Probability and stochastic processes;

Probability and stochastic processes; The Questions 1. Random variables. a) We place uniformly at random n = 200 points in the unit interval [0, 1]. Denote by random variable X the distance between 0 and the first random point on the left. i) Find the probability distribution function FX(x). [3] ii) Derive the limit as ?? ? 8 and comment on your expression. [3] b) The random variable X is uniform in the interval (0, 1). Find the density function of the random variable Y = – lnX. [4] c) X and Y are independent, identically distributed (i.i.d.) random variables with common probability density function ???? (??) = ?? -?? , ???? (??) = ?? -?? , ??>0 ??>0 Find the probability density function of the following random variables: i) Z = XY. [5] ii) Z = X / Y. [5] iii) Z = max(X, Y). [5] Probability and Stochastic Processes © Imperial College London page 2 of 5 2. Estimation. a) The random variable X has the truncated exponential density ??(??) = ???? -??(??-??0 ) , ?? > ??0 . Let x0 = 2. We observe the i.i.d. samples xi = 3.1, 2.7, 3.3, 2.7, 3.2. Find the maximum-likelihood estimate of parameter c. [8] b) Consider the Rayleigh fading channel in wireless communications, where the channel coefficients Y(n) has autocorrelation function ???? (??) = ??0 (2?????? ??) where J0 denotes the zeroth-order Bessel function of the first kind (the function besselj(0,.) in MATLAB), and fd represents the normalized Doppler frequency shift. Suppose we wish to predict Y(n+1) from Y(n), Y(n – 1), …, Y(1). The coefficients of the linear MMSE estimator ??(?? + 1) = ? ?? ???? ??(??) ??=1 are given by the Wiener-Hopf equation ???? = ?? where ?? = [??1 , ??2 , … , ???? ]?? , ?? = [???? (??), ???? (?? - 1), … , ???? (1)]?? , and R is a n-byn matrix whose (i, j)th entry is ???? (?? - ??). i) Give an expression for the coefficient of the first-order MMSE estimator, i.e., n = 1. [4] ii) Let fd = 0.01. Write a MATLAB program to compute the coefficients of the n-th order linear MMSE estimator and plot the mean-square error ????2 = ??0 - ??* ??-?? ?? as a function of n, for 1 = ?? = 20. [10] iii) From the figure, determine whether Y(n) is a regular stochastic process or not, and justify. [3] [As you may imagine, n cannot be greater than 2 for computation of this kind in an exam.] Probability and Stochastic Processes © Imperial College London page 3 of 5 3. Random processes. a) The number of failures N(t), which occur in a computer network over the time interval [0, t), can be modelled by a Poisson process {N(t), t = 0}. On the average, there is a failure after every 4 hours, i.e. the intensity of the process is equal to ? = 0.25. i) What is the probability of at most 1 failure in [0, 8), at least 2 failures in [8, 16), and at most 1 failure in [16, 24) ? (time unit: hour) [7] ii) What is the probability that the third failure occurs after 8 hours? [4] b) Find the power spectral density S( ) if the autocorrelation function i) 2 ??(??) = ?? -???? . 2 ii) ??(??) = ?? -???? cos( c) [3] 0 ??) . [3] The random process X(t) is Gaussian and wide-sense stationary with E[X(t)] = 0. Show that if 2 (??). ??(??) = ?? 2 (??), then autocovariance function ?????? (??) = 2?????? [8] Hint: For zero-mean Gaussian random variables Xk, ??[??1 ??2 ??3 ??4 ] = ??[??1 ??2 ]??[??3 ??4 ] + ??[??1 ??3 ]??[??2 ??4 ] + ??[??1 ??4 ]??[??2 ??3 ] Probability and Stochastic Processes © Imperial College London page 4 of 5 4. Markov chains and martingales. a) Classify the states of the Markov chain with the following transition matrix 0 1/2 1/2 0 1/2) ?? = (1/2 1/2 1/2 0 [2] ??= ( 1/2 1/2 0 0 0 1/2 1/2 0 2/3 0 0 1/3 0 0 2/3 1/3 1/3 1/3 0 0 0 0 0 0 1/3) [3] b) Consider the gambler’s ruin with state space E = {0,1,2,…,N} and transition matrix 1 0 ?? 0 ?? ?? 0 ?? ??= . . . ?? 0 ?? 0 ( 1) where 0 < p < 1, q = 1 – p. This Markov chain models a gamble where the gambler wins with probability p and loses with probability q at each step. Reaching state 0 corresponds to the gambler’s ruin. i) ?? ???? Denote by Sn the gambler’s capital at step n. Show that ???? = (??) is a martingale (DeMoivre’s martingale). [4] ii) Using the theory of stopping time, derive the ruin probability for initial capital i (0 < i < N). c) [4] Let N = 10. Write a computer program to simulate the Markov chain in b). Starting from state i and run the Markov chain until reaching state 0. Repeat it for 100 times, and plot the ruin probabilities as a function of the gambler’s initial capital i (0 < i < N), for i) p = 1/3; [4] ii) p = 1/2; [4] iii) p = 2/3. [4] Also plot the theoretic results of b). [Obviously, such a question cannot be tested in this way in the exam!] Probability and Stochastic Processes © Imperial College London page 5 of 5 PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)