Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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f
ξ
(x) =
1
π cosh x
F
ξ
(x) =
0
x < 0,
1−cos x
2
0 6 x < π,
1
x > π.
F
ξ
(x) =
0
x < 1,
1 −
1
x
5
x > 1.
(Ω, F,
P
)
ξ = (ξ
1
, ..., ξ
`
) : Ω 7→ R
`
{ξ
1
6 x
1
, ..., ξ
`
6 x
`
} ∈ F
x = (x
1
, ..., x
`
) ∈ R
`
X
R
`
ξ
P
{ξ ∈ X} = 1
ξ ξ
p
ξ
(x) ,
P
{ξ = x}, x ∈ X,
ξ
X
R
`
(p(x))
x∈X
`
(p(x))
x∈X
p(x) > 0 ∀x ∈ X
P
x∈X
p(x) = 1
` = 2
p
ij
2 ξ =
(ξ
1
, ξ
2
) 0 ≤ p
ij
≤ 1 i, j
P
ij
p
ij
= 1
p
i
=
P
{ξ
1
= x
i
} =
X
j
p
ij
q
j
=
P
{ξ
2
= y
j
} =
X
i
p
ij
ξ
1
ξ
2
ξ
1
, ..., ξ
`
ξ = (ξ
1
, ..., ξ
`
)
F
ξ
1
,...,ξ
`
(x
1
, ..., x
`
) ,
P
{ξ
1
6x
1
, ..., ξ
`
6x
`
}, x = (x
1
, ..., x
`
) ∈ R
`
.
` = 2
(ξ
1
, ξ
2
)
K
x
1
,x
2
= (−∞, x
1
] × (−∞, x
2
].
S
v
= (x
0
1
, x
00
1
] × (−∞, x
2
]
P
{(ξ
1
, ξ
2
) ∈ S
v
} = F
ξ
1
,ξ
2
(x
00
1
, x
2
) − F
ξ
1
,ξ
2
(x
0
1
, x
2
);
S
g
= (−∞, x
1
] × (x
0
2
, x
00
2
]
P
{(ξ
1
, ξ
2
) ∈ S
g
} = F
ξ
1
,ξ
2
(x
1
, x
00
2
) − F
ξ
1
,ξ
2
(x
1
, x
0
2
);
S = (x
0
1
, x
00
1
] × (x
0
2
, x
00
2
]
P
{(ξ
1
, ξ
2
) ∈ S} =
= F
ξ
1
,ξ
2
(x
00
1
, x
00
2
) − F
ξ
1
,ξ
2
(x
0
1
, x
00
2
) − F
ξ
1
,ξ
2
(x
00
1
, x
0
2
) + F
ξ
1
,ξ
2
(x
0
1
, x
0
2
).
∆
j
(a
j
,b
j
]
F (x
1
, ..., x
j−1
, ·, x
j+1
, ..., x
`
) ,
, F (x
1
, ..., x
j−1
, b
j
, x
j+1
, ..., x
`
) − F (x
1
, ..., x
j−1
, a
j
, x
j+1
, ..., x
`
).
F
ξ
1
,...,ξ
`
(x
1
, ..., x
`
)
lim
y
1
↓x
1
,...,y
`
↓x
`
F
ξ
1
,...,ξ
`
(y
1
, ..., y
`
) = F
ξ
1
,...,ξ
`
(x
1
, ..., x
`
);
lim
x
j
→−∞
F
ξ
1
,...,ξ
`
(x
1
, ..., x
`
) = 0 j ∈ {1, ..., `}
lim
x
1
→+∞,...,x
`
→+∞
F
ξ
1
,...,ξ
`
(x
1
, ..., x
`
) = 1
lim
x
k+1
→+∞,...,x
`
→+∞
F
ξ
1
,...,ξ
k
,ξ
k+1
,...,ξ
`
(x
1
, ..., x
k
, x
k+1
, ..., x
`
) =
= F
ξ
1
,...,ξ
k
(x
1
, ..., x
k
);
a
j
< b
j
j = 1, ..., `
∆
1
(a
1
,b
1
]
...∆
`
(a
`
,b
`
]
F
ξ
1
,...,ξ
`
(·) =
P
{a
1
< ξ
1
6 b
1
, ..., a
`
< ξ
`
6 b
`
} > 0.
F : R
`
7→ R
`
F
∆
1
(a
1
,b
1
]
...∆
`
(a
`
,b
`
]
F > 0 a
j
, b
j
∈ R a
j
< b
j
j = 1, ..., `
lim
y
1
↓x
1
,...,y
`
↓x
`
F (y
1
, ..., y
`
) = F (x
1
, ..., x
`
);
lim
x
j
→−∞
F (x
1
, ..., x
`
) = 0 j ∈ {1, ..., `}
lim
x
1
→+∞,...,x
`
→+∞
F (x
1
, ..., x
`
) = 1.
ξ `
f
ξ
(·) : R
`
7→ R
+
F
ξ
(x
1
, ..., x
`
) =
x
`
Z
−∞
...
x
1
Z
−∞
f
ξ
(u
1
, ..., u
`
)du
1
...
du
`
x = (x
1
, ..., x
`
) ∈ R
`
f
ξ
(x)
ξ
R
R
`
f
ξ
(x)dx = 1
R
R
`
f
ξ
(x)dx =
+∞
R
−∞
...
+∞
R
−∞
f
ξ
(x
1
, ..., x
`
)dx
1
...
dx
`
;
f
ξ
(x
1
, ..., x
`
) =
∂
`
F
ξ
1
,ξ
2
,...,ξ
`
(x
1
,x
2
,...,x
`
)
∂x
1
∂x
2
...∂x
`
(x
1
, ..., x
`
)
f
ξ
f
ξ
1
,...,ξ
k
(x
1
, ..., x
k
) =
+∞
R
−∞
...
+∞
R
−∞
f
ξ
1
,...,ξ
k
,...,ξ
`
(x
1
, ..., x
`
)dx
k+1
...dx
`
.
f : R
`
7→ R
+
R
R
`
f(x)dx = 1
ξ `
f
ξ
g : R
`
7→ R
M
g(ξ) =
Z
R
`
g(x)f
ξ
(x)dx
ξ `
X g : X 7→ R
X X
P
x∈X
|g(x)|
P
{ξ = x} < ∞
M
g(ξ) =
X
x∈X
g(x)
P
{ξ = x}.
cov(ξ, η) =
M
(ξ −
M
ξ)(η −
M
η)
ξ η
%(ξ, η) =
cov(ξ,η)
√
D
ξ
D
η
ξ η
ξ η %(ξ, η) = 0
cov(ξ, η) =
M
(ξη) −
M
ξ
M
η;
D
(ξ + η) =
D
ξ +
D
η + 2 cov(ξ, η).
ξ
1
, ..., ξ
`
F
ξ
1
,...,ξ
`
(x
1
, ..., x
`
) = F
ξ
1
(x
1
) · ... · F
ξ
`
(x
`
)
(x
1
, ..., x
`
) ∈ R
`
ξ
j
X
j
ξ
j
j = 1, ..., `
ξ
1
, ..., ξ
`
P
{ξ
1
= x
1
, ..., ξ
`
= x
`
} =
P
{ξ
1
= x
1
} · ... ·
P
{ξ
`
= x
`
}
x
1
∈ X
1
, ..., x
`
∈ X
`
ξ
1
, ..., ξ
`
f
ξ
1
,...,ξ
`
(x
1
, ..., x
`
)
f
ξ
1
,...,ξ
`
(x
1
, ..., x
`
) = f
ξ
1
(x
1
) · ... · f
ξ
`
(x
`
).
ξ
1
ξ
2
M
(ξ
1
ξ
2
) =
M
ξ
1
M
ξ
2
ξ
1
ξ
2
cov(ξ
1
, ξ
2
) = 0
ξ
1
ξ
2
D
(ξ
1
+ ξ
2
) =
D
ξ
1
+
D
ξ
2
ξ
1
ξ
2
ξ
2
ξ
1
φ(ξ
1
) = aξ
1
+ b
M
[ξ
2
− φ(ξ
1
)]
2
a =
cov(ξ
1
, ξ
2
)
D
ξ
1
= ρ(ξ
1
, ξ
2
)
s
D
ξ
2
D
ξ
1
,
b =
M
ξ
2
−
cov(ξ
1
, ξ
2
)
D
ξ
1
M
ξ
1
.
a
ξ
2
ξ
1
M
[ξ
2
− φ(ξ
1
)]
2
=
D
ξ
2
(1 − ρ
2
(ξ
1
, ξ
2
))
y =
M
ξ
2
+ a(x −
M
ξ
1
)
ξ
2
ξ
1
(
M
ξ
1
,
M
ξ
2
)
ξ
1
ξ
2
ξ
1
ξ
2
(ξ, η) f
ξ,η
(x, y) = c(x + y)
(x, y) ∈ [0, 1]
2
c
F
ξ,η
(x, y)
P
(ξ 6
1
2
, η 6 1)
M
ξη
c
f
ξ,η
(x, y) > 0,
+∞
Z
−∞
+∞
Z
−∞
f
ξ,η
(x, y) dxdy = 1.
c > 0
1
Z
0
1
Z
0
c(x + y) dxdy = c
1
Z
0
x
2
2
+ yx
1
0
dy = c
1
Z
0
1
2
+ y
dy =
= c
y
2
+
y
2
2
1
0
= c.
c = 1
F
ξ,η
(x, y) =
x
Z
−∞
y
Z
−∞
f
ξ,η
(s, t) dsdt =
x
Z
0
y
Z
0
(s + t) dsdt =
=
y
Z
0
s
2
2
+ ts
x
0
dt =
y
Z
0
x
2
2
+ tx
dt =
x
2
t
2
+
t
2
x
2
y
0
=
x
2
y
2
+
y
2
x
2
(x, y) ∈ [0, 1]
2
x > 1, y ∈ [0, 1]
F
ξ,η
(x, y) =
x
Z
−∞
y
Z
−∞
f
ξ,η
(s, t) dsdt =
1
Z
0
y
Z
0
(s + t) dsdt =
=
y
Z
0
s
2
2
+ ts
1
0
dt =
y
Z
0
1
2
+ t
dt =
t
2
+
t
2
2
y
0
=
y
2
+
y
2
2
.
y > 1, x ∈ [0, 1]
F
ξ,η
(x, y) =
x
Z
−∞
y
Z
−∞
f
ξ,η
(s, t) dsdt =
x
Z
0
1
Z
0
(s + t) dsdt =
=
1
Z
0
s
2
2
+ ts
x
0
dt =
1
Z
0
x
2
2
+ tx
dt =
x
2
t
2
+
t
2
x
2
1
0
=
x
2
+
x
2
2
.
x < 0 y < 0 F
ξ,η
(x, y) = 0
x > 1 y > 1
F
ξ,η
(x, y) =
1
Z
0
1
Z
0
(s + t) dsdt = 1.
F
ξ
(x) =
P
{ξ 6 x} =
P
{ξ 6 x, η < +∞} = F
ξ,η
(x, +∞) =
x
2
+
x
2
2
x ∈ [0, 1] F
ξ
(x) = 1 x > 1 F
ξ
(x) = 0 x < 0
F
η
(y) =
y
2
+
y
2
2
y ∈ [0, 1],
1
y > 1,
0
y < 0.
P
(ξ 6
1
2
, η 6 1) = F
ξ,η
(
1
2
, 1) =
3
8
.
M
ξη
M
g(ξ, η) =
+∞
Z
−∞
+∞
Z
−∞
g(x, y)f
ξ,η
(x, y) dxdy,
g(ξ, η) = ξη
M
ξη =
1
Z
0
1
Z
0
xy(x + y) dxdy =
1
Z
0
1
Z
0
(x
2
y + y
2
x) dxdy =
=
1
Z
0
x
3
3
y + y
2
x
2
2
1
0
dy =
1
Z
0
y
3
+
y
2
2
dy =
y
2
6
+
y
3
6
1
0
=
1
3
.
(ξ, η) f
ξ,η
(x, y) =
c
(4+x
2
)(9+y
2
)
c F
ξ,η
(x, y)
P
(ξ < 2, η <
√
3)
(ξ, η) f
ξ,η
(x, y) = c sin(x + y)
(x, y) ∈ [0,
π
2
]
2
c
F
ξ,η
(x, y)
P
(ξ >
π
4
, η <
π
2
)
ξ, η
F
ξ,η
(x, y) =
1
x >
π
2
y >
π
2
,
sin x sin y
0 6 x <
π
2
0 6 y <
π
2
,
0
x < 0 y < 0.
(ξ, η) D = (−∞,
π
2
]×(
π
4
,
π
2
]∪
(
π
4
,
π
2
] × (−∞,
π
2
]
ξ, η
F
ξ,η
(x, y) =
(1 − 2
−x
)(1 − 3
−y
) x > 0, y > 0,
0
x < 0 y < 0.
(ξ, η) D = (−∞, 2] ×
(−∞, 3] ∪ [1, +∞) × [2, +∞]
ξ, η
[2, 4] [3, 5]
(ξ, η)
D = (−∞, 3] × (−∞, 4] ∪ [2, 4] × [3, 5].
A R
ξ A
(ξ, η)
y
j
x
i
ξ
p
1
=
P
{ξ = 3} = 0, 17 + 0, 10 = 0, 27;
p
2
=
P
{ξ = 10} = 0, 13 + 0, 30 = 0, 43;
p
3
=
P
{ξ = 12} = 0, 25 + 0, 05 = 0, 30.
ξ
x
i
p
i