Кудрявцев А.А. Пособие по теории вероятностей
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ξ
i
i = 1, . . . , n
ξ
ξ F
i
(x) = P(ξ
i
< x)
F (x
1
, . . . , x
n
)
F
i
(x
i
) = F (+∞, . . . , +∞, x
i
, +∞, . . . , +∞).
ξ
1
, . . . , ξ
n
B
i
∈ B i = 1, . . . , n
P(ξ
1
∈ B
1
, . . . , ξ
n
∈ B
n
) = P(ξ
1
∈ B
1
) · . . . · P(ξ
n
∈ B
n
). (10.1)
([0, 1], B
[0, 1]
, λ) λ
ξ
1
, ξ
2
, . . .
ξ
n
(ω) =
1,
ω ∈ A
n
;
0, ω ∈ A
n
,
A
n
=
2
n−1
[
i=1
2i − 2
2
n
,
2i − 1
2
n
,
n = 1, 2, . . .
ω
ξ
1
(ω)
1
1
1
2
0
ω
ξ
2
(ω)
1
1
1
2
1
4
3
4
0
n
B
i
= {x
i
}
x
1
= . . . = x
n
= 1
P(ξ
1
= 1, . . . , ξ
n
= 1) = P
0,
1
2
n
=
1
2
n
= P(ξ
1
= 1) · . . . · P(ξ
n
= 1).
x
1
, . . . , x
n
ξ
1
, . . . , ξ
n
x
1
, . . . , x
n
∈ IR
F (x
1
, . . . , x
n
) = F
1
(x
1
) · . . . · F
n
(x
n
).
ξ = (ξ
1
,
. . . , ξ
n
)
ξ
1
, . . . , ξ
n
x
1
, . . . , x
n
∈ IR
f(x
1
, . . . , x
n
) = f
1
(x
1
) · . . . · f
n
(x
n
).
ξ η
H(x) G(x)
F (x) = P(ξ + η < x) = P(ξ < x − η) =
+∞
Z
−∞
H(x − y) dG(y).
F (x) =
+∞
Z
−∞
G(x − y) dH(y).
H(x) G(x)
F (x) =
+∞
Z
−∞
H(x − y) dG(y) =
+∞
Z
−∞
G(x − y) dH(y).
F (x) = (H ∗ G)(x) = (G ∗ H)(x) F = H ∗ G = G ∗ H
H(x)
h(x) F (x) = (H ∗ G)(x)
f(x) =
+∞
Z
−∞
h(x − y) dG(y). (10.2)
H(x)
F (x)
G(x) F = G ∗ H
F (x)
F
(s)
(x) =
+∞
Z
−∞
F (x + y) dF (y).
F (x)
F (x)
η
P(η < x) = P(η > −x)
x ∈ IR η
d
= ξ −ξ
1
ξ ξ
1
P(η < x) = P(ξ − ξ
1
< x) = P(ξ
1
− ξ < x) = P(−η < x) = P(η > −x).
ξ η
[0, 1]
ξ + η
h(x) ξ η
f(x) ξ + η
h(x − y) =
1
x − y ∈ [0, 1];
0 x − y 6∈ [0, 1],
h(y) =
1
y ∈ [0, 1];
0 y 6∈ [0, 1].
f(x) =
+∞
Z
−∞
h(x − y)h(y) dy.
x 6∈ [0, 2] x ∈ [0, 2]
x −y y [0, 1] x −1 ≤ y ≤ x
x ∈ [0, 1]
f(x) =
x
Z
0
h(x − y)h(y) dy = x,
x ∈ [1, 2]
f(x) =
1
Z
x−1
h(x − y)h(y) dy = 2 − x.
f(x) = 0
ξ η ξ η
ζ ξ
Eξ = Eζ = 0 η = ξζ ξ η
ξ
Eξη = Eξ
2
ζ = Eξ
2
Eζ = 0 = EξEη.
ρ(ξ, η)
ξ η
ρ(ξ, η) = 0 ξ η
|ρ(ξ, η)| ≤ 1
|ρ(ξ, η)| = 1 ξ η
ξ
1
=
ξ − Eξ
√
Dξ
η
1
=
η − Eη
√
Dη
.
Eξ
1
= Eη
1
= 0 Dξ
1
= Dη
1
= 1
0 ≤ D(ξ
1
± η
1
) = E(ξ
1
± η
1
)
2
= Eξ
2
1
+ Eη
2
1
± 2ρ(ξ, η) = 2 ± 2ρ(ξ, η),
ξ η P(η = a + bξ) = 1
a b 6= 0
ρ(ξ, η) = E
ξ − Eξ
√
Dξ
·
a + bξ − a − bEξ
|b|
√
Dξ
=
b.
|ρ(ξ, η)| = 1
D(ξ
1
± η
1
) = 2 ± 2ρ(ξ, η) = 0.
ξ
1
± η
1
c
P
ξ = ∓
√
Dξ
√
Dη
η ±
√
Dξ
√
Dη
Eη + c
q
Dξ + Eξ
!
= 1.
[0, 1]
ξ [0, 1]
ξ η
ξ
2
+ η
2
= 1
ξ η P(ξ > 0) = P(η >
0) = 3/4 P(ξ + η > 0) = 1/2 ξ η
ξ
1
ξ
2
λ
1
λ
2
λ
1
≤ λ
2
t ≥ 0
P(ξ
1
≤ t) ≥ P(ξ
2
≤ t)
ξ
1
ξ
2
λ
1
λ
2
P(ξ
1
= k|ξ
1
+ ξ
2
= n) k = 1, . . . , n
ξ η
ξ 1 η
[0, 1] ξ + η
ξ η
ξ + η
ξ η
ξ η
ξ η
ξ + η ξ
η
ξ
F (x) F
2
(x, y)
(ξ, ξ)
ξ
F (x) F
2
(x, y)
(ξ, |ξ|)
ξ
[0, 1]
−λ
−1
ξ λ > 0
ξ
ξ |ξ|
ξ P(ξ > 0) = α
P(ξ < 0) = β Eξ = a E|ξ| = b (ξ, ξ)
(Ω, F, P) ξ η
(Ω, F, P)
ξ, ξ
1
, ξ
2
, . . .
ξ
1
, ξ
2
, . . .
ξ n → ∞
P {ω|ξ
n
(ω) −→ ξ(ω)} = 1.
ξ
n
−→ ξ
ξ
1
, ξ
2
, . . .
ξ
ε > 0 n → ∞
P(|ξ
n
− ξ| > ε) −→ 0.
ξ
n
P
−→ ξ
ξ
1
, ξ
2
, . . .
r r = 1
r = 2 ξ
n → ∞
E|ξ
n
− ξ|
r
−→ 0.
ξ
n
(r)
−→ ξ
ξ
1
, ξ
2
, . . .
ξ
n → ∞
F
ξ
1
(x), F
ξ
2
(x), . . . F
ξ
(x)
ξ
n
d
−→ ξ
ξ
1
, ξ
2
, . . .
ξ
f(x) n → ∞
Ef(ξ
n
) −→ Ef(ξ).
ξ
n
=⇒ ξ
F
1
, F
2
, . . . F
f(x) n → ∞
Z
f(x) dF
n
(x) −→
Z
f(x) dF (x).
F
n
(x) =⇒ F (x)
F
ξ
1
, F
ξ
2
, . . . F
ξ
ξ
1
, ξ
2
, . . . ξ
ξ, ξ
1
, ξ
2
, . . .
ξ
1
, ξ
2
, . . . ξ
ϕ
ξ
1
(t), ϕ
ξ
2
(t), . . .
ϕ
ξ
(t) n → ∞ t ∈ IR
ξ
n
−→ ξ
ξ
n
−→ ξ
ξ
n
−→ ξ
ξ
n
=⇒ ξ
ξ
n
−→ ξ
F
n
=⇒ F
ϕ
n
−→ ϕ
P
r
d
( )
П Н
..
ξ
F
ξ, θ
G
θ
F
ξ, θ
(x) ≈ G(x)
F
ξ, θ
G
T
n
(X)
θ n → ∞
T
n
(X)
P
−→ θ
ξ
n
(r)
−→ ξ
ξ
n
P
−→ ξ n → ∞
ε > 0 r > 0
P(|ξ
n
− ξ| ≥ ε) ≤
E|ξ
n
− ξ|
r
ε
r
.
ξ
n
P
−→ ξ
ξ
n
(r)
−→ ξ
n → ∞
([0, 1], B
[0, 1]
, λ) λ
ξ
n
=
e
n
,
0 ≤ ω ≤ 1/n;
0, 1/n < ω ≤ 1.
ξ
n
P
−→ 0
ξ
n
−→ 0
E|ξ
n
|
r
=
e
rn
n
−→ ∞
n → ∞ r > 0
ξ
n
P
−→ ξ
ξ
n
−→ ξ n → ∞
A
0
A
1
A
2
A
3
ω
0
(Ω, F, P) Ω
F σ Ω P
F
A
0
A
0
1/2 A
1
A
1
1/3
A
2
A
2
A
2
A
3
1/4
ξ
n
n = 1, 2, . . .
ξ
n
=
1, ω A
n−1
A
n
;
0, ω A
n−1
A
n
.
ξ
n
P
−→ 0 n → ∞
ω
0
∈ Ω
A
i−1
A
i
ξ
1
(ω
0
), ξ
2
(ω
0
), . . .
ω
0
ξ
1
(ω), ξ
2
(ω), . . .
ω ∈ Ω
ξ
n
P
−→ ξ
ξ
n
P
−→ η
P(ξ = η) = 1
P(|ξ − η| > δ) < ε
ε > 0 δ > 0
P (|X + Y | > x) ≤ P
|X| >
x
2
+ P
|Y | >
x
2
, (11.1)