Session11 Solution

Exercise Session 11, November 29th ,2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua Chen (Cissy), Natalia Guseva Solutions Problem 1 Find the moment generating function for the following distributions: (a) bernoulli(p), (b) exp( ), (c) N ormal( ; 2 ) and N ormal(0; 2 ) and compute the …rst and second moments. Ans: a). P (X = 0) = 1

p; P (X = 1) = p:

X (u)

E [X] =

@ @u

E X2 =

X (u) u=0 2

@ @u2

X (u)

V ar(X) = p(1

p)

b). fX (x) = 1 e

x

= E euX = eu 1 p + eu 0 (1 = 1 + p [eu 1] = eu pju=0 = p

u=0

= eu pju=0 = p

> 0 and u < 1 Z 1 1 uX = eux e X (u) = E e

; x > 0;

x

0

E [X] =

@ @u

E X2 = V ar(X) =

X (u) u=0 2

@ @u2 2

p)

X (u)

=

u=0

(1

=

=

u)2 u=0 2 2 (1 u)3 u=0

1

=2

2

dx =

1 1

u

c). X

N( ;

2

);

uX

X (u)

= E(e

)=

1 2

+1 Z

1 p 2

+1 Z

1 p 2

+1 Z

=

=

=

p

= e

+1 Z

p

1

e

1 2

(x )2 2 2

eux e

x2 + 2

2 2 ux

2x 2 2

dx

dx

1

(

x2

2x

2 + 2u

) (

e

E [X] =

@ @u

dx

1

(x (

+ 2u

e

+1 Z

1 p 2

2 u2 2

2

))

2 2

1

u+

E X2 =

2

@ @u2

X (u)

V ar(X) = E X 2

X

N (0;

X (u) = e E [X] = 0 E X2 =

u+

X (u) u=0

2

)

2 2

|

+

u+

(x ( e

2 u2 2

+ 2u

1

2 u2 2

=( +

u=0

2

2

= (( +

(E [X])2 =

u+

u)e

2

u)2 +

+

);

2 u2 2

2

2

2

2

dx

{z

2 u2 2

dx

))

2 2

=1(b ecause integral of pdf N ( +

= e

2 + 2u + 2

) (

+ 2u +

}

2 u; 2 ))

=

u=0 2 u+

)e

2

=

2

2 u2 2

u=0

=

2

+

2