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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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M (ξ) = 1, 9, M (ξ2) = 7, 3 p1, p2, p3

ξ		x1
p1 = 0, 1 x2	x1 < x2	M (ξ)
D(ξ)	M (ξ) = 3, 9, D(ξ) = 0, 09

n p

A n

P {ξ = m} = P (m) = Cnmpmqn−m, q = 1 − p, m = 0, 1, . . . , n.

ξ				. . .	k	. . .	n
p	qn	npqn−1	Cn2p2qn−2		Cnkpkqn−k		pk

n p

(p + q)n = Cnmpmqn−m.

m=0

(p + q)n = p + (1 − p) n = 1

1, 2, . . . , n) i

ξ	n	ξi (i =
ξi = 1	i	A ξi = 0
	¯
	A
	ξ = ξ1 + . . . + ξn.
	A	n

ξi

pp 1 − p

M (ξi) = 1 · p + 0 · (1 − p) = p,

D(ξi) = M (ξi2) − M (ξi) 2 = 12 · p + 02 · (1 − p) − p2 = p − p2 = p(1 − p) = p · q.

M (ξ) = M (ξ1) + . . . + M (ξn) = n · p,

D(ξ) = D(ξ1) + . . . + D(ξn) = n · p · q.

M (ξ) = np; D(ξ) = npq.

n = 4 p = 0, 5

P4(0) = 0, 54 ≈ 0, 06; P4(1) = 4 · 0, 5 · 0, 53 ≈ 0, 25; P4(2) = C42 · 0, 52 · 0, 52 ≈ 0, 38; P4(3) = p4(1) ≈ 0, 25;

P4(4) = p4(0) ≈ 0, 06.

M (ξ) = n · p = 4 · 0, 5 = 2; D(ξ) = nqp = 4 · 0, 5 · 0, 5 = 1. M (ξ) = 2, D(ξ) = 1

n → ∞, p → 0			np → a
		Pn(m)
	am
P {ξ = m} =		e−a, a > 0.
P {ξ = m} =	m!	e−a, a > 0.

. . .

e−a

ae−a

e−a

∞

m=0

e−a = e−a

= e−a · ea = 1.

m=0

∞

−

M (ξ) =

e−a =

1)!

e−a =

m=0

m=1

∞

am−1

∞

−

= a · e−a

1)!

= a · e−a

= a · e−a · ea = a.

m=1

k=0

D(ξ) = a

M (ξ) = D(ξ) = a.

. . .

C0 Ck

C1 Ck−1

C2 Ck−2

L K−L

K−L

. . .

L K−L

M (ξ) = k · KL .

ξ					. . .

p	p	p(1 − p)	p(1 − p)2	p(1 − p)3	. . .

ξ A A p = p(A)

M (ξ) =	1 − p	, D(ξ) =	1 − p	.
	p		p2

[a; b]
ϕ(x) =	C	x [a; b],
	0	x / [a; b].

	+∞				b
	ϕ(x)dx = 1 =				Cdx = 1 =
	−∞			a	1
	b				1

=	Cx a	= 1 = C(b − a) = 1 = C =				b − a	.
					[a; b]
		1
	ϕ(x) =				x [a; b],
	ϕ(x) =		b − a		x [a; b],
		0			x / [a; b].

x < a

a x b

F (x) = ϕ(t)dt.

−∞

F (x) = 0dt = 0;

−∞	a	b − a		b − a
	a	b − a		b − a
a	x	1		x − a
F (x) = 0dt +			dt =		;
−∞

x > b F (x) = 0dt +

−∞

[a; b]

x − a F (x) = b − a

a	b − a	b	b − a
	b	x	b − a
	1	dt + 0dt =	b − a	= 1.
		dt + 0dt =		= 1.

x < a,

a x b,

b < x.

a b

ϕ(x)
a	b	x

F(x)
1
a	b	x

+∞

−

dx x

−

= a + b

−∞

dx =

M (ξ) = xϕ(x)dx =

− a)

b a

2 (b a)

+∞

(a + b)2

+ b)2

D(ξ) =

x2ϕ(x)dx −

dx −

b − a

−∞

(a + b)2

(b − a)3

(a + b)2

−

−2

−

b a

· 3

3 (b a)

4 (a

+ ab + b ) 3 (a + b)

(b a)

[a; b]

M (ξ) =

a + b

; D(ξ) =

(b − a)2

a x < x+Δx b

P {x ξ < x +

x+Δx

P {x ξ < x + x} =

ϕ(x)dt =

dt =

x + x − x

b − a

[a; b]

M (ξ) = a+b

[0; 4]

		[0; 4]
	1					0			x < 0,
	1	x [0; 4],					x		0 x 4,
ϕ(x) =			F (x) =
	4
	4
	0	x / [0; 4],	(4			4
	0	x / [0; 4],				4			x > 4,
						1			x > 4,
					0)2			4
		M (ξ) = 2; D(ξ) =		−			=		.
		M (ξ) = 2; D(ξ) =		12			=		.
				12				3

M (ξ) = 2; D(ξ) = 34

ϕ(x) =	λe−λx	x 0,
ϕ(x) =	0	x < 0,	λ > 0.

λ > 0

x				0	x
−∞	ϕ(t)dt =					λe−λtdt = −e−λt		x	= 1	− e−λx;
−∞
x 0 F (x) =								0
				x		x
x < 0 F (x) =					ϕ(t)dt =		0dt = 0.
				−∞		−∞
F (x) =		1	−	e−λx		x 0,
F (x) =		0	−			x < 0.

+∞		+∞		+∞
M (ξ) =	xϕ(x)dx =		xλe−λxdt = λ		xe−λxdx =
−∞		0		0

ϕ(x)

F(x)

= x

du = dx

/ =

= . dvu= e−λx

v = −

e−λx

∞

+∞

∞

= −xe−λx 0

e−λxdx = −

e−λx 0

+∞			+∞
D(ξ) =	x2ϕ(x)dx −	1	=	x2λe−λxdx −	1	=	1	.

		λ2			λ2		λ2
−∞			0

M (ξ) =	1	; D(ξ) =	1	.
	λ		2
			λ

ξ1, ξ2, ξ3, . . .

t a = λt

ξ F (x)

							ξ

		ξ
		p
x 1
x 1			F (x) = 0			ξ
		1 < x 2 F (x) = P (ξ < x) = P (ξ = 1) = 1/6.
2 < x 3
ξ				ξ < x
			ξ < 2 2 ξ < 3
	F (x) = P (ξ < x) = P {(ξ < 2) + (2 ξ < 3)} =
			= P (ξ < 2) + P (2 ξ < 3).
P (ξ < 2) = 1/6 P (2 ξ < 3)
2 x < 3							ξ
					2 < x 3 F (x) = 1/6 + 1/6 = 1/3
	3 < x 4