35361 Probability and Stochastic Processes, Spring 2015
Assignment 1 (due to September 9, 2015)
The assignment should be handed in to Tim Ling on 9/09/2015. You
may use any software for calculations (Mathematica is preferable).
Problem 1.
1) Let Xi, i = 1, 2 be independent random variables (i.r.v.s) having a
Chi-square distribution, EXi = 4.
(i) Using characteristic functions and the inversion formula find the
probability density function (pdf) of Y = X1 X2.
(ii) Find E(Y 8).1Problem 2. 1) Using a variance reduction technique (e.g. control
variates) find the Monte-Carlo approximation for the integral
J=Z 8
02
-x
e
(1 + x2)dx.Use the sample sizes n = 106, n = 107 and compare the results with
the exact value.
2) Using the 3-sigma rule estimate a sample size n required for obtaining a Monte-Carlo approximation with a control variate for J with
an absolute error less than ? = 10-6.2Problem 3.
Let B0(t), t ? [0, 1] be a Brownian Bridge that is a Gaussian process
with E(B0(t)) = 0 and the covariance function
R(t, s) = min(t, s) ts.
1) Using simulations with a discrete-time process approximation for
B0(t) (e.g. use N=1000 trajectories and n=1000 discretisation points)
find an approximation for the distribution function of the random variable
X = max |B0(t)|
0=t=1
at the points {0.2, 0.6, 2.0}. Hint: use a representation for B0(t) in
terms of a standard Brownian motion.
2) Verify the results using the analytical expression for the distribution
function of X :
P {X < x} = 1 + 28 X2 x2 k -2k (-1) e .k=1 3

35361 Probability and Stochastic Processes, Spring 2015
Assignment 1 (due to September 9, 2015)
The assignment should be handed in to Tim Ling on 9/09/2015. You
may use any software for calculations (Mathematica is preferable).
Problem 1.
1) Let Xi, i = 1, 2 be independent random variables (i.r.v.’s) having a
Chi-square distribution, EXi = 4.
(i) Using characteristic functions and the inversion formula find the
probability density function (pdf) of Y = X1 – X2.
(ii) Find E(Y 8).

1

Problem 2. 1) Using a variance reduction technique (e.g. control
variates) find the Monte-Carlo approximation for the integral
J=

Z 8
0

2
-x
e
(1 + x2)dx.

Use the sample sizes n = 106, n = 107 and compare the results with
the exact value.
2) Using the 3-sigma rule estimate a sample size n required for obtaining a Monte-Carlo approximation with a control variate for J with
an absolute error less than ? = 10-6.

2

Problem 3.
Let B0(t), t ? [0, 1] be a Brownian Bridge that is a Gaussian process
with E(B0(t)) = 0 and the covariance function
R(t, s) = min(t, s) – ts.
1) Using simulations with a discrete-time process approximation for
B0(t) (e.g. use N=1000 trajectories and n=1000 discretisation points)
find an approximation for the distribution function of the random variable
X = max |B0(t)|
0=t=1
at the points {0.2, 0.6, 2.0}. Hint: use a representation for B0(t) in
terms of a standard Brownian motion.
2) Verify the results using the analytical expression for the distribution
function of X :
P {X < x} = 1 + 2

8
X

2 x2
k
-2k
(-1) e
.

k=1
3

35361 Probability and Stochastic Processes, Spring 2015
Assignment 1 (due to September 9, 2015)
The assignment should be handed in to Tim Ling on 9/09/2015. You
may use any software for calculations (Mathematica is preferable).
Problem 1.
1) Let Xi, i = 1, 2 be independent random variables (i.r.v.’s) having a
Chi-square distribution, EXi = 4.
(i) Using characteristic functions and the inversion formula find the
probability density function (pdf) of Y = X1 – X2.
(ii) Find E(Y 8).

1

Problem 2. 1) Using a variance reduction technique (e.g. control
variates) find the Monte-Carlo approximation for the integral
J=

Z 8
0

2
-x
e
(1 + x2)dx.

Use the sample sizes n = 106, n = 107 and compare the results with
the exact value.
2) Using the 3-sigma rule estimate a sample size n required for obtaining a Monte-Carlo approximation with a control variate for J with
an absolute error less than ? = 10-6.

2

Problem 3.
Let B0(t), t ? [0, 1] be a Brownian Bridge that is a Gaussian process
with E(B0(t)) = 0 and the covariance function
R(t, s) = min(t, s) – ts.
1) Using simulations with a discrete-time process approximation for
B0(t) (e.g. use N=1000 trajectories and n=1000 discretisation points)
find an approximation for the distribution function of the random variable
X = max |B0(t)|
0=t=1
at the points {0.2, 0.6, 2.0}. Hint: use a representation for B0(t) in
terms of a standard Brownian motion.
2) Verify the results using the analytical expression for the distribution
function of X :
P {X < x} = 1 + 2

8
X

2 x2
k
-2k
(-1) e
.

k=1
3