MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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Problem Set 8
Fall 2008
due 11/12/2008
Readings: Notes for Lectures 14-16.
Optional Readings:
[GS] Section 4.9 (multivariate normal)
[GS] Sections 5.7-5.8 (characteristic functions)
Exercise 1. Let X be

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 5
Fall 2008
due 10/10/2008
Readings: Notes for lectures 8 and 9
Recommended readings: Chapter 3 of [BT]; and Sections 4.1-4.6 of [GS].
Exercise 1. Let X1 , X2 , X3 be independent random vari

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2008
Midterm exam, 7-9pm (120 mins/100 pts)
6.436J/15.085J
10/21/08
Problem 1: (15 points)
Let cfw_Xn be a sequence of random variables (i.e., measurable functions) dened
on the same probability space (, F , P).

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2007
Midterm exam, 7-9pm, (120 mins/70 pts)
6.436J/15.085J
10/23/07
Possibly useful facts:
(a) If X is uniform on [a, b], then the variance of X is (b a)2 /12.
(b) n=1 1/n is innite when 1, and nite when > 1.
Pro

6.436/15.085
Midterm Exam
Date: October 23, 2006
Problem 1 Which of the following functions is a distribution function? For those which are
compute the density function. For those which are not explain what fails.
A.
F (x) =
2
1 ex , x 0;
0,
otherwise.
B.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 9
Fall 2008
due 11/19/2008
Exercise 1. Let cfw_Xn be a sequence of random variables dened on the same proba
bility space.
(a) Suppose that limn E[|Xn |] = 0. Show that Xn converges to zero,

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 7
Fall 2008
due 10/29/2008
Readings: Notes for lectures 11-13 (you may skip the proofs in the notes for
lecture 11).
Optional additional readings:
Adams & Guillemin, Sections 2.2-2.3, skim S

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 6
Fall 2008
due 10/17/2008
Readings: Notes for lecture 10.
Recommended readings: Sections 3.6, 4.2 of [BT, 1st edition], or Section 4.1
of [BT, 2nd edition]; Sections 4.7-4.8 of [GS].
Exerci

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 4
Fall 2008
due Friday 10/3/2008
Readings: Notes for lectures 6 and 7. Pay special attention to the section on
indicator variables in the lecture 7 notes.
Optional readings:
(a) Sections 3.1

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 3
Fall 2008
due 9/24/2008
Readings: Notes from Lectures 4 and 5, and Recitation 3.
Optional additional readings:
(a) Sections 2.1 and 2.3 of [Grimmett & Stirzaker]
(b) Chapter 3.1-3.12 of [W

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 2
Fall 2008
due 9/15/2008
Readings: Notes from Lectures 2 and 3 (not responsible for the Appendix in
Lecture 2). To better understand the material, try the various exercises in the
lecture n

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Problem Set 1
Fall 2008
due 9/8/2008
Readings:
(a) Notes from Lecture 1
(b) Handout on background material on sets and real analysis (Recitation 1).
Supplementary readings:
[GS], Sections 1.1-1.3.
[W],

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2008
Final exam, 1:304:30pm, (180 mins/100 pts)
6.436J/15.085J
12/18/08
Whenever asked to explain or justify an answer, a formal proof is not needed, but
just a brief explanation.
Problem 1: (30 points)
Consider

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2007
Final exam, 1:304:30pm, (180 mins/100 pts)
6.436J/15.085J
12/19/07
Problem 1: (24 points)
During the time interval [0, t], men and women arrive according to independent Poisson
processes with parameters 1 an