Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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D(Y
n
) = D
Ã
1
n
n
X
k=1
X
k
!
=
1
n
2
n
X
k=1
D(X
k
) =
d
n
,
(Y
n
)
ε > 0
P (|Y
n
− a| < ε) > 1 −
D(Y
n
)
ε
2
= 1 −
d
nε
2
.
lim
n→∞
P (|Y
n
− a| < ε) > 1.
lim
n→∞
P (|Y
n
− a| < ε ) = 1.
(Y
n
)
a ¤
X
1
, X
2
, . . . , X
n
, . . . n M (X
n
) =
a C > 0 n D(X
n
) 6 C
Y
1
, Y
2
, . . . , Y
n
, . . . Y
n
=
1
n
n
P
k=1
X
k
a ε > 0
lim
n→∞
P
Ã
¯
¯
¯
1
n
n
X
k=1
X
k
− a
¯
¯
¯
< ε
!
= 1.
Y
n
M(Y
n
) = a
X
1
, X
2
, . . . , X
n
C
D
Ã
1
n
n
X
k=1
X
k
!
=
1
n
2
n
X
k=1
D(X
k
) 6
1
n
2
n
X
k=1
C =
C
n
.
P
Ã
¯
¯
¯
1
n
n
X
k=1
X
k
− a
¯
¯
¯
< ε
!
> 1 −
1
ε
2
D
Ã
1
n
n
X
k=1
X
k
!
> 1 −
C
nε
2
.
lim
n→∞
P
Ã
¯
¯
¯
1
n
n
X
k=1
X
k
− a
¯
¯
¯
< ε
!
= 1,
¤
X
1
, X
2
, . . . , X
n
, . . .
M(X
n
) = a D(X
n
) = d ε > 0
lim
n→∞
P
Ã
¯
¯
¯
1
n
n
X
k=1
X
k
− a
¯
¯
¯
< ε
!
= 1.
n
a
(X
1
, X
2
, . . . , X
n
) X
k
k = 1, n
M(X
k
) = a D(X
k
) =
σ
2
1
n
n
X
k=1
X
k
n a
x
1
, x
2
, . . . , x
n
n a
a ≈
1
n
n
X
k=1
x
k
.
A p µ
n
A n ε > 0
lim
n→∞
P
³
¯
¯
¯
µ
n
n
− p
¯
¯
¯
< ε
´
= 1.
n
µ
n
n
p M(µ
n
) = np D(µ
n
) = npq
M
³
µ
n
n
´
= p, D
³
µ
n
n
´
=
pq
n
.
ε > 0
P
³
¯
¯
¯
µ
n
n
− p
¯
¯
¯
< ε
´
> 1 −
pq
nε
2
,
lim
n→∞
P
³
¯
¯
¯
µ
n
n
− p
¯
¯
¯
< ε
´
= 1. ¤
n
A (X
1
, X
2
, . . . , X
n
)
X
k
k = 1, n
X
k
P 1 − p p
1
n
n
X
k=1
X
k
=
µ
n
n
n p
n
A m
P (A) ≈
m
n
.
X
1
, X
2
, . . . , X
n
, . . .
X
n
P
1
6
1
6
1
6
1
6
1
6
1
6
M(X
n
) = 3, 5 D(X
n
) =
35
12
P
Ã
¯
¯
¯
1
1000
1000
X
k=1
X
k
− 3, 5
¯
¯
¯
< 0, 1
!
> 1 −
35
12 · 1000 · 0, 01
≈ 0, 708.
1 −
35
12 · n · 0, 01
> 0, 95
n >
35
12 · 0, 01 · 0, 05
≈ 5833, 3.
¤
X
1
, X
2
, . . . , X
n
, . . .
M(X
n
) = a
n
D(X
n
) = d
n
n = 1, 2, . . .
S
n
=
n
X
k=1
X
k
∼
S
n
=
n
P
k=1
X
k
−
n
P
k=1
a
k
s
n
P
k=1
d
k
=
S
n
− M (S
n
)
σ(S
n
)
.
lim
n→∞
P
µ
S
n
− M (S
n
)
σ(S
n
)
< x
¶
=
1
√
2π
x
Z
−∞
e
−
t
2
2
dt,
∼
S
n
n → ∞
∼
S
n
S
n
∼
S
n
c
n
X
n
c
n
= M(|X
n
− a
n
|
3
)
lim
n→∞
n
P
k=1
c
k
µ
n
P
k=1
d
k
¶
3
2
= 0,
(8.7)
X
1
, X
2
, . . . , X
n
, . . .
n
M(X
n
) = a, D(X
n
) = σ
2
, M(|X
n
− a|
3
) = c
n
P
k=1
c
k
µ
n
P
k=1
σ
2
k
¶
3
2
=
nc
(nσ
2
)
3
2
=
c
σ
√
n
→ 0
n → ∞
lim
n→∞
P
n
P
k=1
X
k
− na
σ
√
n
< x
=
1
√
2π
x
Z
−∞
e
−
t
2
2
dt.
X
n
X
n
P q p
,
0 < p < 1 q = 1 − p
n
P
k=1
c
k
µ
n
P
k=1
σ
2
k
¶
3
2
=
npq(p
2
+ q
2
)
(npq)
3
2
=
p
2
+ q
2
√
npq
→ 0
n → ∞
lim
n→∞
P
µ
µ
n
− np
√
npq
< x
¶
=
1
√
2π
x
Z
−∞
e
−
t
2
2
dt
X
X
P
1
4
1
2
1
4
X
S
12
=
12
X
k=1
X
k
,
X
1
, X
2
, . . . , X
12
X
P (10 6 S
12
6 14) = P (S
12
= 10) + P (S
12
= 11) +
+P (S
12
= 12) + P (S
12
= 13) + P (S
12
= 14) ≈
≈ 0, 1169 + 0, 1473 + 0, 1612 + 0, 1473 + 0, 1169 = 0, 6869.
S
12
= 12
X
1
, X
2
, . . . , X
12
P (S
12
= 12) =
12!
6!0!6!
µ
1
4
¶
6
µ
1
2
¶
0
µ
1
4
¶
6
+
+
12!
5!2!5!
µ
1
4
¶
5
µ
1
2
¶
2
µ
1
4
¶
5
+
12!
4!4!4!
µ
1
4
¶
4
µ
1
2
¶
4
µ
1
4
¶
4
=
=
12!
3!6!3!
µ
1
4
¶
3
µ
1
2
¶
6
µ
1
4
¶
3
+
12!
2!8!2!
µ
1
4
¶
2
µ
1
2
¶
8
µ
1
4
¶
2
+
+
12!
1!10!1!
µ
1
4
¶
1
µ
1
2
¶
10
µ
1
4
¶
1
+
12!
0!2!0!
µ
1
4
¶
0
µ
1
2
¶
12
µ
1
4
¶
0
=
= 0, 1612.
M(X) = 1 D(X) =
1
2
M(S
12
) = 12
D(S
12
) = 6
a = 12 σ =
√
6
Φ
µ
14, 5 − 12
√
6
¶
− Φ
µ
9, 5 − 12
√
6
¶
= 2Φ
µ
2, 5
√
6
¶
= 0, 6922,
S
12
∼
S
12
=
S
12
√
6
.
X
S
n
=
n
X
k=1
X
k
,
k = 1, n X
k
X
n
2
n
12
∼
S
n
=
r
12
n
n
X
k=1
µ
X
k
−
1
2
¶
M
³
∼
S
n
´
= 0, D
³
∼
S
n
´
= 1
n → ∞
n = 12
∼
S
12
=
12
X
k=1
X
k
− 6
X
a α > 0
P (X < α) > 1 −
a
α
.