Random Variables
chris bemis
January 22, 2011
chris bemis
Random Variables
We begin by dening three mathematical objects:
A set
of possible events
A collection of subsets, , of
And a function P
01
chris bemis
Random Variables
What are we doing?
With , we
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 12 (April 22)
(1) (10 points) Consider the following piece of code:
Z=[0 randn(1,n)];
B=cumsum(Z);
t=linspace(0,10,n+1);
X=60*exp(0.995*t+0.1*sqrt(10)*B);
This is simulating Geomet
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 11 (April 15)
(1) (10 points) If X1 , X2 are independent N (0, 1) variables,
(a) What is the distribution of the vector (X1 , X2 )?
SOLUTION:
X1
0
10
N
,
X2
0
01
(b) What is the di
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 10 (April 8)
(1) (10 points) With discretization parameters x, y and t, write the equation
corresponding to an interior node (i, j ) for the time step tn tn+1 when we apply
the Cra
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 9 (April 1)
1. (10 points) Construct in MATLAB, directly as a sparse matrix an N N identity
matrix.
SOLUTION:
sparse(1:N,1:N,ones(1,N);
2. (5 points) If we are given two real numbe
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 8 (Mar 25)
1. (10 points) Write down the equations of
following problem
ut uxx + u = 0
u(x, 0) = 0 (x)
u(a, t) = u(b, t) = 0
the CrankNicolson discretization of the
a < x < b,
0 <
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 7 (Mar 11)
(1) (10 points) Using Taylor expansions, prove that
f (x + h) 2f (x) + f (x h)
+ O(h2 )
f (x) =
h2
SOLUTION: We see that
h2
h3
f (x + h) = f (x) + hf (x) + f (x) + f (x)
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 6 (Mar 4)
1. (8 points) You are given three time values of two solutions of equations of the form
ut + c ux + r u = 0.
The value of c is the same for both, but the value of r is di
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 5 (Feb 25)
(1) (10 points) A certain curve tting problem leads us to solving the following incompatible system of linear equations
1
1 1.1 (1.1)2
1 1.2 (1.2)2 A1
0.8
1 1.3 (1.
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz #4 (Feb 18)
(1) (5 points) We have run the following lines of MATLAB code for a given square
matrix A:
> singv=svd(A);
> singv(1)
ans =
33.0010
> singv(end)
ans =
0.1000
What are A 2
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz #3 (Feb 11)
(1) (5 points) Given the matrix
3 5 1
A = 8 5 0 ,
9 5 1
if we apply partial pivoting in Gaussian elimination in the rst column, what
is the pivot and what are the multipl
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz #2 (Feb 4)
(1) (10 points) Consider the iteration
xn+1 =
1
2
xn +
a
xn
where a > 0 is given.
(a) Show that a is the only positive xed point.
(b) Show that a is a stable xed point and
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz #1 (Jan 28)
(1) (10 points) The nth base2 Halton number xn is dened as follows: we write n
in base 2, ip the digits and use them as the digits of a fractional base 2 number;
nally we
NAME:
ID#:
FM 5012: Mathematical Background for Finance II
Spring10
Quiz # 13 (April 29)
(1) (10 points) Write the integral
x2 e3x dx
1
as E [h(X )] where X Exp(3). Specify what h is. Write an algorithm (or sketch
a program) to compute this expected value