Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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A
a b
F (x) = b−e
a−x
x ∈ [1, ∞) F (x) = 0 x 6∈ [1, ∞)
ξ
c
e
2x
+e
−2x
x ∈ R c
P
{ξ ∈
[0, ln
√
3]}
a b
F (x) = a − e
−bx
x ∈ [0, ∞) F (x) = 0 x 6∈ [0, ∞)
ξ
P
{ξ = n} = a
−n
, n ∈ N a
M
ξ.
ξ
[0, e
−λ
], λ > 0
η = −ln ξ
M
η
D
η
ξ
λ > 0
η = sign(ξ − λ)
M
ξ
D
ξ
ξ
n p
M
ξ = 5
M
ξ
2
= 29
n p
ξ N(a, σ
2
)
M
ξ
2
= 4,
M
(ξ − a)
4
= 3. a σ
2
n
n p 0 ≤ p ≤ 1.
λ > 0
1 A ξ
A
ξ
M
ξ
n
D
ξ
{(x, y) : x
2
+ (y − 1)
2
= 1}
A ξ A
ξ
M
ξ
0.95
0.9
0.8
0.4
x y
xy y/x
x y
x+ y xy
0.09
0.2
0.9
p
P
(B | A) ≥ 1 −
P
(B
c
)/
P
(A)
P
(A) > 0
N
6
36
6
(ξ
1
, ξ
2
)
y
j
x
i
η = ξ
1
ξ
2
n m
k
r
k
n n
K = {(x, y) : |x| < 2a, |y| < a}
A
A r (r < a)
K
[−1, 1]
p q
x
2
+ px + q = 0
A
A x
A
a
A
A
n
n
n
ξ
P
{ξ = k} =
C
k(k+1)
k = 1, 2, . . .
C
P
{ξ ≤ 3}
P
{n
1
≤ ξ ≤ n
2
}
n
ξ
λ > 0
η =
√
ξ ζ =
1
λ
ln ξ
f
ξ
(x) =
1
π(1+x
2
)
η =
2ξ
1−ξ
2
ξ
1
, ξ
2
f
ξ
1
,ξ
2
(x, y) = C(x + y)
(x, y) ∈ [0, 1]
2
f
ξ
1
,ξ
2
(x, y) = 0
C ξ
1
ξ
2
η = max(ξ
1
, ξ
2
)
A
B
1
, B
2
, . . . B
i
∩ B
j
= ∅ i 6= j
A C =
∞
S
j=1
B
j
ξ η
[0, a]
ξη
ξ
η
ξ
1
, ξ
2
, ξ
3
[0, 1]
ξ
1
+ ξ
2
+ ξ
3
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
M
ξ
1
M
ξ
2
D
ξ
1
D
ξ
2
ξ
λ =
1
2
P
{ξ > 2}
P
{ξ > t + 2 | ξ > t}
ξ
f
ξ
(x) =
1
π
1
1+x
2
P
{ξ ≤ 1}
P
{|ξ| ≥ 1}
ξ
N(0, 1)
M
ξ cos ξ
M
ξ
1+ξ
2
ξ
f
ξ
(x) = a sin x x ∈
[0, π]
a
P
(|ξ| <
π
4
)
M
ξ
ξ
N(0, 1)
η =
ξ,
|ξ| ≤ 1,
−ξ,
|ξ| > 1.
η
M
η
D
η
ξ
λ > 0
M
ξ
2
,
M
ξ
1+ξ
N ξ
M
ξ
ξ η ξ ∈ N(0, 1)
η ∈ N(0, 4)
(ξ, η) 0 ≤ x ≤ 1, −4 ≤ y ≤ 0
ξ η ξ ∈ N(0, 4)
η ∈ N(0, 4)
(ξ, η) D = {(x, y) : 1 ≤ min(|x|, |y|), max(|x|, |y|) ≤ 2}
k n
1, 2, . . . , k
n
1
, . . . , n
k
(n
1
+ . . . + n
k
= n)
P
(A | B)
P
(A) = 0.5
P
(B) = 0.4
P
(A \
B) = 0.3.
[0, l]
n
1
n
2
k
1
k
2
n
ξ + η
λ
1
λ
2
x
ξ =
(ξ
1
, ξ
2
, ξ
3
)
1 x −x
x 1 x
−x x 1
.
ξ
[−λ, λ] λ > 0
|ξ| ξ
2
ξ
η
1
= cos ξ, η
2
= sin ξ cov(η
1
, η
2
)
η
1
η
2
ξ
η = 2ξ + 1
ξ
f
ξ
(x) =
1
π(1+x
2
)
η =
ξ
2
1+ξ
2
ζ =
1
ξ
x
2
+ y
2
= 1
A ξ
ξ
(x−1)
2
+y
2
= 1
A A
O ξ OA
P
{|ξ| ≤ 1}
AB a O
ξ OA
ξ
ζ
[0, 1]
η
1
= sin
π
2
ζ
η
2
= cos
π
2
ζ
AB a O
OA ξ
ξ
{1, 2, . . . , 2n}
n
m
x
1
+ x
2
+ ··· + x
k
= n n
{1, 2, . . . , n}
m ≤ M
{1, 2, ··· , M }
n
{a
1
, . . . , a
n
} a
1
a
n
k
{1, 2, . . . , n}
m
2
{1, 2, . . . , 2n}