x 3 , x > 2, 0 otherwise;

Exerise 8, Autumn 2009

1. Findtheexpetation

E(X)

^{, ifthe}probabilitydensityfuntionof random variable

X

^is

f

^{, and}

a)

f(x) = ( ₈

x ³ , x > 2, 0

^otherwise;

b)

f(x) = ¹ ₂ e ^{−| x |}

^,

x ∈ R

^;

)

f(x) =

( xe ^{− 1 2 x 2} , x > 0,

0

^otherwise.

2. Asssume that

X ¹

^,

X ²

^and

X ³

^are independent normally distributed ran- dom variables and that eah has distribution

N(1, 3)

^{. Find}

P {X ¹ + X ² + X ³ > 0}.

3. 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 averageof}

X i

^ie

X ¯ = ¹ / n

P ⁿ

i=1 X i

^{. Find}

n

^{suh that}

P {| X| ¯ < a

100 } = 0, 95?

4. Determine the distribution of random variable

X − Y

^{, if}

X

^and

Y

^are

independent and both follow exponential distribution with parameter

λ

^,

where

λ > 0

^.

5. Find thedistribution of random variable

2X ² + 1

^{, if}

X ∼ N(0, 1)

^.

6. A ray of light originating from point

(0, 1) ∈ R ²

^{forms angle}

Θ

^with

x

^-

axel.Assumethat

Θ

^{is uniformly}distributed 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