MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 9 Solutions
15.085/6.436
Total: 100
Problem 1. Let c = i ci < and bi = limj aij . Consider random
variables Xn on a probability measure space (N, 2N , P), such that for each
n = 1, 2, . . ., Xn (i) = ain

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 3 Solutions
15.085/6.436
Problem 1. We have to show that PX is a probability measure.
(i) PX (B) = P(X 1 (B) [0, 1].
(ii) PX (R) = P(X 1 (R) = P() = 1.
(iii) If Bi B(R) are disjoint, then X 1 (Bi ) are di

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 10
1
15.085/6.436
Fishy Poisson process
Consider two independent Poisson processes, one slow and the other fast,
with rate s < f arrivals per minute.
1. What is the expected time until the rst arr

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 8
1
15.085/6.436
Modes of convergence
The main four modes of convergence of random variables are
1. Xn X almost surely, P(lim supn |Xn X| = 0) = 1 or equivalently
, P(lim sup |Xn X| > ) = 0.
n
2.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
6.436J/15.085J
This document is a brief summary of the material covered in the class. It is meant to be a listing, not an
exposition, and it is presented in rather abstract terms. For the nal exam, you shoul

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 9
1
15.085/6.436
Product Measures and the Tonelli-Fubini Theorem
A measure space (, A, ) is -nite if there exists An A, n = 1, 2, . . .
such that n An = and (An ) < for all n. Clearly, if is a pro

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 4
1
15.085/6.436
Random Variables
Given a probability space (, F, P), recall that a random variable X in this
space is a measurable function X : R, i.e., for all B B(R), X 1 (B)
F. More generally

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 5
1
15.085/6.436
Expectation of ratios
Let X1 , X2 , . . . , Xn be independent identically distributed random variables
1
for which E(X1 ) and E(X1 ) exists. Show that, if m n, then E(Sm /Sn ) =
m

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 2
15.085/6.436
Problem 1. Coin Tosses I. Recall F is the -eld generated by F0 dened in Lecture 2 for the innite coin toss model. Show that the set S of
sequences where the average number of heads

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 3
15.085/6.436
Problem 1. Let An be a sequence of independent events with P(An ) < 1
for all n, and P(n An ) = 1. Show that P(An i.o.) = 1.
Solution: Suppose 0 pi < 1, for all i. Then, we have pi

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 11 Solutions
15.085/6.436
Total: 70
Problem 1. First, we can compute that the steady state distribution is
A = B = D = E = 1/6, and C = 1/3. We can do this either by solving
a system of linear equations (

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 10 Solutions
15.085/6.436
Total: 60
Problem 1. For any > 0,
P(|Y2n Yn | > ) P(|Y2n Y | + |Y Yn | > )
P(|Y2n Y | > /2) + P(|Y Yn | > /2) 0,
p
as n . Therefore |Y2n Yn | 0.
There is no loss in generality i

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 8 Solutions
15.085/6.436
Total: 60
Problem 1. We can assume that X 0 a.s., since min(X + X , n) =
min(X + , n) X . Clearly Xn X, so lim supn EXn EX. To prove
the other direction, consider Zn X a.s., Zn n

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 6 Solutions
15.085/6.436
Total: 50
Problem 1. Probably the simplest solution is to use characteristic functions, however since this has not been formally discussed, we will resort to
variable transformati

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 4 Solutions
15.085/6.436
Total: 80
Problem 1. Let Ai be the event that the ith couple survives, and let
X = n 1Ai be the total number of surviving couples.
i=1
n
E[X] =
P(Ai )
i=1
n
2n 2
2n
m
m
i=1
m
m
=n

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 2 Solutions
15.085/6.436
Problem 1.
(a) Let An = cfw_ : n = 0 Fn . Then, A = n0 A2n+1 (F0 ).
However, A F0 , because otherwise there exists some n such that
/
A Fn . But the event A cannot be determined i

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
HW 7 Solutions
15.085/6.436
Total: 70
Problem 1. Note that in this problem, vectors are 1 n. Since V is
symmetric and non-singular, there exists unitary matrix U (U T = U 1 ) and
diagonal matrix D such that

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2006
Recitation 1
15.085/6.436
Background
Version of 9/6/061
1
Sets
A set is a collection of objects, which are the elements of the set. If A is a set
and x is an element of A, we write x A. If x is not an elemen