Кудрявцев А.А. Пособие по теории вероятностей
Подождите немного. Документ загружается.
B
4
B
i
∈ B
i
P(ξ ∈ B
i
)
P(ξ ∈ B
1
) = P(Ω) P(ξ ∈ B
2
) = P(∅)
P(ξ ∈ B
3
) = P(ξ = 0) P(ξ ∈ B
4
) = P(ξ = 1).
p
i
= P(ξ ∈ {x
i
})
p
1
p
2
p
3
1
F (x)
xx
1
x
2
x
3
0
x
i
x
F (x) = P(ξ < x) =
X
i: x
i
<x
P(ξ = x
i
) =
X
i: x
i
<x
p
i
.
x P(ξ = x) > 0
F (x + 0) −F (x) = P(ξ = x)
λ{F
0
(x) 6=
0} = λ{x|P(ξ = x) > 0} = 0
ξ
a P(ξ = a) = 1
ξ = a
ξ
P(ξ = x
i
) = 1/n x
i
∈ IR
i = 1, . . . , n
ξ
n ∈ IN p ∈ (0, 1) P(ξ = k) = C
k
n
p
k
(1 −
p)
n−k
k = 0, . . . , n ξ ∼ Bi(n, p)
Bi(1, p)
ξ
n p
n p
ξ
η
1 p
η = 1
ξ
λ > 0
P(ξ = k) = e
−λ
λ
k
k!
,
k = 0, 1, . . . ξ ∼ P ois(λ)
ξ
p ∈ (0, 1) P(ξ = k) = (1 − p)p
k
k =
0, 1, . . . ξ ∼ G(p)
n = 3
1 p
1
F (x)
xa
0
1
3
1
3
1
3
1
F (x)
xx
1
x
2
x
3
0
1 p
p
1
1
F (x)
x
0
1 2 3 4 5 6
p
i
i
0
e
λ
1 2 3 4 5 6
p
i
i
0
1
p
ξ
P(ξ = i) = 0.2 i = −2, −1, 0, 1, 2
η = −ξ ζ = |ξ|
P(ξ = x
i
) η P(η =
i) = P(ξ = −i) = 0.2 i = −2, −1, 0, 1, 2 ζ
P(ζ = i) = P({ξ = i}∪{ξ = −i}) = 0.4 i = 1, 2 P(ζ = 0) = P(ξ = 0) = 0.2
P(A) = P(B)
A = B
ξ 0 1
η −1 0 1 P(ξ = i, η = j)
i = 0, 1 j = −1, 0, 1
P(·, ·) η = −1 η = 0 η = 1
ξ = 0 1/16 1/4 1/16
ξ = 1 1/16 1/4 5/16
ζ = ξη
ζ
−1 0 1
P(ζ = k) k = −1, 0, 1
P(ζ = −1) = P(η = −1, ξ = 1) =
1
16
,
P(ζ = 0) = P({η = 0, ξ = 0} ∪ {η = 0, ξ = 1}∪
∪{η = −1, ξ = 0} ∪ {η = 1, ξ = 0}) =
5
8
,
P(ζ = 1) = P(η = 1, ξ = 1) =
5
16
.
ξ η
ξ + η ξ − η max{ξ, η} min{ξ, η} ξη |ξ|
ξ 0 1
η −1 0 1 P(ξ = i, η =
j) i = 0, 1 j = −1, 0, 1
ζ
1
= ξ + η ζ
2
= ξ − η
ξ 0 1
η −1 0 1 P(ξ = i, η =
j) i = 0, 1 j = −1, 0, 1
ξ η
P(ζ
1
= i, ζ
2
= j)
ζ
1
ζ
2
P ois(1.5)
Bi(3, 1/3)
G(1/3)
ξ
P(ξ = k) = C/[k(k + 1)] k = 1, 2, . . . C
P(ξ ≤ 3) P(n
1
≤ ξ ≤ n
2
) n
1
, n
2
∈ IN
ξ
P(ξ = k) = C/[k(k + 1)(k + 2)] k = 1, 2, . . .
C P(ξ ≥ 3) P(n
1
≤ ξ ≤ n
2
) n
1
, n
2
∈ IN
ξ −1 1
η −1 0 1 P(ξ = i, η = j)
i = −1, 1 j = −1, 0, 1
P(·, ·) η = −1 η = 0 η = 1
ξ = −1 1/8 1/12 7/24
ξ = 1 5/24 1/6 1/8
ξ η
ξ −1 1
η −1 0 1 P(ξ = i, η =
j) i = −1, 1 j = −1, 0, 1
ζ
1
= ξ + η ζ
2
= ξη
P(ζ
1
= i, ζ
2
= j)
ζ
1
ζ
2
N
P(N = n) = e
−λ
λ
n
n!
, λ > 0, n = 0, 1, . . .
p
0.5
s
S
i
1
S
i
S
i+1
g(s)
0
(S, H)
µ
H µ
P
µ(S) < ∞
g : S −→ IR
g(s) = y
n
y
n
6= y
k
n 6= k s ∈ S
n
S
n
∈ H ∪
∞
n=1
S
n
= S
g
(S, H) g
∞
X
n=1
y
n
µ(S
n
)
Z
S
g dµ
Z
S
g(s) µ(ds). (7.3)
f : S −→ IR S
f
g
n
lim
n→∞
Z
S
g
n
dµ = I
I
(S, H) S = IR H = {IR
∅ }
(IR, B)
(Ω, F, P)
P
ξ
ξ
f(x) B
P
ξ
(B) =
Z
B
f(x) dx. (7.4)
f(x)
ξ
f(x)
+∞
Z
−∞
f(x) dx = 1.
λ
Z
B
f(x) dx =
Z
B
f(x) λ(dx). (7.5)
F
ξ
(x) =
x
Z
−∞
f(u) du, (7.6)
F
ξ
(x)
f(x) =
dF (x)
dx
. (7.7)
IR
P
λ P(B) = 0 B
λ(B) = 0
B
F (x) IR
ε > 0 δ > 0
(a
k
, b
k
) ⊂ IR k = 1, . . . , n
n
X
k=1
(b
k
− a
k
) < δ,
n
X
k=1
|F (b
k
) − F (a
k
)| < ε.
IR
F
0
(x) IR
F (x) = F (−∞) +
x
Z
−∞
F
0
(u) du.
λ
P
µ
f(x)
P
ξ
(B) =
Z
B
f(x) µ(dx),
f(x) = e
−λ
λ
x
x!
x f(x) = 0
F
0
(x)
f(x)
f(x)
B
ξ
a ∈ IR b ∈ IR a < b
[a, b]
f(x) =
1
b − a
x ∈ [a, b];
0 x /∈ [a, b].
ξ ∼ R[a, b]
f(x)
1
b−a
x
b
a
0
F (x)
1
x
b
a
0
ξ
a ∈ IR σ
2
> 0
f(x) =
1
√
2πσ
2
· e
−
(x−a)
2
2σ
2
, x ∈ IR.
ξ ∼ N(a, σ
2
)