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Московский государственный индустриальный университет

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	n
γ	τ γ
	2
	γ
	τ γ
	2

ξi N (a; σ), i = 1, . . . , n

ξ = (ξ1 + . . . ξn)/n

¯	¯	σ
		√		.
M (ξ) = a, σ(ξ) =
			n
a
¯	− a\| < ε} = γ,
P {\|ξ

γ ε

P {|ξ¯ − a| < ε} = 2Φ

σ/√

σ/√n

· √n

σ/√n

2Φ

= γ

= τ γ

ε = τ γ

σ = 2 n = 64 x¯ = 5, 2

γ = 0, 95

γ	= 0, 475 τ γ	= 1, 96
2	2
	2

x¯ − tγ	S			S		,
	√		; x¯ + tγ	√
		n			n
tγ					k = n − 1
α = 1 − γ

Fst(x) n − 1

Fst(tγ ) = 1 + γ , 2

x¯ S

i = 1, . . . , n

ξi N (a; σ)

−√a

t =

S / n

tγ

n − 1

P {|t| < tγ } = γ.

ϕst(t)

tγ

P t < t

= γ

P t > t

= 1

−

{

t > t

1 − γ

{| |

γ }

{| |

γ }

−

(t ) =

1 − γ

(t ) =

1 + γ

S /√n

− γ

S /√n

− a

< t = γ

t <

ξ − a

< t = γ,

Fst(x)

1 − γ

1 −

tγ

x¯ = 10, 5 S = 1, 6 n = 16 γ = 0, 99

k = n−1 = 15 α = 1−γ = 0, 01

tγ = 2, 98

(9, 308; 11, 692)

			A
	5%	B	2%
94%	ξ
			A ζ
B			(ξ, ζ)
			A B
			ξ
		A
	ζ
	B	x1 = 1, x2 = 0; y1 = 1, y2 = 0.
	pij = P {ξ = xi, ζ = yj }.		p22 = P {ξ = 0, ζ = 0} = 0, 94.
	x2 = 0 y1 = 1 y2 = 0
P (x2) = p21 +p22		P (x2) = 0, 95 p22 = 0, 94 p21 = P (x2)−p22 =
= 0, 01		p12 = P (y2) − p22 = 0, 98 − 0, 94 = 0, 04
	P (x1) = 1 − P (x2) = 0, 05		P (x1) = p11 + p12
	p11 = 0.01

ζ\ξ

ξ				ζ

p				p

M (ξ) = 0, 05, M (ξ2) = 0, 05, D(ξ) = 0, 0475, M (ζ) = = 0, 02, M (ζ2) = 0, 02, D(ζ) = 0, 0196.

(ξ, ζ)

ξ · ζ

ξ · ζ

M (ξ · ζ) = 0, 01.

0, 01 − 0, 001

0, 009

0, 295.

ξζ

√0, 0475

0, 0196

≈

0, 0305

≈

(ξ, ζ)

ϕ(x, y) =

cos(x

−

[0,

π ],

]

2 ],

x / [0,

y / [0,

M (ξ) =

xϕ(x, y)dxdy,

M (ζ) =

yϕ(x, y)dxdy.

+∞

−∞

π/2

xdx 0

cos(x − y)dy =

M (ξ) =

cos(x − y) M (ζ) = M (ξ) = π/4

+∞

D(ξ) = x2ϕ(x, y)dxdy − M 2(ξ).

			−∞
	1	π/2	π/2		π		π2		π
D(ξ) =		0	x2dx 0	cos(x − y)dy − (		)2 =		+		− 2.
D(ξ) =	2	0	x2dx 0	cos(x − y)dy − (	4	)2 =	16	+	2	− 2.
		x								D(ζ) = D(ξ)

+∞

M (ξ · ζ) = xyϕ(x, y)dxdy,

−∞

	1	π/2	π/2		π2		π
M (ξ · ζ) =		0	xdx 0	y cos(x − y)dy =		−		+ 1.
M (ξ · ζ) =	2	0	xdx 0	y cos(x − y)dy =	8	−	2	+ 1.

x y

= (

π2

+ 1

π2

)/(

π2

2) =

π2 − 8π + 16

ξζ

8 −

−

16 16 2

π2 + 8π

−

(ξ, ζ)

M (ξ) = 2, M (ζ) = −5, D(ξ) = 9, D(ζ) = 4.

(ξ, ζ)

rξζ = 0.

	1	e−	1		(x−2)2		(y+5)2
ϕ(x, y) =			2	(	9	+	4	).
	12π

σξ = 3, σζ = 2

x y

F (x, y) =

ϕ(x, y)dxdy,

−∞ −∞

(x−2)2

(y+5)2

−∞ e−

F (x, y) =

dx ·

dy =

12π

(x−2)2

(y+5)2

( −∞ e−

dx +

e−

dx) · ( −∞ e−

dy + 0

e−

12π

+∞

√

−∞ e−x

dx = 0

e−x

dx =

e−

xΦ(x) =

√

dz.

2π

dy).

u = (x − 2)/3, v = (y + 5)/2

x−2

(x−2)

e−

dx = 3

y+5

e−

du = 3√2πΦ(

−

e−

dy = 2 ·

e− 2

dv = 2√

Φ( y 2

2π

12π

(y+5)2

+ 5

√18π

√

−

√

y + 5

F (x, y) =

+ 3 2πΦ(

)

2π + 2 2πΦ(

) =

+ Φ(

x −

)

+ Φ(

y + 5

) .

(ξ, ζ)

ϕ(x, y) = axy

ϕ(x, y) = 0

− 1 = 0, x = 0, y = 0.

x + y −

rξζ

ϕ(x, y)dxdy = 1.

+∞

−∞

1−x

a 0

dx 0

xydy = 1 a ·

= 1 a = 24.

M (ξ) =

xϕ(x, y)dxdy = 24

x2dx 0

ydy = 5 .

+∞

−

−∞

M (ζ) = 2/5

x3ydy − ( 5 )2 = 5 − 25 = 25 .

D(ξ) =

x2ϕ(x, y)dxdy = 24

+∞

−

−∞

D(ζ) = 1/25

M (ξ · ζ) =

ξ · ζ

x2dx 0

y2dy =

15 .

xyϕ(x, y)dxdy = 24

+∞

−

−∞

rξζ = (

−

)/(

) = −

M (ξ) = M (ζ) = 0, σ(ξ) = σ(ζ) = 1

(ξ, ζ)

R = 2

ϕ(x, y) = ϕξ(x) · ϕζ (y)

ϕξ(x) =

√

e−x2/2,

ϕζ (y) =

√1

e−y2/2.

2π

e−

ϕ(x, y) = ϕξ(x) ·

ϕζ (y) =

2 (x

2π

P = G

ϕ(x, y)dxdy = 2π G e− 2

)dxdy.

P = 2π G

e− 2 r rdrdϕ = 2π

dϕ

e− 2 r rdr =

2π

= 1 − e−2 ≈ 1 − 0, 135 = 0, 865.

M (ξ) = M (ζ) = 0, σ(ξ) = σ(ζ) = 1

R (ξ, ζ)

P = e− 12 r2 rdr = −e− 12 r2 |R0 = 1 − e− 12 R2 .

R 1 − e−R2/2 = 0, 9. R = 2, 145

					(ξ, ζ)
		M (ξ) = M (ζ) = 0, σξ, σζ , rξζ = 0
								G
a = kσξ, b = kσζ
				x2/(kσξ)2 + y2/(kσζ )2 = 1
P ((ξ, ζ) G) = G					ϕ(x, y)dxdy,
						2	2
				1	− 21 ( σx2 + σy		2 )
ϕ(x, y) =					e	ξ	ζ .
ϕ(x, y) =			2πσξσζ		e	ξ	ζ .
			2πσξσζ
x = σξr cos ϕ, y = σζ r sin ϕ
I = σξσζ r
1			2π	k
1		0		0	2		2
P ((ξ, ζ) G) =					re−r	/2dr = 1 − e−k		/2.
P ((ξ, ζ) G) =	2π				re−r	/2dr = 1 − e−k		/2.

ξ ζ

ζ\ξ

ξ ζ

β 4, 5%

95% α β

0 x π, 0 y π				(ξ, ζ)	ϕ(x, y)		=	(1/4) sin x sin y
0 x π, 0 y π					ϕ(x, y) = 0
						(ξ, ζ)
	1 sin(x + y)			x [0,	π ],	y	[0,	π ],
ϕ(x, y) =	2				2			2
ϕ(x, y) =		x / [0, π2 ]		y / [0, π2 ].
0		x / [0, π2 ]		y / [0, π2 ].
				σξ, σζ
rξζ
ϕ(x, y) =	a2	2	y2	x2 + y2		a2(a > 0),
ϕ(x, y) =	0	− xx−2 + y2 > a2.