6.262.PS3

# 6.262.PS3 - 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 #3 Issued: February 19, 2010 Due: February 26, 2010 Reading: For this week, Finish Chapter 2 and read Sections 1.5 – 1. 7 in Chapter 1. For next week, read Sections 1.1 – 1.5 of Chapter 4. (We will only use the results of the Perron - Frobenius Theory in Section 4.4, e.g., the statements of Theorems 4.6 – 4.8.) 1. Exercise 2.12 in the class notes. (Exercises 2.12 (f) and (g) are subtle, and some of the answers are surprising. For each pmf you find in Exercise 2.12, find the expectation and argue whether it is reasonable.) (h) A bus arrives at 10:30 am and a second bus arrives at 11:00 am. Let k W be the waiting time of the k-th person to enter the second bus, and let 1 1 N k k WW N = = be the average waiting time for all N passengers who enter the second bus. Find E[W|N > 0] and explain your reasoning. (i) Find E[W|N > 0] for the 5 th bus to arrive that day. 2. Problem 2.23 in the class notes. Skip part d). Hint: All the parts are easy if you use merging and splitting of Poisson processes the right way. 3. This problem continues the line of thought begun in Problem 3 of Pset #2 and applies it to the case of stochastic convergence. Let Y 1 , Y 2 , --- be a sequence of random variables and g(y) be a deterministic function that is continuous at y = y 0 . Please give a rigorous proof for those of the following

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6.262.PS3 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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