Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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2n 1, 2, . . . , 2n
k 1, 2, . . . , k
n
m
1
m
2
m
k
k m
1
+ . . . + m
k
= n
2n
n
m
1
m
2
m
1
+ m
2
< n
n k
k m
n
r r ≤ n
ABC
B AC
(2, 0)
{(x, y) : x
2
+ y
2
≤ 1, y ≥ 0}
(0, 1) [0,
√
3]
{(x, y) : x
2
+ y
2
= R
2
}
(R, 0) r
(r < 2R)
x y [0, 1]
x + iy
1
[0, 1]
p q
x
2
+px + q = 0
[−1, 2]
A B {(x, y) : |x| <
1, y = 0}
{(x, y) : |y| < 1, x = 0}
A B
K = {(x, y) : |x| + |y| < a}
A A
r (
√
2r < a) K
P
(A) = a
P
(B) = b
P
(A ∩B) =
c
P
(A
c
∪ B
c
)
P
(A
c
| B)
P
(A
c
4B)
P
(A) = a
P
(B) = b
P
(A ∩B) =
c
P
(A
c
∩ B
c
)
P
(A | B
c
)
P
(A
c
4B
c
)
P
(A) = a
P
(B) = b
P
(A ∩B) =
c
P
(A
c
∩ (A ∪ B))
P
(A
c
| B
c
)
P
(A4B)
P
(A) = a
P
(B) = b
P
(A ∩B) =
c
P
(A ∪ (A
c
∩ B))
P
(A
c
∪ B)
P
(A ∪ B | B
c
)
P
(A) = a
P
(B) = b
P
(A ∩B) =
c
P
(A
c
∩ B)
P
((B \ A)
c
)
P
(A4B | A
c
)
n
n m
α β
A
B
B
A B
C α β γ
B
α β
γ
A B C
A B4C
A B A ⊂ B
P
(A) = 0
P
(B) = 1
A B A C C ⊂
B
A B \ C
P
(B) > 0
P
(A | B) +
P
(A
c
) = 1
A B
A B A C A
B ∩ C A B ∪ C
m
n
p A = {
}
B = {
} C = { n
}
n
k
A = { k }
B = { } C = {
}
n m
A = { }
B = {
} C = { n
}
n m
p A =
{
} B = {
} C = {
m }
l n m l ≤
min(n, m)
k
A = { l
} B = { k
} C = { l
}
A B
B A
A B
A
B A
A B
A B A
1, 2, . . .
n
1
n+1
ξ
f
ξ
(x) = ln x, x ∈ [1, λ] λ
M
ξ.
ξ
F
ξ
(x) =
a − be
−x
− e
−x
2
x ≥ 0 a ≥ 0 b ≥ 0
a b
M
ξ
ξ
f
ξ
(x) = cx
2
e
−x
3
, x > 0 c
M
ξ
n
c
c
k!
, n ∈ N
ξ
M
2
ξ
ξ
f
ξ
(x) = cx
−2/3
exp{−x
2/3
}, x > 0 c
M
ξ.
[0, 1] A B
ξ A B
0 1 ξ
M
ξ
D
ξ
{(x, y) : x
2
+ y
2
≤ 1}
A ξ A
ξ
M
ξ
2 A ξ
A
ξ
M
ξ
D
ξ
ξ N(0, 1/2)
η = |ξ|
3
M
η
ξ
[0, 1]
η =
1
ξ
M
√
ξ
M
ξ
[0, 1]
1/3 ξ
ξ
M
ξ
n
n p 0 ≤ p ≤ 1
ξ
0 < p < 1
M
ξ = 2
p
ξ
λ > 0
M
ξ
2
= 1 λ
(ξ, η)
y
j
x
i
(ξ, η)
y
j
x
i
ξ
P
{ξ = k} =
1
5
k =
−2, −1, 0, 1, 2
η
1
=
−ξ
η
2
= |ξ|
ξ η
P
{ξ = η}
(ξ, η)
y
j
x
i
η
1
= ξ
1
+ ξ
2
η
2
= ξ
1
ξ
2
ξ
1
ξ
2
[0, 1] [0, 2]
η = ξ
1
ξ
2
ξ
1
ξ
2
[0, 1]
η = ξ
1
+ ξ
2
ξ
1
ξ
2
λ = 1
P
{|ξ
1
− ξ
2
| ≤ 1}
ξ + η ξ η
[0, 1] λ = 1
ξ
1
ξ
2
N(0, 1)
P
{|ξ
1
− ξ
2
| ≤
1}
ξ
1
, ξ
2
f
ξ
1
,ξ
2
(x, y) = x + y
(x, y) ∈ [0, 1]
2
f
ξ
1
,ξ
2
(x, y) = 0
cov(ξ
1
, ξ
2
)
ξ
[0, 1] ρ(aξ, bξ)
a, b > 0
ξ η
M
ξ =
M
η = 0
M
ξ
3
< ∞
M
η
3
< ∞
M
(ξ + η)
3
=
M
ξ
3
+
M
η
3
ξ
[0, 1] ρ(ξ −
1
2
, (ξ −
1
2
)
2
)
x
ξ = (ξ
1
, ξ
2
, ξ
3
)
1 x x
x 1 x
x x 1
.
ξ, η
F
ξ,η
(x, y) =
0
min(x, y) < 0,
min(x, y)
0 ≤ min(x, y) ≤ 1,
1
min(x, y) > 1.
(ξ, η)
D = ([−2, 2] × [−3, 3]) \ ([−1, 1] × [−2, 2]).
ξ η ξ ∈ N(0, 1)
η ∈ N(0, 4)
(ξ, η) |x| ≤ 1, |y| ≤ 2
ξ η ξ ∈ N(0, 1)
η ∈ N(0, 4)
(ξ, η) 0 ≤ x ≤ 2, |y| ≤ 2
ξ η ξ ∈ N(0, 1)
η ∈ N(0, 4)
(ξ, η) 0 ≤ x ≤ 2, 0 ≤ y ≤ 4
ξ η ξ ∈ N(0, 4)
η ∈ N(0, 4)
(ξ, η) D = {(x, y) : 2 ≤ min{|x|, |y|}, max{|x|, |y|} ≤ 3}
ξ
(n, p) 0 6 p 6 1 n ∈ N
P(ξ = k) = C
k
n
p
k
(1 − p)
n−k
, k = 0, ..., n.
ξ
p 0 < p < 1
P(ξ = k) = p(1 − p)
k
, k = 0, 1, ....
ξ
λ > 0
P(ξ = k) =
λ
k
k!
e
−λ
, k = 0, 1, ....
[a, b] a < b
f(x) =
1
b − a
χ(x, [a, b]).
a ∈ R σ
2
σ > 0
f(x) =
1
√
2πσ
exp
n
−
(x − a)
2
2σ
2
o
.
λ > 0
f(x) = λe
−λx
χ(x, [0, ∞[).
α > 0 β > 0
f(x) =
β
α
Γ(α)
x
α−1
e
−βx
χ(x, ]0, ∞[).