Кудрявцев А.А. Пособие по теории вероятностей
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f(x)
1
√
2πσ
x
6σ
a
0
F (x)
1
1
2
xa
0
0 1
N(0, 1)
Φ(x) ϕ(x)
a σ
2
Φ
a, σ
2
(x) ϕ
a, σ
2
(x)
Φ
a, σ
2
(x) = Φ
x − a
σ
. (7.8)
Φ(x) =
1
√
2π
x
Z
−∞
e
−
u
2
2
du
ξ
λ > 0
f(x) =
(
λe
−λx
x ≥ 0;
0 x < 0.
ξ ∼ exp(λ)
f(x)
λ
x
0
F (x)
1
x
0
ξ
a ∈ IR σ > 0
f(x) =
1
π
·
σ
σ
2
+ (x − a)
2
, x ∈ IR.
ξ ∼ K(a, σ)
f(x)
1
πσ
xa
0
F (x)
1
x
1
2
a
0
(Ω, F, P) ξ
[0, 1]
Ω = [0, 1] F σ
[0, 1] P
F P(Ω) = 1 ω [0, 1] P{ω} = 0
P F
0 1 B = (a, b) 0 < a < b < 1
0 < P
ξ
(B) = b − a < 1.
P
ξ
x
F (x) ε > 0
ε x
F (x + ε) − F (x − ε) > 0.
F
ξ
(x)
dF (x)/dx = 0
F (+∞) − F (−∞) = 1
x
1
1
3
2
3
1
9
2
9
7
9
8
9
F (x)
1
1
2
1
4
3
4
0
F (x) = 0 x < 0
F (x) = 1 x > 1 [0, 1]
[0, 1/3], [1/3, 2/3]
[2/3, 1] F (x)
[1/3, 2/3] 1/2
F (x) = 1/4 x ∈ [1/9,
2/9] F (x) = 3/4 x ∈ [7/9, 8/9]
K = [0, 1]\
∞
[
n=1
n
[
k=1
"
3k − 2
3
n
,
3k − 1
3
n
#
K
K
K
F (x)
F (x) = p
1
F
1
(x) + p
2
F
2
(x) + p
3
F
3
(x),
p
i
≥ 0 i = 1, 2, 3 p
1
+ p
2
+ p
3
= 1 F
i
(x)
p
i
A = (u, v)
Ω = {(u, v)|u, v ∈ [0, 1]}
ξ
1
= ξ
1
(u, v) = u
ξ
2
= ξ
2
(u, v) =
1
u ≥ v;
−1 u < v.
ξ
1
ξ
2
x > 1
P(ξ
1
< x) = P{(u, v)|u < x, (u, v) ∈ Ω} = P(Ω) = 1.
x ≤ 0
P(ξ
1
< x) = P{(u, v)|u < x, (u, v) ∈ Ω} = P(∅) = 0.
0 < x ≤ 1
A(x) = {(u, v)|u < x, (u, v) ∈ Ω}
1 x
P(ξ
1
< x) =
mes A(x)
mes Ω
= x.
0 1
ξ
1
f(x) =
1
x ∈ [0, 1];
0 x /∈ [0, 1].
1
1
x
0
u
v
ξ
2
P(ξ
2
< x) =
0
x ≤ −1;
1/2 − 1 < x ≤ 1;
1 x > 1,
{ξ
2
< x} = {ξ
2
= −1} = {(u, v)|
u < v, (u, v) ∈ Ω} −1 < x ≤ 1
P(ξ
2
= −1) = P(ξ
2
= 1) =
1/2 ξ
2
ξ
2
=
−1, 1/2;
1, 1/2.
η = ξ
2
1
ξ
1
η 0 1 (−∞, 0] (1, +∞)
P(η < x) 0 < x ≤ 1
P(η < x) = P(ξ
2
1
< x) = P(−
√
x < ξ
1
<
√
x) = P(ξ
1
<
√
x) =
√
x.
ξ
Φ
a, σ
2
(x) Φ
a, σ
2
(ξ)
ξ
e
−ξ
ξ
ξ
2
ξ
ξ
1
η = [ξ]
2
[x] x
ξ
|ξ|
k > 0
f(x) =
(
ax
2
e
−kx
x > 0;
0 x ≤ 0.
a
ξ
λ
√
ξ ξ
2
ξ
λ
1 − exp{−λξ} λ
−1
ξ
ξ
[0, 1] 2ξ + 1
− (1 − ξ)
ξ
0 1
ξ
2
/(1 + ξ
2
) 1/(1 + ξ
2
)
ξ
0 1
2ξ/(1 − ξ
2
) 1/ξ
C > 0
f(x) =
C/x
4
x > 1;
0 x ≤ 1.
C
ξ
1/ξ
(Ω, F, P)
ξ
ξ
Eξ =
Z
Ω
ξ(ω) P(dω). (8.1)
f(x) u(x) [a, b] a =
x
0
< x
1
< . . . < x
n
= b
σ = f(y
1
)[u(x
1
) − u(x
0
)] + . . . + f(y
n
)[u(x
n
) − u(x
n−1
)],
x
i−1
≤ y
i
≤ x
i
i = 1, . . . , n
max
i
{x
i
− x
i−1
} → 0
σ I I
f(x) u(x)
I =
b
Z
a
f(x) du(x).
f(x) u(x)
[a, b]
ξ
Eξ =
+∞
Z
−∞
x dF
ξ
(x), (8.2)
F
ξ
(x) ξ
σ = y
1
[u(x
1
) − u(x
0
)] + . . . + y
n
[u(x
n
) − u(x
n−1
)],
x
i−1
≤ y
i
≤ x
i
i = 1, . . . , n σ
max
i
{x
i
−
x
i−1
} → 0 x y
i
[u(x
i
)−u(x
i−1
)]
Eξ = S
1
− S
2
S
1
y = 1 y = F
ξ
(x) x ≥ 0 S
2
y = F
ξ
(x) x < 0
Z
x dF
ξ
(x) = −
0
Z
−∞
F
ξ
(x) dx +
+∞
Z
0
(1 − F
ξ
(x)) dx. (8.3)
1
F (x)
x
x
i
1
y
i
x
i
0
1
F (x)
x
0
S
1
S
2
Z
x dF
ξ
(x)
+∞
Z
−∞
x dF
ξ
(x)
Eξ = Eη ξ
d
= η
Eξ ξ
x
1
, x
2
, . . .
p
1
, p
2
, . . .
Eξ =
X
i
x
i
p
i
. (8.4)
Eξ ξ
f(x)
Eξ =
Z
xf(x) dx. (8.5)
n
x =
P
n
i=1
x
i
m
i
P
n
i=1
m
i
, (8.6)