Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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µ(ω) A n
Ω =
{ω
1
, ω
2
, . . . , ω
16
}
ω
1
= (1, 1, 1, 1), ω
2
= (1, 1, 1, 0), ω
3
= (1, 1, 0, 1), ω
4
= (1, 1, 0, 0),
ω
5
= (1, 0, 1, 1), ω
6
= (1, 0, 1, 0), ω
7
= (1, 0, 0, 1), ω
8
= (1, 0, 0, 0),
ω
9
= (0, 1, 1, 1), ω
10
= (0, 1, 1, 0), ω
11
= (0, 1, 0, 1), ω
12
= (0, 1, 0, 0),
ω
13
= (0, 0, 1, 1), ω
14
= (0, 0, 1, 0), ω
15
= (0, 0, 0, 1), ω
16
= (0, 0, 0, 0),
µ(ω
1
) = 4, µ(ω
2
) = µ(ω
3
) = µ(ω
5
) = µ(ω
9
) = 3,
µ(ω
4
) = µ(ω
6
) = µ(ω
7
) = µ(ω
10
) = µ(ω
11
) = µ(ω
13
) = 2,
µ(ω
8
) = µ(ω
12
) = µ(ω
14
) = µ(ω
15
) = 1, µ(ω
16
) = 0.
P (ω
1
) =
µ
1
6
¶
4
,
P (ω
2
) = P (ω
3
) = P (ω
5
) = P (ω
9
) =
µ
1
6
¶
3
µ
5
6
¶
1
,
P (ω
4
) = P (ω
6
) = P (ω
7
) = P (ω
10
) = P (ω
11
) = P (ω
13
) =
=
µ
1
6
¶
2
µ
5
6
¶
2
,
P (ω
8
) = P (ω
12
) = P (ω
14
) = P (ω
15
) =
1
6
µ
5
6
¶
3
,
P (ω
16
) =
µ
5
6
¶
4
.
n
A p
n A
k
C
k
n
p
k
(1 − p)
n−k
.
B
n A k
ω µ(ω) = k
B
ω µ(ω) = k
p
µ(ω)
(1 − p)
n−µ(ω)
= p
k
(1 − p)
n−k
,
k k n−
k
k n
n k
P (B) = C
k
n
p
k
(1 − p)
n−k
. ¤
n A k
p
n
(k)
p
n
(k) = C
k
n
p
k
q
n−k
,
q = 1 − p
n k
0, 1, 2, . . . , n
k = 0, 1, . . . , n
p
n
(k
1
, k
2
) n
A k
1
k
2
p
n
(> 1)
n A
p
n
(k
1
, k
2
) =
k
2
X
k=k
1
p
n
(k),
p
n
(> 1) = p
n
(1, n) =
n
X
k=1
p
n
(k) = 1 − p
n
(0) = 1 − q
n
.
p
n
(k) k
A [np + p] np + p
A
k
0
= np + p k
0
− 1 np + p
p
n
(k) k = 0, 1, . . . , n
p
n
(k)
p
n
(k − 1)
=
C
k
n
p
k
q
n−k
C
k−1
n
p
k−1
q
n−k+1
=
(n − k + 1)p
kq
=
=
kq + np + p − kp − kq
kq
= 1 +
np + p − k
kq
.
k < np + p k
p
n
(k)
p
n
(k − 1)
> 1 p
n
(k − 1) < p
n
(k),
k p
n
(k) k > np + p
k
p
n
(k)
p
n
(k − 1)
< 1 p
n
(k − 1) > p
n
(k),
k p
n
(k) np + p
k
0
p
n
(k
0
)
p
n
(k
0
− 1)
= 1 +
np + p − k
0
k
0
q
> 1,
k < k
0
p
n
(k) < p
n
(k
0
),
p
n
(k
0
+ 1)
p
n
(k
0
)
= 1 +
np + p − k
0
− 1
(k
0
+ 1)q
< 1,
k > k
0
p
n
(k) < p
n
(k
0
).
p
n
(k) (k = 0, 1, 2, . . . , n)
p
n
(k
0
) k
0
= [np+p] np+p
k
0
= np + p
p
n
(k
0
)
p
n
(k
0
− 1)
= 1 +
np + p − k
0
k
0
q
= 1,
p
n
(k
0
− 1) = p
n
(k
0
)
p
n
(k) (k = 0, 1, 2, . . . , n) P
n
(k
0
−
1) p
n
(k
0
) ¤
A p =
1
6
p
10
(1) = C
1
10
µ
1
6
¶
1
µ
5
6
¶
9
≈ 0, 323
p
10
(4) = C
4
10
µ
1
6
¶
4
µ
5
6
¶
6
≈ 0, 054
p
10
(0, 2) = p
10
(0) + p
10
(1) + p
10
(2) =
= C
0
10
µ
5
6
¶
10
+ C
1
10
µ
1
6
¶
1
µ
5
6
¶
9
+ C
2
10
µ
1
6
¶
2
µ
5
6
¶
8
≈ 0, 776
p
10
(> 1) =
10
X
k=1
p
10
(k) = 1 −
µ
5
6
¶
10
≈ 0, 838. ¤
n
n
1%
A
p = 0, 01
n
[n · 0, 01] = 10,
10 6 0, 01n < 11.
n
1000 6 n < 1100.
n
p
n
(> 1) > 0, 095 1 −(1 − p)
n
> 0, 95.
p = 0, 01
1 − (0, 99)
n
> 0, 95
(0, 99)
n
6 0, 05
n >
ln 0, 05
ln 0, 99
n > 298.
¤
n p
n
(k)
p
n
(k
1
, k
2
)
n > 100 0 < np < 10
p
n
(k) ≈
(np)
k
k!
e
−np
( ).
np > 10
p
n
(k) ≈
1
√
npq
ϕ(x
k
) ( ),
ϕ(x) =
1
√
2π
e
−
x
2
2
x
k
=
k − np
√
npq
,
p
n
(k
1
, k
2
) = Φ(x
2
) − Φ(x
1
),
Φ(x) =
1
√
2π
x
Z
0
e
−
t
2
2
dt
x
1
=
k
1
− np
√
npq
, x
2
=
k
2
− np
√
npq
.
ϕ(x)
Φ(x) x
x < 0 ϕ(x) = ϕ(−x), Φ(x) = −Φ( −x)
x > 4 ϕ(x) < 0, 0001 x > 5 0, 5 − 10
−7
< Φ(x) < 0, 5
n = 20, p = 0, 01; 0, 1; . . . ; 0, 5; k = 0, 1, 2, 5
p
n
(k)
k p
k p
A
A
n
p
n
(75, n) > 0, 9,
Φ
µ
n − 0, 8n
√
n · 0, 8 · 0, 2
¶
− Φ
µ
75 − 0, 8n
√
n · 0, 8 · 0, 2
¶
> 0, 9.
n − 0, 8n
0, 4
√
n
=
√
n
2
,
n > 75
√
n
2
> 4, 3 Φ(
√
n
2
)
0, 5 − Φ
µ
75 − 0, 8n
0, 4
√
n
¶
> 0, 9
Φ
µ
75 − 0, 8n
0, 4
√
n
¶
6 −0, 4
Φ
µ
0, 8n − 75
0, 4
√
n
¶
> 0, 4.
Φ(x) = 0, 4
n
0, 8n − 75
0, 4
√
n
> 1, 285.
0, 8n − 0, 514
√
n − 75 > 0,
√
n > 10 n > 100 A
¤
n
p A
p
p
α
1
β
1
α
2
β
2