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 kth 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[WN >
0] and explain your reasoning.
(i) Find E[WN > 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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 Spring '08
 Moon,J
 Computer Science, Electrical Engineering, Probability theory

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