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Московский государственный индустриальный университет

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Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

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<<< < Предыдущая 16 17 18 19 20 21 22 23 24 25 26 2728 / 5028 29 30 31 32 33 34 35 36 37 38 39 40 > Следующая >>>

m	Pn(m)
m	m
A n
(n + 1)p − 1 m (n + 1)p.	m
	m
9 · 0.6 − 1 m 9 · 0, 6
m = 5

n = 1 p = 12

12 · 12 − 1 m 12 · 12 .

m = 5 m = 6

	1	5	1	6		1	6	1	5
P11(5) = P11(6) = C115					= C116				≈ 0, 23.

	2		2			2		2

Pn(m)

n → ∞, p ≈/ 0, p ≈/ 1

m − np

e−

P (m)

ϕ(x) =

2 .

≈ √

npq

√

npq

√2π

ϕ(x)

ϕ(x) 0 x 4 ϕ(x)

ϕ(−x) = ϕ(x), ϕ(+∞) = ϕ(−∞) = 0.

		n = 100		m = 50 p = 0,		6
P (40)	1			ϕ	50 − 100 · 0, 6		≈

100	≈ √100 · 0, 6 · 0, 4		·		√100 · 0, 6 · 0, 4

≈ 0, 20 · ϕ(2, 04) = 0, 20 · 0, 05 = 0, 01.

A n

					Φ(x)
			x
1			0	e−	t2
Φ(x) =	√				2	dt.
		2π

Φ(x)

D(y) = (−∞; +∞) Φ(0) = 0

Φ(−x) = −Φ(x)

Φ(+∞) = 0, 5, Φ(−∞) = −0, 5

1			0
				e−	t2
Φ(0) =	√					dt = 0.
					2
		2π

−	√2π				1 dt = −du
	1	−x	2		t = −u
Φ( x) =	1	e−	t	dt =
			2
			2
		0

+∞

e− t22 dt = √2π = Φ(+∞) = √12π

−∞

+∞	t2		1	+∞	t2
e−		dt =		e−		dt
	2				2
			2

2 = −	1			x
	1				e−		u2
									du = −Φ(x).
							2
	√
	√		2π
				0
+∞	t2				1					1		1
+∞	t2				1					1		1
e−	2	dt		=	√			·			√2π =
										2		2
						2π				2		2
0

0−∞

0 x 5

Φ(x)

Pn(m1 m m2)

n → ∞, p ≈/ 0, p ≈/ 1

−

√npq

≈

√npq

P (m

m )

m2 − np

m1 − np

e−

Φ(x) =

√

2π

p A

Pn(m) n

p A n n

Pn(m)	A		m n
n → ∞, p → 0, np → a
	(np)m
Pn(m) ≈		e−np.
	m!
Pn(m)	p ≈ 1		Pn(n − m)
	¯
n − m		A

q = 1 − p ≈ 0

¯
P (B)
np = 100 · 0, 01 = 1				10			11
¯			≈	10	−1		11	−1	≈ 0, 74.
P (B) = P100	(0) + P100	(1)	≈	0! e		+	1! e		≈ 0, 74.

P (B) ≈ 1 − 0, 74 = 0, 36

p A A m n

m n

P	n		− p	ε
		m

P	p		− p	ε
P	n		− p	ε
		m

= P − ε n − p ε

= P np − nε m np + nε ≈

≈

√npq

−

√npq

np + nε − np

np − nε − np

= Φ ε

− Φ − ε

= 2Φ ε

n → ∞, p ≈/ 0, p ≈/ 1

− p

≈ 2Φ ε

p = 0, 1

p = 0, 1 q = 0, 9 ε = 0, 03

− 0, 1

0, 03 = 0, 9544.

√

2Φ 0, 03

= 0, 9544 = Φ(0, 1

√n) =

0, 1

0, 9

= 0, 4772 = 0, 1 n = 2 = n = 400.

400 · 0, 1 − 400 · 0, 03 = 28 400 · 0, 1 + 400 · 0, 03 = 52

P (Hi)

A A

				A
									H1
				H2			H3
P (H1) =	50	=	5	, P (H2) =	34	=	17	,	P (H3) =	36	=	3	.

	12)		12		120		60			120		10

P (A/H1) = 0, 96, P (A/H2) = 0, 93, P (A/H3) = 0, 95.

P (A) = 125 · 0, 96 + 1760 · 0, 93 + 103 · 0, 95 = 0, 949.

P (A) ≈ 0, 95

	A																			H1
						H2
	25			5								5					1
P (H1) =			=		,			P (H2) =								=		,
P (H1) =	30		=	6	,			P (H2) =						30		=	6	,
P (A/H1) =	22	=		11		,			P (A/H2) =							21	=		3	.

	28			14												28			4
A
			5		11			1		3		131
P (A) =				·			+			·		=			.
P (A) =			6	·	14		+	6		·	4	=	168		.

P (A) = 131/168

				Hi	i = 0, 1, 2, 3,
			i
	A		P (Hi) = C20i · C53−i/C253 .
	A
	1		P (A/Hi) = C203	−i/C253		3439
P (A) =	1		(C53 ·C203 +C201 ·C52 ·C193 +C202		·C51 ·C183 +C203 ·C173 ) =	3439	≈ 0, 260.

	(C3	)3				13225
	25

P (A) ≈ 0, 26

A H1 H2 H3

P (H1) = p1p2 = 0, 8 · 0, 9 = 0, 72,

P (H2) = p1q2 + p2q1 = 0, 8 · 0, 1 + 0, 9 · 0, 2 = 0, 26, P (H3) = q1q2 = 0, 2 · 0, 1 = 0, 02;

p1 = 0, 8, p2 = 0, 9, q1 = 1 − p1 = 0, 2, q2 = 1 − p2 = 0, 1.

P (A/H1) = 1, P (A/H2) = 1 − 0, 7 = 0, 3, P (A/H3) = 0.

P (A) = 0, 72 · 1 + 0, 26 · 0, 3 + 0, 02 · 0 = 0, 798.

P (A) ≈ 0, 80

P (Hi)

		Hi(i = 1, 2, 3)			i
				P (Hi) = 1/3		A
P (A/H1) = 5/12,				P (A/H2) = 4/7,	P (A/H3) = 1.
				3		3
	1	P (A/Hi)/	1	j
P (Hi/A) =	3	P (A/Hi)/	3	P (A/Hj ) = P (A/Hi/		P (A/Hj ).
				=1		j=1

								A
P (H1/A) =	5/12	=	35	,	P (H2/A) =	48	,	P (H3/A) =	84	.

	5/12 + 4/7 + 1		167			167			167

P (H1/A) = 35/167 P (H2/A) = 48/167 P (H3/A) = 84/167

40%, 35% 25% 4%, 3% 5%

A H1, H2, H3

P (H1) = 0, 4, P (H2) = 0, 35, P (H3) = 0, 25,

P (A/H1) = 0, 04, P (A/H2) = 0, 03, P (A/H3) = 0, 05.

P (A) = P (Hi)P (A/Hi) = 0, 4 · 0, 04 + 0, 35 · 0, 03 + 0, 25 · 0, 05 = 0, 039.

i=3

P (H1/A) =		P (H1) · P (A/H1)			=	0, 4	· 0, 04			=	16	,
											39
			P (A)			0, 039
P (H2	/A) =			P (H2) · P (A/H2)			=	7	,
								26
				P (A)
P (H3	/A) =			P (H3) · P (A/H3)			=	25	.
								78
				P (A)

P (A) ≈ 0, 04 P (H1/A) = 16/39 P (H2/A) = 7/26 P (H3/A) = 25/78

95%

H1 H2

P (H1) = 0, 95, P (H2) = 0, 05

A
		P (H1/A) = 0, 97,		P (H2/A) = 0, 08.
P (H	/A) =	P (H1) · P (A/H1)	=			0, 95 · 0, 97		0, 08 ≈	0, 9957.
1		P (A)		0, 95	·	0, 97 + 0, 05	·	0, 08 ≈
					·		·

P (H1/A) ≈ 0, 996

60% 40%

3% 5%

85%, 90% 95%

H1 H2 H3

90% 8% 2%

60% 10%

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