µ 1 andµ 2,and

(1)

Exerise 2, Autumn 2009

1. Let

X

^and

Y

^beindependentrandomvariableswithmeans

µ 1

^and

µ 2

^,^and

varianes

σ 1

^and

σ 2

^, respetively. Present, using these, following a)

E(aX + bY)

^, ^where

a

^and

b

^are ^onstants;

b)

D ² (aX + bY)

^, ^where

a

^and

b

^are ^onstants;

)

E

X−Y 2

2

.

2. Determine the

p

^fratile ^of ^random ^v^ariable

X

^for

p = 0.5

^,

p = 0.75

^and

p = 0.99

^, ^when

a)

X ∼ Tas(0, 1)

^,

b)

X ∼ Exp(2)

^,

)

X ∼ N( ¹ / ₂ ¹ / ₄ )

^.

3. Let

P(A) = p

^. ^Determine ^the probabilitygenerating funtion of the indi- ator

1 A

^and ^use ^this ^to^determine ^the probabilitygenerating funtionof distribution

Bin(n, p)

^.

4. Let

X

^be ^a

N

valued random variable and

G

^the probability generating funtionof

X

^.

a) Calulate

G(0)

^and

G(1)

^.

b) Expressthe probabilitythat

X

îs êven ûsing

G

^.

5. Let

X

^and

Y

^beindependentrandomvariables.Determinetheonditional distribution

P { X = k | X + Y = n }

^, ^for

k = 0

^,

1

^,

. . .

^,

n

^,,

when

a)

X ∼ Bin(n 1 , p)

^and

Y ∼ Bin(n 2 , p)

^,

b)

X, Y ∼ Geom(p)

^.

6. (Jenssen's inequality)Suppose that

g

^is dierentiable andits derivative is inreasing. Show that, if random variables

X

^and

g(X)

^have ^expeted

value, then

g(E(X)) ≤ E(g(X)).

Hint:Prove rst following lemma:

If

g ^′

^is ^inreasing, ^then

g ^′ (y)(x − y) ≤ g(x) − g(y).

for every

x, y ∈ R

.

µ 1 andµ 2,and

X

Y

µ 1

µ 2

σ 1

σ 2

E(aX + bY)

a

b

D 2 (aX + bY)

a

b

E

X−Y 2

2

p

X

p = 0.5

p = 0.75

p = 0.99

X ∼ Tas(0, 1)

X ∼ Exp(2)

X ∼ N( 1 / 2 1 / 4 )

P(A) = p

1 A

Bin(n, p)

X

N

G

X

G(0)

G(1)

X

G

X

Y

P { X = k | X + Y = n }

k = 0

1

. . .

n

X ∼ Bin(n 1 , p)

Y ∼ Bin(n 2 , p)

X, Y ∼ Geom(p)

g

X

g(X)

g(E(X)) ≤ E(g(X)).

g ′

g ′ (y)(x − y) ≤ g(x) − g(y).

x, y ∈ R

D ² (aX + bY)

X ∼ N( ¹ / ₂ ¹ / ₄ )

g ^′

g ^′ (y)(x − y) ≤ g(x) − g(y).