Колодий Н.А., Мазепа Е.А. Сборник задач и упражнений по теории вероятностей
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1
η
q
1
=
P
{η = 4} = 0, 17 + 0, 13 + 0, 25 = 0, 55;
q
2
=
P
{η = 5} = 0, 1 + 0, 3 + 0, 05 = 0, 45.
η
y
j
q
j
1
(ξ, η)
η ξ
ξ
1
ξ
2
P
{ξ
1
ξ
2
= 0} = 1
P
{ξ
i
= 1} =
P
{ξ
i
= −1} = 1/4, i = 1, 2
M
ξ
1
,
M
ξ
2
,
D
ξ
1
,
D
ξ
2
, cov(ξ
1
, ξ
2
)
ξ η
M
ξ =
1
M
η = 2
D
ξ = 1
D
η = 4
ξ
2
+ 2η
2
−ξη −4ξ + η + 4 (ξ + η + 1)
2
ξ η
η = aξ + b a b
ξ η
ρ(ξ, η) =
cov(ξ,η)
√
D
ξ
√
D
η
cov(ξ, η) =
M
[(ξ −
M
ξ)(η −
M
η)] =
M
(ξη) −
M
ξ
M
η =
=
M
(aξ
2
+ bξ) −
M
ξ
M
(aξ + b) = a
M
(ξ
2
) + b
M
ξ −
M
ξ(a
M
ξ + b) =
= a
M
(ξ
2
) + b
M
ξ −a(
M
ξ)
2
− b
M
ξ = a
D
ξ.
D
η
D
η =
D
(aξ + b) = a
2
D
ξ.
ρ(ξ, η) =
a
D
ξ
√
D
ξ
p
a
2
D
ξ
=
a
|a|
,
a 6= 0 ρ(ξ, η) = 0 a = 0
n
2 ξ
η ξ
η
ξ
[−λ, λ] λ > 0
ξ ξ
2
n
ξ η
a
σ
2
ζ
1
= αξ + βη ζ
2
= αξ − βη
ξ η
P
(ξ = 1) =
P
(ξ = −1) = 1/2,
P
(η = 1) =
P
(η = −1) = 1/4,
P
(η = 0) = 1/2.
ηξ η
ηξ η
ξ
ξ
sign ξ
ξ
F
ξ
(x) = 1−F
ξ
(−x−0) f
ξ
(x) = f
ξ
(−x)
ξ |ξ|
ξ
1
ξ
2
η
1
= ξ
1
+ ξ
2
η
2
= ξ
1
−ξ
2
ξ
[0, 2π], η
1
= cos ξ, η
2
= sin ξ cov(η
1
, η
2
)
η
1
η
2
ξ
1
ξ
n
D
(ξ
1
+. . .+
ξ
n
) =
D
ξ
1
+. . .+
D
ξ
n
ξ
i
ξ
j
i 6= j
ξ η
D
ξ = σ
2
1
D
η = σ
2
2
D
(ξ + η)
ξ, η f
ξ,η
(x, y) = (x +
y)
(x, y) ∈ [0, 1]
2
η ξ
M
ξ,
M
η,
D
ξ,
D
η
ξ, η
f
ξ,η
(x, y) =
3
√
3
π
e
−4x
2
−6xy−9y
2
.
η
ξ
M
ξ =
M
η = 0
σ
1
=
1
√
6
σ
2
=
q
2
27
ρ = −
1
2
ξ, η f
ξ,η
(x, y) =
c cos(x + y)
(x, y) ∈ [0,
π
4
]
2
c
ξ η
ξ
x
i
x
1
x
2
. . . x
n
p
i
p
1
p
2
. . . p
n
η = g(ξ)
η
y
i
y
1
y
2
. . . y
n
q
i
p
1
p
2
. . . p
n
y
i
= g(x
i
)
η = g(ξ)
x
i1
, x
i2
, . . . x
ik
ξ
y
i
= g(ξ
i1
) = g(ξ
i2
) = . . . g(ξ
ik
) q
i
=
P
{η = y
i
} =
p
i1
+ p
i2
+ ··· + p
ik
ξ
F
ξ
(x) η = g(ξ) y = g(x)
F
η
(y) = F
ξ
(g
−1
(y))
ξ g(x)
η
f
η
(y) =
f
ξ
(g
−1
(y))
g
0
(g
−1
(y))
.
F
η
(y)
P
{η ∈ B} =
P
{ξ ∈ g
−1
(B)},
g
−1
(B) B ⊂ R
g
ξ, η
f
ξ
(x) f
η
(y)
ζ
1
= ξ + η ζ
2
= ξη ζ
3
=
ξ
η
f
ζ
1
(x) =
+∞
Z
−∞
f
ξ
(y)f
η
(x − y) dy =
+∞
Z
−∞
f
ξ
(x − y)f
η
(y) dy;
f
ζ
2
(x) = −
0
Z
−∞
1
y
f
ξ
(y)f
η
x
y
dy +
+∞
Z
0
1
y
f
ξ
(y)f
η
x
y
dy;
f
ζ
3
(x) =
+∞
Z
−∞
|y|f
ξ
(xy)f
η
(y) dy.
F
ξ
(x)
ξ
η = sign(ξ)
M
η
η
−1, 0, 1
P
{η = −1} =
P
{ξ < 0} = F
ξ
(0−)
P
{η = 0} =
P
{ξ = 0} = F
ξ
(0) − F
ξ
(0−)
P
{η =
1} =
P
{ξ > 0} = 1 −
P
{ξ 6 0} = 1 − F
ξ
(0)
M
η = −1·F
ξ
(0−)+0·(F
ξ
(0)−F
ξ
(0−))+1·(1−F
ξ
(0)) = 1−F
ξ
(0)−F
ξ
(0−).
ξ
F (x)
1
2
(ξ + |ξ|)
ξ
η = −ln(1 − ξ)
ξ
f
ξ
(x) =
1
x ∈ [0, 1],
0
x 6∈ [0, 1].
f
ξ
(x) 6= 0 x ∈ [0, 1] f
η
(y) 6= 0 y > 0
η = −ln(1 − ξ)
y > 0
x
y = −ln(1 − x) x = 1 − e
−y
x ∈ [0, 1)
ξ (−∞, x]
η
F
η
(y) =
P
{η 6 y} =
P
{ξ 6 x} =
x
Z
−∞
f
ξ
(s) ds =
x
Z
0
1 ds = x = 1 − e
−y
y > 0 y < 0 x < 0 f
ξ
(x) = 0 F
η
(y) =
0
f
η
(y) = F
0
η
(y) =
e
−y
y > 0,
0
y < 0.
ξ
f
ξ
(x) =
1
π(1+x
2
)
η = arctg(ξ)
ξ N(0, 1)
η = ξ
2
ξ
[−1, 2]
η = sign(ξ)
ξ
[0, 1]
η = −
1
λ
ln ξ, λ > 0.
ξ
1
e
−ξ
ξ
b
π((x−a)
2
+b
2
)
, −∞ < x < ∞.
1
ξ
ξ
[0, 1]
η = ln
ξ
1−ξ
.
ξ
[0, 1]
η =
1
ξ
α
α > 0.
AB a O
OA ξ
ξ
AB
A O
η
[0, a]
f
η
(x) =
1
a
x ∈ [0, a],
0
x 6∈ [0, a].
η
η ξ
ξ = 2πη η [0, a]
ξ [0, 2πa]
F
ξ
(y)
F
ξ
(y) =
P
{ξ 6 y} =
P
{η 6
y
2π
} =
y
2π
Z
0
1
a
dx =
y
2πa
,
y ∈ [0, 2πa]
F
ξ
(y) =
P
{ξ 6 y} =
P
{η 6
y
2π
} =
a
Z
0
1
a
dx = 1,
y > 2πa
F
ξ
(y) =
P
{ξ 6 y} =
P
{η 6
y
2π
} = 0,
y < 0 f
ξ
(y)
f
ξ
(y) =
1
2πa
y ∈ [0, 2πa],
0
y 6∈ [0, 2πa].
ξ
1
, ξ
2
AB a O
ξ OA
ξ
ξ
1
, ξ
2
, . . . , ξ
n
[0, θ]
η = max(ξ
1
, . . . , ξ
n
)
M
η
F (x)
[0, θ]
η ξ
i
x ∈
[0, θ]
F
η
(x) =
P
{η 6 x} =
P
{max(ξ
1
, . . . , ξ
n
) 6 x} =
=
P
{ξ
1
6 x, ξ
2
6 x, . . . , ξ
n
6 x} =
P
{ξ
1
6 x}
P
{ξ
2
6 x}. . .
P
{ξ
n
6 x} =
= F
n
(x) =
x
θ
n
.
F
η
(x) = 0 x < 0 F
η
(x) = 1 x > θ
f
η
(x) = F
0
η
(x) =
nx
n−1
θ
n
x ∈ [0, θ],
0
x 6∈ [0, θ].
M
η =
+∞
Z
−∞
xf
η
(x) dx =
θ
Z
0
nx
n
θ
n
dx =
nx
n+1
(n + 1)θ
n
θ
0
=
nθ
n + 1
.
ξ
1
, ξ
2
, . . . , ξ
n
[0, θ]
η =
min(ξ
1
, . . . , ξ
n
)
M
η
ξ η
λ >
0
ζ =
min(ξ, η)
M
ζ
ξ η
λ >
0
ζ =
max(ξ, η)
M
ζ
ξ
1
ξ
2
η = min(ξ
1
, ξ
2
)