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# INTRODUCTION TO RANDOM PROCESSES AND APPLICATIONS ELEC 533

Rice University

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This 2 page Class Notes was uploaded by Deondre Ullrich on Monday October 19, 2015. The Class Notes belongs to ELEC 533 at Rice University taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/224981/elec-533-rice-university in Electrical Engineering & Computer Science at Rice University.

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Date Created: 10/19/15

31 32 33 Electrical and Computer Engineering 533 Introduction to Random Processes and Applications Problem Set 111 Problem 111 in the notes Chemoff Bound Chebyshev s Inequality is a very loose bound for the probability a random variable deviates from its mean A much tighter and therefore more useful result is the ChernoiT Bound To derive the bound assume X is a random varibale with mean m and variance 02 We seek an expression for PrX 7 m 2 k0 where k is a constant a Write PrX 7 m 2 w as the integral of some function from w to in nity b Show that for any 5 2 0 because 64 2 1 when x 2 k0 that PrX 7 m 2 1w 3 6 9 E 69X m 7 520 c d Derive an equation for the optimal value of s the value that produces the tightest bound e When X N Jm7 02 nd the ChernoiT bound using the optimal value for 5 How does it compare to the Chebyshev bound V How is the expected value related to the momentgenerating function of X And if the sequence is dependent The Law of Large Numbers typically assumes the sequence of random variables X1L7 X27 consists of identically distributed and statistically independent components What happens if we relax the second assumption a For this part assume the sequence of random variables are statistically independent Show that the relative frequency of occurrence converges in at least four senses in distribution in probability in mean square and almost surely 1 V L MM E Z 1AXk k1 b Suppose we consider the sample average of correlated random variables If each ran dom variable is correlated with the others with correlation coe icient p does the average converge to the mean in the meansquare sense c Suppose the correlation among the random variables extends only to adjacent pairs 1 239 39 covXXj j T P ll Jl 1 0 239 7 jl gt 1 Does the sample average converge to the mean in the meansquare sense ELEC 533 Problem Set III 1 Does the value of the correlation coef cient affect the rate at which the sample average converges to the mean in part c How does the rate as a function of p with the statistically independent case 34 Martingales 35 A sequence ofrandom variables X07 X17 7 Xn7 is said to be a martingale ifthe conditional expectation of the present value given the entire past satis es Eanan717Xn727 39 39 39 7X0 Xn71 a Let Xn 220 Wk where Wk i1 with equal probability and the sequence is meansquare independent ElVkHVk17 7 W0 Show that Xn is a martingale b An independent increments sequence has the property that rst di erences are indepen dent random variables Thus for Xn to have independent increments Xn 7 Xikl is statistically independent of Xn1 7 Xikg Show that if Xn has independent increments then Xn 7 EXn is a martingale A P V The likelihood ratio is a very important quantity in statistical hypothesis testing and in detection theory In such problems we are trying to decide if the data X07 7Xn X were produced by probability law pX MO xiV10 or pX M1X Jl1 where Mo and M1 describe alternative models for the data generation process The likelihood ratio is de ned to be M ANX E p X M1Xi 1 pXMlo XiV10 Show that under model M0 ANX is a martingale Gaussian Random Vector and Linear Transformations Let X be a jointly Gaussian random vector having mean In and covariance matrix K a Let Y AX In other words Y is a linear transformation of a Gaussian random vector Assuming the matrix A is invertible show that Y is also a jointly Gaussian random vector Find its mean and covariance b Suppose A is not invertible having a smaller rank than dimX Is Y still jointly Gaus sian If so demonstrate a proof if not why not

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