6.262.PS1

6.262.PS1 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 – Discrete Stochastic Processes Problem Set #1 Issued: February 5, 2010 Due: February 12, 2010 Reading: For this week, please study sections 1.1 – 1.4, section 1.5.3, and sections 1.6 – 1.7 of Gallager’s notes. For next week, please read Chapter 2 of Gallager’s notes. 1) Exercise 1.11 from course notes. 2) Exercise 1.23 from course notes. 3) a) Exercise 1.31 from course notes. b) Use the central limit theorem to find lim ( ) n n P Xnn →∞ ≤+ where X n is a Poisson random variable with expectation n. (Hint: This question will become easier after you have completed part c) of Problem 5 and thought about how to use your answer for the Poisson distribution.) 4) Exercise 1.37 from course notes. (This problem introduces a theme we will develop in much more detail as the course proceeds.) 5) A class C of probability distributions is said to be stable if, for any two independent random variables X and Y with probability distributions in C, the probability distribution for the sum Z = X + Y also lies in C. a) What does the central limit theorem suggest about classes of stable distribution? In particular does it suggest that the family of all Gaussian distributions is stable? Does it suggest that the family of all Gaussian distributions is the only family of stable distributions?
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6.262.PS1 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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