Probability and Stochastic Processes
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 ∞
e
−x
2
(1 + x
2
)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
0≤t≤1
|B0(t)|
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 X
∞
k=1
(−1)k
e
−2k
2x
2
.
3