HW6_131A - ╬ given by the arrival rate This result follows...

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Online EE 131A Homework #6 Fall 2010 Due Nov. 10th K. Yao Read Leon-Garcia (3rd edition), pp. 155-180; 255-278. 1. Problem 4.68, p. 222. 2. Problem 4.88, p. 223. 3. Let X be a Laplacian rv with a pdf given by f X ( x ) = ( α/ 2) exp( - α | x | ) , 0 < α, -∞ < x < . Suppose X is the input to the eight-level uniform quantizer in Fig. 4.8(a) on page 162 (3rd edition). Find the pmf of the quantizer output levels. Find the probability that the input X exceeds the [-4d, 4d] of the dynamic range of the quantizer. 4. Problem 4.99, parts a, b, and c, p. 224. 5. Consider the application of the Poisson distribution to the evaluation of arrival prob- lems in a telephone system. Let λ denote the average number of calls per unit time (also called the arrival rate). Assume the calls arrival independently. Then the proba- bility of the number of calls per unit time is modeled by the Poisson distribution with
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Unformatted text preview: ╬╗ given by the arrival rate. This result follows from the approximation of the Binomial distribution by the Poisson distribution. Speci´Čücally, divide each unit time into a large number n of smaller intervals, so that for a small enough interval, there is either 0 or at most 1 call in that interval. Then these n time intervals are Bernoulli trials with success being a call in each interval. We do not know n or p but we do know ╬╗ = np . Thus, k calls per unit time is given by e-╬╗ ╬╗ k /k !. Now, suppose we know on the average there are 2 calls per second. Find the probability of having at most 3 calls in 5 seconds. (Hint: Find the average number of arrivals in 5 seconds.)...
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