20105ee132B_1_hw7

20105ee132B_1_hw7 - UCLA Electrical Engineering Department...

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1 UCLA Electrical Engineering Department EE132B HW Computer Workout #1 Professor Izhak Rubin Note. Please turn in source codes in separate files and specify what software/language you use. A. Continuous-Time Information Streams 1) Generate a stream of i.i.d. random variables U = {U n , 1 n N} which are uniformly distributed over [0, 1], for N = 1000, 5000, 10000. Use the resulting streams to draw probability distribution histograms which show the probabilities that U is in the interval (0.1i, 0.1i+0.1], for i = 0,1,…, 9. Compute the sample means and variances, for each of the three runs. Compare the results with the computed means and variances. nN 2 N 2 i i N 2 11 NN Note: When the variables {X ,1 } are observed, N>1, the sample mean m and the sample variance are given by: X (X m ) m ; . 1 ii    2) Generate a stream of i.i.d. random variables T={T n , 1 n N} which are exponentially distributed with parameter (intensity) λ = 0.1 for N = 1000, 5000, 10000. The random variable T n is governed by the distribution       1 t n P T t e U t , where U(t) is the unit step function. Note that the mean and variance of T n are equal to 1/ λ and (1/ λ ) 2 , respectively. Show that if U n is uniformly distributed over (0, 1], then the random variable T defined by the transformation   1 ln n TU  is exponentially distributed with parameter λ and mean 1/ λ .
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2 3) Consider a traffic stream represented by the stochastic point process A={A n , n 0}, A 0 =0, where A n denotes the time of the occurrence of the n-th message arrival. The n-th inter-arrival time is 1 ,1 n n n T A A n
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This note was uploaded on 03/29/2011 for the course ELEC ENGR 132B taught by Professor A during the Winter '10 term at UCLA.

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20105ee132B_1_hw7 - UCLA Electrical Engineering Department...

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