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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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F (x)		ϕ(x)
F (x)		ϕ(x)
	P {x1 ξ < x2} = F (x2) − F (x1) = x2		ϕ(x)dx.
		x1
	+∞
		ϕ(x)dx = P {−∞ < ξ < +∞} = 1.
	−∞
[x1; x2)		ϕ(x)
		ϕ(x)

x1 x2 x

x2 x1

(x; x + x)

ϕ(x) · x

ϕ(x)

ϕ(x) ξ (x; x + x)

ϕ(x) =	C	x [0; 4];
	0	x / [0; 4].
C		P {0 < ξ < 3}

ϕ(x)

a	x	b	x
	ϕ(x)	ϕ(x)

0	4				1
	Cdt = 1 =		C · 4 = 1 =	C =	1	.

					4
ϕ(x) =		1/4	x [0; 4];
		0	x / [0; 4].

			0	3
		P {0 < ξ < 3} =		1		1	3	3

					4 dt =	4 t	0	= 4 .
					4 dt =	4 t	0	= 4 .
C =	1	; P {0 < ξ < 3} =	3

	4		4

ξ [a; b] n

	x1, x2, . . . , xn
	Ci (i = 1, 2, . . . , n)
ξ	Ci	Pi = ϕ(Ci)Δxi

Ciϕ(Ci)Δxi.

xϕ(x)dx

ξ ϕ(x)

+∞

M (ξ) = xϕ(x)dx.

−∞

D(ξ) = M ξ − M (ξ) 2.

+∞

D(ξ) = x − M (ξ) ϕ(x)dx.

−∞

+∞

D(ξ) = x2ϕ(x)dx − M (ξ) 2.

−∞

σ(ξ) = D(ξ).

x [0; 4] M (ξ) =

ϕ(x)

1		x2		4
			0

4	xdx =	8		= 2

	ϕ(x)

x [0; 4]	M (X) = 2
4	1						x3					4		16	4
0	1						x3					4		16	4

D(ξ) =	x2 ·	4	dx − (2)2 =				12			0			− 4 =	3	− 4 =	3	.

			σ(x)
			σ(x)

	4					2
	σ(ξ) =				=	√					≈ 1, 2.
				3
				3					3
	4				≈	2
M (ξ) = 2, D(ξ) = 3 , σ(ξ)						√
M (ξ) = 2, D(ξ) = 3 , σ(ξ)						√		3

xi pi

M1(4; 0, 1), M2(6; 0, 3), M3(7; 0, 2), M4(10; 0, 3), M5(11; 0, 1)

						ξ
P {ξ = 0} = 1 −		1	=	5	,	P {ξ = 1} =	1	.

		6		6			6

	ξ

	p

x1 = 0, x2 = 1, x3 = 2, x4 = 3.

P {ξ = k} = Cnk · CNm−−nk/CNm,

{

ξ = 0

}

C100 · C53

, P

{

ξ = 1

}

C101 · C52

C153

{

ξ = 2

}

C102 · C51

, P

{

ξ = 3

}

C103 · C50

C153

0 + 1, 0 + 2, 0 + 3, 1 + 2, 1 + 3, 2 + 3.

p = 0, 7, q = 0, 3. P {ξ = 0}

P {ξ = 0} = 0, 7.

P {ξ = 1}

P {ξ = 1} = p · q = 0, 7 · 0, 3 = 0, 21.

P {ξ = 2} = p · q2 = 0, 063, P {ξ = 3} = p · q3 = 0, 0189,

P {ξ = 4} = q4 = 0, 0081.

1 − p = q

P {ξ = 1} = q

P {ξ = 2} = = p · q.

P {ξ = 3} = p2 · q, ..., P {ξ = m} = pm−1 · q, ...

ξ p q pq p2q

p q

M (ξ) = xi · pi

i=1

M (ξ) = 1, 2 · 0, 2 + 1, 6 · 0, 4 + 2, 3 · 0, 1 + 3, 2 · 0, 2 + 4, 5 · 0, 1 = 2, 2.

D(ξ) = M (ξ2) − M (ξ)2

M (ξ2)

M (ξ2) = 1, 22 ·0, 2 + 1, 62 ·0, 4 + 2, 32 ·0, 1 + 3, 22 ·0, 2 + 4, 52 ·0, 1 = 1, 44 ·0, 2 + + 2, 56 · 0, 4 + 5, 29 · 0, 1 + 10, 24 · 0, 2 + 20, 25 · 0, 1 = 5, 914.


	D(ξ) = 5, 914 − 2, 22 = 1, 074.

	σ(ξ) =		D(ξ) =		1, 074 ≈ 1, 036.
M (ξ) = 2, 2;	D(ξ) =	1, 074; σ(ξ)		≈		1, 036
	D(ξ) =

ζ = 4ξ + 3η, M (ξ) = 11, M (η) = 8.

M (ζ) = M (4ξ + 3η) = M (4ξ) + M (3η) = = 4 · M (ξ) + 3 · M (η) = 4 · 11 + 3 · 8 = 68.

M (ξ) = 68

ξ	η
ζ = 5ξ − 6η	D(ξ) = 3, D(η) = 2.

D(ζ) = D(5ξ − 6η) = D(5ξ) + D(6η) =

= 25 · D(ξ) + 36 · D(η) = 25 · 3 + 36 · 2 = 147. D(ξ) = 147

			ξ
	x1	p1 x2 = 5	p2 = 0, 4
x3 = 4	p3 = 0, 5	x1 p1
	M (ξ) = 6

p1 = 1 − p2 − p3 = 0, 1

M (ξ) = x1p1 + x2p2 + x3p3

6 = x1 · 0, 1 + 5 · 0, 4 + 4 · 0, 5 x1 = 20

x1 = 20; p1 = 0, 1

x1 = 2, x2 = 5, x3 = 6

					M (ξ) = 4, 4; M (ξ2) = 22
p1, p2, p3							x1, x2, x3
	p1 + p2 + p3 = 1
	2 p1 + 5 p2 + 6 p3 = 4, 4
	·	p1 + 5·2	·	p2 +	·62	·	p3 = 22.
	22·	p1 + 5·2		p2 +	·62		p3 = 22.
p1 = 0, 3 p2 = 0, 4,						p3 = 0, 3.
p1 = 0, 3; p2 = 0, 4; p3 = 0, 3
				n
	M (ξ)		D(ξ)				ξ

p1 = 1/n

n − 1

· n − 1

n − 1

n − 2

· n − 1

· n −

1 + 2 + 3 + ... + n = n(n + 1) , 2

12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1) . 6

n
i	1		n + 1
M (ξ) = xipi =		(1 + 2 + 3 + ... + n) =		.
M (ξ) = xipi =	n	(1 + 2 + 3 + ... + n) =	2	.
=1

n
i			1									(n + 1)(2n + 1)
M (ξ2) =	xi2pi =			(12 + 22 + 32 + ... + n2) =										,
=1			n									6
=1
D(ξ) =		(n + 1)(2n + 1)				−	(	n + 1	)2	=	n2 − 1		.
D(ξ) =		6					(		)2	=			.
		6			n2−1		2					12
M (ξ) =	n+1 ; D(ξ) =				n2−1
	2				12

ξi

												ξ
							ξ = ξ1 + ξ2 + ξ3
M (ξ) = M (ξ1 + ξ2 + ξ3) = M (ξ1) + M (ξ2) + M (ξ3).
									ξi
D(ξ) = D(ξ1 + ξ2 + ξ3) = D(ξ1) + D(ξ2) + D(ξ3).
	ξi
M (ξ) = 3 · M (ξ1),							D(ξ) = 3 · D(ξ1).
ξ1
	ξ1
	p

	1										7
M (ξ1) =					(1 + 2 + 3 + 4 + 5 + 6) =									,
M (ξ1) =				6	(1 + 2 + 3 + 4 + 5 + 6) =								2	,
M (ξ) = 3 · (7/2) = 21/2.
M (ξ2) =		1	(12		+ 22 + 32	+ 42 + 52 + 62) =										91		,
M (ξ2) =			(12		+ 22 + 32	+ 42 + 52 + 62) =												,
1	6													6
	6													6
						91					7			35
D(ξ1) = M (ξ12) − M 2(ξ1) =										− (		)2 =					.
D(ξ1) = M (ξ12) − M 2(ξ1) =								6		− (	2	)2 =			12		.