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2007 Anna University Chennai B.E Electronics & TeleCommunication Engineering RANDOM PROCESSES Question paper
MODEL PAPER B.E. DEGREE EXAMINATION. Fourth Semester Electronics and Communication Engineering MA 034 — RANDOM PROCESSES Time : Three hours Maximum : 100 marks (Use of Statistical Tables allowed.) Answer ALL questions. PART A — (10 ´ 2 = 20 marks) Let X be a random variable taking values –1, 0 and 1 such that Find the mean of The moment generating function of a random variable X is given by Find A continuous random variable X has probability density function given by Find k such that The joint probability density function of two random variables X and Y is given by and otherwise. Find Define Poisson random process. Is it a stationary process. Justify the answer. Define (a) Wide sense stationary random process (b) ergodic random process. A stationary random process with mean 3 has auto–correlation function Find the standard deviation of the process. Define cross–covariance function of two random processes and state two of its properties. The auto–correlation function of a wide sense stationary random process is given by Determine the power spectral density of the process. The power spectral density function of a wide–sense stationary process is given by . Find the auto–correlation function of the process. PART B — (5 ´ 16 = 80 marks) (i) If X and Y are independent Poisson random variables, show that the conditional distribution of X given is a binomial distribution. (ii) A random variable X is exponentially distributed with parameter 1. Use Tchebycheff's inequality to show that Also find the actual probability. (a) (i) If the joint p.d.f. of two random variables X and Y is given by Find the p.d.f. of XY. (ii) If are Poisson variates with parameter use the central limit theorem to estimate where and n = 75. Or (b) (i) If X, Y and Z are uncorrelated random variables with zero mean and standard deviation 5, 12 and 9 respectively, and if and find the correlation coefficient between U and V. (ii) If X and Y are independent exponential distributions with parameter 1, find the p.d.f of (a) (i) State the conditions under which the Poisson distribution is a limiting case of the Binomial distribution and show that under these conditions the Binomial distribution is approximated by the Poisson distribution. (ii) Check whether the two random processes given by and where A and B are uncorrelated, zero mean and equal variance random variables are wide sense stationary. Or (b) (i) Show that X is a discrete random variable taking positive integer values with memoryless property, then X is the geometric distribution. (ii) If the wide sense stationary process is given by where is uniformly distributed in , prove that X is ergodic with respect to the auto–correlation function. (a) (i) For the jointly wide sense processes X and Y prove that (ii) For an input–output linear system show that (1) (2) Or (b) (i) Show that the inter–arrival time of a Poisson process with intensity obeys an exponential law. (ii) The power spectral density of a zero mean wide sense stationary process is given by where k is a constant. Show that and are uncorrelated. (a) (i) The power spectral density of a wide sense stationary process is given by . Find the auto–correlation function of the process. (ii) The power spectrum of a wide sense stationary process is given by Find the auto–correlation function and average power of the process. Or (b) (i) Show that for an input–output system where is the system transfer function, and the input X is wide sense stationary. (ii) Find the power spectral density of a wide sense stationary process with auto–correlation function where is a constant. ———————
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