09hw12

09hw12 - EE 464, Mitra Homework 12 Due FRIDAY, April 24,...

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EE 464, Mitra Spring 2009 Homework 12 Due FRIDAY, April 24, 2009, 10am This problem set investigates random sequences: convergence of random sequences and limit theorems. 1. L-G 7.21. 2. L-G 7.45 (Hint: use the Cauchy-Schwarz inequality). 3. Let ω U [0 , 1] . In what senses do the following sequences converge, it at all? For each sequence, identify the limit and show whether they converge in distribution, in probability, in mean-square, almost surely or surely . (a) X n = sin ( ω + 1 n ) (b) X n = cos n ( ω ) 4. We have a sequence of independent random variables, whose probability density functions for each n are given by f X n ( x ) = ± 1 - 1 n 1 2 πσ exp " - 1 2 σ 2 ± x - n - 1 n σ 2 # + 1 n σ exp( - σx ) u ( x ) where u ( x ) is the unit step-function. Determine whether or not the sequence converges in (i) mean-square (ii) probability or (iii) distribution. 5. Consider an i.i.d. sequence of uniform random variables: X n U [0 , 1] . Define the random sequence Y n = min
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This note was uploaded on 02/02/2010 for the course EE 464 taught by Professor Caire during the Spring '06 term at USC.

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09hw12 - EE 464, Mitra Homework 12 Due FRIDAY, April 24,...

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