3x 3 , 2 < x < 4, 0 otherwise;

(1)

Exerise 8, Autumn 2008

1* Findtheexpetation

E(X)

^, ^if^theprobabilitydensityfuntionof random variable

X

^is

f

^, ^and

a)

f(x) = ( ₃₂

3x ³ , 2 < x < 4, 0

^otherwise;

b)

f(x) = ¹ ₂ e ^−| ^x ^|

^,

x ∈ R

^;

2. Let

X

^be ^the^greatestôf ^numbers ^thatâppears^whenâ ^dieîs ^thrown^four

times.Find the expetation

E(X)

^.

3. Asssume that

X ¹

^,

X ²

^and

X ³

^are independent normally distributed random variables and that eah has distribution

N(1, 3)

^. ^Find

P {X ¹ + X ² + X ³ > 0}.

4. Let

X ¹

^,

X ²

^, ^.^.^.^,

X ⁿ

^be ^measuring ^errors ⁱⁿ ^repeated measurements. As- sume that random variables

X ¹

^,

X ²

^, ^.^.^.^,

X n

^are independent, normally distributed,eah has distribution

N(0, σ ² )

^and

P (|X ⁱ | < a) = 0, 95

^for ^every

i = 1, 2, . . . , n.

Let

X ¯

^be ^the ^average^of

X ⁱ

^ie

X ¯ = ¹ / n

P ⁿ

i=1 X ⁱ

^. ^Find

n

^suh ^that

P {| X| ¯ < a

100 } = 0, 95?

5* a) Determinethedistributionofrandomvariable

2X ² + 1

^,^if

X ∼ N(0, 1)

^.

b) Determine the distribution of random variable

4X ³ + 3

^, ^if

X

^follows

geometri distributionwith parameter

3 4

^.

6. A ray of light originating from point

(0, 1) ∈ R ²

^forms ^angle

Θ

^with

x

^-

axel.Assumethat

Θ

^is ^uniformlydistributed on interval

] − ^π ₂ , ^π ₂ [

^. ^Let

X

be the

x

^oordinate ^of ^the intersetion of light ray and

x

^-axel. ^Find ^the

probabilityfuntionand probabilitydensity funtionof

X

^.^Does

X

^have

3x 3 , 2 < x < 4, 0 otherwise;

E(X)

X

f

f(x) = ( 32

3x 3 , 2 < x < 4, 0

f(x) = 1 2 e −| x |

x ∈ R

X

E(X)

X 1

X 2

X 3

N(1, 3)

P {X 1 + X 2 + X 3 > 0}.

X 1

X 2

X n

X 1

X 2

X n

N(0, σ 2 )

P (|X i | < a) = 0, 95

i = 1, 2, . . . , n.

X ¯

X i

X ¯ = 1 / n

P n

i=1 X i

n

P {| X| ¯ < a

100 } = 0, 95?

2X 2 + 1

X ∼ N(0, 1)

4X 3 + 3

X

3 4

(0, 1) ∈ R 2

Θ

x

Θ

] − π 2 , π 2 [

X

x

x

X

X

f(x) = ( ₃₂

3x ³ , 2 < x < 4, 0

f(x) = ¹ ₂ e ^−| ^x ^|

X ¹

X ²

X ³

P {X ¹ + X ² + X ³ > 0}.

X ¹

X ²

X ⁿ

X ¹

X ²

N(0, σ ² )

P (|X ⁱ | < a) = 0, 95

X ⁱ

X ¯ = ¹ / n

P ⁿ

i=1 X ⁱ

2X ² + 1

4X ³ + 3

(0, 1) ∈ R ²

] − ^π ₂ , ^π ₂ [