Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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20
◦
C
2
◦
C
4
◦
C
X
a σ
P (|X − a| > 2σ).
A n
1
3
A
n = 9000
n = 75000
A n
2
3
A
A n
A
X
1
, X
2
, . . . , X
n
, . . .
K > 0 n D(X
n
) 6 K
ε > 0
lim
n→∞
P
Ã
¯
¯
¯
1
n
n
X
k=1
X
k
−
1
n
n
X
k=1
M(X
k
)
¯
¯
¯
< ε
!
= 1.
Y
X
k
k = 1, 3200 M(X
k
) = 3 D(X
k
) = 2
P (2, 95 < Y < 3, 075)
a X
a
n x
1
, x
2
, . . . , x
n
a ≈
1
n
n
X
k=1
x
k
.
X
1
, X
2
, . . . , X
n
X n
S
n
=
n
X
k=1
X
k
na nσ
2
σ
2
X
P
Ã
¯
¯
¯
1
n
n
X
k=1
X
k
− a
¯
¯
¯
<
3σ
√
n
!
≈ 2Φ(3) ≈ 0, 997.
a
1
n
n
X
k=1
x
k
3σ
√
n
•
•
•
•
U
f
0
(x) =
(
0, x < 0,
x, 0 6 x 6 1,
0, x > 1.
a b 0 6 a < b 6 1
[a; b] b − a
U
U
U
U = 0, α
1
α
2
. . . α
n
. . . ,
α
1
, α
2
, . . . , α
n
, . . .
P (α
n
= 0) = P (α
n
= 1) =
1
2
,
α
1
, α
2
, . . . , α
n
, . . .
0, 1, . . . , 9
P (α
n
= 0) = P (α
n
= 1) = ··· = P (α
n
= 9) =
1
10
.
U
U
U = 0, α
1
α
2
. . . α
n
,
n−1
X
k=1
α
k
2
k
+
U
2
n−1
[0; 1]
n = 1
U
n = 2
P
µ
α
1
2
+
U
2
< x
¶
=
= P (α
1
= 0)P
µ
1
2
U < x
¶
+ P (α
1
= 1)P
µ
1
2
+
1
2
U < x
¶
=
=
1
2
P (U < 2x) +
1
2
P (U < 2x − 1).
P (U < 2x) =
0, x 6 0,
2x, 0 < x 6
1
2
,
1, x >
1
2
,
P (U < 2x − 1) =
0, x 6
1
2
,
2x − 1,
1
2
< x 6 1,
1, x > 1,
P (
1
2
α
1
+
1
2
U < x) =
(
0, x 6 0,
x, 0 < x 6 1,
1, x > 1.
β
n
=
n−1
X
k=1
α
k
2
k
0,
1
2
n−1
,
1
2
n−2
,
1
2
n−2
+
1
2
n−1
, ···,
1
2
+
1
2
2
+ ··· +
1
2
n−1
P (β
n
= 0) = P
µ
β
n
=
1
2
n−1
¶
= P
µ
β
n
=
1
2
n−2
¶
=
= P
µ
β
n
=
1
2
n−2
+
1
2
n−1
¶
= ··· =
= P
µ
β
n
=
1
2
+
1
2
2
+ ··· +
1
2
n−1
¶
=
1
2
n−1
,
P
Ã
n−1
X
k=1
α
k
2
k
+
U
2
n−1
< x
!
= P (β
n
)P (U < 2
n−1
x) +
+P
µ
β
n
=
1
2
n−1
¶
P (U < 2
n−1
x − 1) +
+P
µ
β
n
=
1
2
n−2
¶
P (U < 2
n−1
x − 2) +
+P
µ
β
n
=
1
2
n−2
+
1
2
n−1
¶
P (U < 2
n−1
x − 3) + ··· +
+P
µ
β
n
=
1
2
+
1
2
2
+ ··· +
1
2
n−1
¶
P (U < 2
n−1
x − 2
n−1
+ 1). ¤
n = 2, 3, . . .
P (β
n
< x) −F
0
(x) 6
1
2
n−1
,
F
0
(x) U
β
n
β
n
= U
0
−
U
2
n−1
,
U
0
U
α
1
, α
2
, . . . , α
n−1
U = u
0
P (β
n
< x) = P
³
U
0
−
u
0
2
n−1
< x
´
= P
³
U
0
< x +
u
0
2
n−1
´
=
=
0, x 6 −
u
0
2
n−1
,
x +
u
0
2
n−1
, −
u
0
2
n−1
< x 6 1 −
u
0
2
n−1
,
1, x > 1 −
u
0
2
n−1
6
6
0, x 6 −
u
0
2
n−1
,
1
2
n−1
, −
u
0
2
n−1
< x 6 0,
F
0
(x) +
1
2
n−1
, 0 < x 6 1 −
u
0
2
n−1
,
1 +
1
2
n−1
, 1 −
u
0
2
n−1
< x 6 1,
1, x > 1
6
6 F
0
(x) +
1
2
n−1
.
n = 2, 3, . . .
P (β
n
< x) −F
0
(x) 6
1
2
n−1
. ¤
lim
n→∞
P
Ã
n−1
X
k=1
α
k
2
k
< x
!
= F
0
(x),
k
2
n
k = 0, 1, . . . , 2
n
−1
n
lim
n→∞
P
Ã
n−1
X
k=1
α
k
10
k
< x
!
= F
0
(x),
0, α
1
α
2
. . . α
n
,
α
k
k = 1, n
n