Кудрявцев А.А. Пособие по теории вероятностей
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ε > 0
P
η
n
n
− 3.5
≥ ε
.
ξ
i
i
Eξ
i
3.5
E
η
n
n
= E
P
n
i=1
ξ
i
n
=
3.5n
n
= 3.5,
ξ
i
D
η
n
n
= D
P
n
i=1
ξ
i
n
=
P
n
i=1
Dξ
i
n
2
=
n(91/6 − (21/6)
2
)
n
2
,
P
η
n
n
− 3.5
≥ ε
≤
105
36nε
2
.
ξ
Dξ ≥ Eξ
2
2E|ξ| ≤ Dξ + 1
ξ P(0 < ξ <
1) = 1 Dξ < Eξ
ξ
P(|ξ| ≤ c) = 1 Dξ ≤ cE|ξ|
ξ = ξ(ω)
(Ω, F, P) Ω = [0, 1] F = B
[0, 1]
P = λ ξ = ω
2
ξ = ω − 1/2 ξ = sin πω ξ = sin 2πω
d
0 < a ≤ d ≤ b d
[a, b]
ξ
1
, . . . , ξ
n+1
n + 1 p
η
n
i i = 1, . . . , n ξ
i
= ξ
i+1
= 1
P(|η
n
/n − p
2
| ≥ ε)
ε > 0
ξ η
Dξη ≥ Dξ · Dη
ξ
Eξ
4
(Eξ)
4
ξ η Eξ = 1
Eη = 2 Dξ = 1 Dη = 4
ξ
2
+ 2η
2
− ξη − 4ξ + η + 4 (ξ + η + 1)
2
ξ η
q
Dξ −
q
Dη
2
≤ D(ξ + η) ≤
q
Dξ +
q
Dη
2
.
ξ η
E|ξη| ≤
q
Eξ
2
·
q
Eη
2
.
ξ
Eξ(1 − Eξ) ≤ Dξ
(Ω, F, P)
ξ
ξ
1
(ω) + iξ
2
(ω) (ξ
1
, ξ
2
)
i
ξ
ϕ
ξ
: IR −→ C t ∈ IR
ϕ
ξ
(t) = Ee
itξ
=
Z
e
itx
dF
ξ
(x). (9.1)
ϕ
ξ
(t)
F
ξ
(x)
F
ξ
(x) f
ξ
(x)
f
ξ
(x)
ϕ
ξ
(t) =
Z
e
itx
f
ξ
(x) dx.
|ϕ
ξ
(t)| =
Z
e
itx
dF
ξ
(x)
≤
Z
e
itx
dF
ξ
(x) = 1.
ϕ
ξ
(t) = E cos(tξ) + iE sin(tξ). (9.2)
ξ
ϕ
ξ
(0) = 1 |ϕ
ξ
(t)| ≤ 1 t;
ξ a, b ∈ IR
ϕ
aξ+b
(t) = e
itb
ϕ
ξ
(at);
ξ η
ϕ
ξ+η
(t) = ϕ
ξ
(t)ϕ
η
(t);
ϕ
ξ
(t)
E|ξ|
k
< ∞ k ≥ 1 k
ϕ
ξ
(t)
ϕ
(k)
ξ
(t)
t=0
= i
k
Eξ
k
;
ϕ
ξ
(t) = ϕ
ξ
(−t) = ϕ
−ξ
(t).
a
0
, a
1
, . . .
ψ : C −→ C |z| ≤ 1
ψ(z) =
∞
X
n=0
a
n
z
n
.
ξ
ψ
ξ
: C −→ C |z| ≤ 1
ψ
ξ
(z) = Ez
ξ
.
0, 1, . . .
p
0
, p
1
, . . .
ψ
ξ
(z) = Ez
ξ
=
∞
X
n=0
z
n
p
n
,
p
0
, p
1
, . . .
ψ
ξ
(e
it
) = ϕ
ξ
(t)
ξ η
ψ
ξ+η
(t) = ψ
ξ
(t)ψ
η
(t);
E|ξ|
k
< ∞ k ≥ 1 ψ
(k)
ξ
(z)
z=1
= Eξ
[k]
.
ξ
F
ξ
ϕ
ξ
ψ
ξ
f
ξ
ξ
k
E
ξ
P
P
ξ
ϕ
ξ
(t) ψ
ξ
(z)
ξ = a e
ita
z
a
Bi(n, p) (1 + p(e
it
− 1))
n
(1 + p(z − 1))
n
P ois(λ) e
λ(e
it
−1)
e
λ(z−1)
G(p)
1 − p
1 − pe
it
1 − p
1 − pz
R[a, b]
e
itb
− e
ita
it(b − a)
R[−a, a]
sin at
at
N(a, σ
2
) e
ita−
σ
2
t
2
2
exp(λ)
λ
λ − it
K(a, σ) e
ita−σ|t|
a
cos t
cos t
cos t =
e
it
+ e
−it
2
=
1
2
· e
1·it
+
1
2
· e
−1·it
.
e
−i|t|
e
−i|t|
ξ
ϕ
−ξ
(t) = ϕ
ξ
(−t) = e
−i|t|
, ∀t ∈ IR,
ξ −ξ
ξ
ϕ
ξ
(t) = ϕ
ξ
(t).
e
−i|t|
ξ
ψ(z) Eξ Dξ
Eξ = Eξ
[1]
= ψ
0
(z)
z=1
,
Dξ = Eξ
2
− (Eξ)
2
= Eξ(ξ − 1) + Eξ − (Eξ)
2
= Eξ
[2]
+ Eξ(1 − Eξ) =
= ψ
(2)
(z)
z=1
+ ψ
0
(z)
z=1
1 − ψ
0
(z)
z=1
.
ξ ∼ P ois(λ
1
) η ∼ P ois(λ
2
)
ζ = ξ + η
ϕ
ζ
(t)
ξ
ϕ
ξ
(t) = Ee
itξ
=
∞
X
k=0
e
itk
e
−λ
1
λ
k
1
k!
= e
−λ
1
∞
X
k=0
(λ
1
e
it
)
k
k!
= e
−λ
1
e
λ
1
e
it
= e
λ
1
(e
it
−1)
.
ϕ
ζ
(t) = ϕ
ξ
(t)ϕ
η
(t) = e
(λ
1
+λ
2
)(e
it
−1)
.
ψ(z) = |z|
ξ η ξ
0 1 1/2 η 0 1 2 3
1/8 1/4 1/2 1/8
ζ ξ ξ + ζ = η
ξ η
ξ
2
+ η
2
1/2
cos
2
t exp{−t
2
}
ξ η
ξ + η 0 1 2 1/3
ξ η
(1 + z)
2
/4 p(1 − (1 − p)z)
−1
F (x)
F (x) = 1 − F (−x + 0)
ϕ(t)
ϕ(t) =
+∞
Z
−∞
cos tx dF (x),
F (x)
ξ η
ϕ(t)
ξ − η
F (x)
ϕ(t) ϕ(t)
F (x)
ϕ(t) |ϕ(t)|
2
ξ η
ξ−η
|ξ| − |η| ξ
(Ω, F, P)
ξ
1
, . . . , ξ
n
ξ(ω) = (ξ
1
(ω), . . . , ξ
n
(ω))
n
IR
n
P(ξ ∈ B) = P{ω|(ξ
1
(ω), . . . , ξ
n
(ω)) ∈ B},
B ∈ B
n
B
n
σ IR
n
ξ
ξ
1
, . . . , ξ
n
x
1
, . . . , x
n
F (x
1
, . . . , x
n
) = P(ξ
1
< x
1
, . . . , ξ
n
< x
n
)
ξ
ξ
1
, . . . , ξ
n
f(x) = f(x
1
, . . . , x
n
) x ∈ IR
n
B ∈ B
n
P(ξ ∈ B) =
Z
B
f(x) dx ≡
Z
. . .
Z
B
f(x
1
, . . . , x
n
) dx
1
. . . dx
n
,
f(x) ξ
ξ
1
, . . . , ξ
n