LF Marketing
Hashim Alhamad Joey Konefal Julius Myers
What is Poros?
https:/www.pinterest.com/rachelhebert92/tabl
eau-projet-%C3%A9cole/
What is Poros?
http:/www.myfrenchlife.org/2014/02/13/frenc
h-grammar-french-politesse/
5Cs
Company
Integration of bag
Team work:
Facilitator/Coordinator
The coordinator/facilitator should
-Focus the team toward the task
- Get all team members to participate
- Keep the team on deadline
- Suggest alternatives
-Help team members confront problems
-Summarize team decisions
S
Guidelines for Effective Writing:
1.
Create a Strategy and use it to inform your structure (purpose, audience,
communicator)
2.
Write clearly and concisely: Identify the who in your sentence, use verbs to
specify actions, make every word tell: because- du
Strategic communication =
Thinking purposefully about your message, which will allow you to both interact effectively w/
others and achieve your goals through communication
Structure can be direct or indirect:
-Informative Direct: main points followed by
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 2
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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
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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
Recitation 1
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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
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 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
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 4 Solutions
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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
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 4
1
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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
HW 6 Solutions
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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 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 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 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
Recitation 9
1
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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
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 8
1
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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
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
15.093 Optimization Methods
Lecture 2: The Geometry of LO
1
Outline
Slide 1
Polyhedra
Standard form
Algebraic and geometric denitions of corners
Equivalence of denitions
Existence of corners
Optimality of corners
Conceptual algorithm
2
Central Prob
15.093: Optimization Methods
Lecture 12: Discrete Optimization
1
Todays Lecture
Slide 1
Modeling with integer variables
What is a good formulation?
Theme: The Power of Formulations
2
2.1
Integer Optimization
Mixed IO
(MIO) max
s.t.
2.2
cx+hy
Ax + By b
15.093: Optimization Methods
Lecture 14: Lagrangean Methods
1
Outline
Slide 1
The Lagrangean dual
The strength of the Lagrangean dual
Solution of the Lagrangean dual
2
The Lagrangean dual
Slide 2
Consider
ZIP = min
s.t.
cx
Ax b
Dx d
x integer
(P )
X
15.093 Optimization Methods
Lecture 7: Sensitivity Analysis
1
Motivation
1.1
Questions
z = min
s.t.
Slide 1
cx
Ax = b
x0
How does z depend globally on c? on b?
How does z change locally if either b, c, A change?
How does z change if we add new constrai
15.093 Optimization Methods
Lecture 10: Network Optimization
Introduction and Applications
1
Network Optimization
What is a network?
2
Networks
2.1
Formally
2.2
Electrical & Power Network
2.3
Subway/Train Network
2.4
Airline Route Map
2.5
Internet US Back