35361 Probability and Stochastic Processes, Spring 2015

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