Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

0,

x 1,

1/6,

1 < x 2,

1/3,

2 < x 3,

1/2,

3 < x

4,

2/3,

4 < x

5,

3/6,

5 < x 6,

1,

6 < x.

F(x)

1

5/6

4/6

3/6

2/6

1/6

1

2

3

4

5

6

x

0

x

−

2,

2 < x

F (x) =

(x + 2)2

−

−

1,

1

x > 1.

(−3/2, −1).

ξ

ξ

ξ

ϕ(x) =

cos x

(0, π/2)

ϕ(x) = 0.

ξ

(π/4, π/3)

a = π/4, b = π/3, ϕ(x) = cos x

π/3

√

√

P (

π

< ξ <

π

) =

cos xdx = sin x

π/3

=

3 − 2

.

|π/4

4

3

π/4

2

( 3 − 2)/2.

ξ

0

x

1,

F (x) =

(x 1)2

1 < x 2,

1

− x > 2.

0

x

1,

ϕ(x) = F (x) =

2(x 1)

1 < x 2,

0

− x > 2.

ϕ(x)

0,

x 0,

ϕ(x) =

3

(4x x2), 0 < x 4,

32

0,

− x > 4.

2

2 3

3

x3

11

ϕ(x)dx =

(4x − x2)dx =

(2x2 −

)|12 =

.

32

32

3

32

0

1

1

D(ξ) = −1 x2(x + 1)dx + 0

x2(−x + 1)dx =

.

6

+∞

1

1

0

1

∞

M (ξ) = −∞

x

λe−λ|x|dx =

λ

−∞ xeλxdx +

λ

0

xe−λxdx.

2

2

2

1

2

x2

·

λe−λ|x|dx =

.

2

λ2

√

σ(ξ) =

D(ξ) = 2/λ.

/λ

0

x < 0,

1 − exp(−

x2

)

x ≥ 0.

2σ2

ϕ

ϕ(x) = F (x) =

x2

· exp(−

x2

)

x 0.

2σ2

2σ2

1

x2

x2

exp(−

) · (1 −

)

σ2

2σ2

σ2

x 0 ϕ (x) = 0

x = σ.

ϕ (x)

√

√

x = σ 2 · ln2

ϕ(x) = 0

ξ

0

F (x) =

1/2

1

ξ

0

x 0,

F (x) =

ax3

0 < x 4,

1

x > 4.

0, x

0

x > π,

+∞

+∞

+∞

1

(σt + a)e−

t2

σ

te−

t2

a

e−

t2

=

√

2

dt =

√

2

dt +

√

2

dt =

2π

2π

2π

−∞

−∞

−∞

σ

t2

+∞

−∞

σ2

+∞

D(ξ) = x2

√

e−

−∞

ξ

a

σ

ϕ(x)

F(x)

1

1

2π σ

0,5

a- σ

a

a+σ

x

a

x

P

x

ξ < x

= F (x

)

F (x

) = 0, 5 + Φ

x2 − a

−

σ

σ

−

σ

0, 5 + Φ

x1 − a

= Φ

x2

− a

Φ

x1 − a

,

{

1

2}

σ

−

σ

P

x

ξ < x

= Φ

x2

− a

Φ

x1 − a

.

Φ(+∞) = 0, 5, Φ(−∞) = −0, 5

{

P

ξ < x

= Φ

x2

− a

+ 0, 5,

{

}

−

σ

P

x

ξ

= 0, 5

Φ

.

ε

ε

ε

P {|ξ − a|

< ε} = 2Φ

ε

.

Φ(0, 3) = 0, 1179

a = 20 = 10

P {|ξ −10| < 3}

10

−

10

−

≈

= Φ

13 −

20

Φ

7 − 20

= Φ(1, 3)

Φ(0, 7)