cos ξ exp{ξ}
ξ ([0, 1],
B
[0, 1]
, λ) λ
ξ ([0, 1], F,
P) F = {Ω, ∅, [0, 1/2], (1/2, 1]} P([0, 1/2]) = 0.1 P((1/2, 1]) = 0.9
ξ
P(ξ = x) = 0
x ∈ IR
ξ
P(ξ = x) = 0
x ∈ IR x P(ξ = x) 6= 0
ξ
P(ξ = x) = 0 x ∈ IR
(Ω, F, P)
ξ
B
P
ξ
(B) = 1
P(ξ = x) > 0
F
ξ
x
x
1
x
2
x
3
. . . x
n
. . .
p
1
p
2
p
3
. . . p
n
. . .
x
i
ξ
x
i
6= x
j
i 6= j p
i
p
i
= P(ξ = x
i
) > 0,
X
i
p
i
= 1.
x
i
p
i
0 1
P(ξ = 0) > 0 P(ξ = 1) > 0 P(ξ ∈ {0, 1}) = 1
σ B B
1
B
2
B
3