Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 9 Notes: General VC Inequality
In Lecture 8, we proved the following Generalized VC
inequality
P f F,
Efn
Ex
f(xi) n 0 log1/2 D(F, , d)d + 27/2 Ex n
d(0,
d(0,
29/
1 n
2
f)2t
f)
i=1
d(f, g)
n
=
1 n
1/
2
2
(f(xi) f(xi ) g(xi) + g(xi )
i
=
1
Definit
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Kristine Herma
Professor Lavery
Reflection Paper #6
April 12th, 2016
In my eyes, family can mean many different things to people. As we all know people
learn their roles and statues in socialization process by family, because family is an institute
where
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Math 123 Homework: Budgets, CPI & Percentages
Show all work, including setup of problems. Write answers using complete sentences when appropriate.
1. By January 2014 the US population had grown to 317.3 million and the US Federal Debt was
a reported $17.3
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Name_
MRS. IMBOREK
Psychoanalysis of Characters in The Cat in the Hat
1. For this assignment, you are to psychoanalyze the characters using Freuds Psychoanalytic Theory. Your task
is to assess which part of the psychological self (ID, EGO, SUPEREGO), best
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Abnormal Psychology Study Guide
HERMA
KRISTINE
Test Wednesday 4/13/16
Chapters 14,16, & 17
Ch. 14 Schizophrenia
1. What is Schizophrenia?
Schizophrenia is a long term mental disorder of a type involving a breakdown
in the relation between thought, emotio
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Math 123 Homework: Proportions, Firearm Deaths And Additional Scientific Notation
Show all work, including setup of problems. Write answers using complete sentences when appropriate.
1.
x 18
=
2 6
36=6x x=6
2.
x
9
=
12 27
108=27x x=4
Math 123 Homework: Pr
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Kristine Herma
Professor Lavery
Reflection Paper #7
13 April 2016
Personally I never saw a benefit to conflict until reading further into it. Every time I had
been involved with a conflict or around conflict, I always became stressed out and my anxiety
wo
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 4 Notes: Chebyshevs Inequality
1
Let Z1, , Zn R be i.i.d. random variables. Were interested in bounds on n
Zi EZ.
(1) Jensens inequality: If is a convex function, then (EZ) E(X).
EZ
(2) Chebyshevs inequality: If Z 0, then P (Zt) t . Proof:
EZ = EZ
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 1 Notes: Intro to Course
Consider a family of weak classifiers
H = cfw_h : X cfw_1, +1.
Let the empirical minimizer be
1
n
i
n
h0= argmin
I(h(Xi)= Yi)
=1
and assume its expected error,
1
2> = Error(h0), >0
Examples:
d
d
X = R , H = cfw_sign(wx +
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 7 Notes: Sauers Dilemma
Theorem 13.1. Assume F is a VCsubgraph class and V C(F) =V . Suppose 1f(x)1 for all f F and x X .
1 n
Let x1, . . . , xn X and define d(f, g) =n
i=1 f(xi)g(xi). Then
D(F, , d)
(which is
K V+
8 e 7 V
log .
for some .)
P
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 2 Notes: Maximizing Margins
d
As in the previous lecture, consider the classification setting. Let X = R , Y = cfw_+1,1, and
d
H = cfw_x + b, R , b R
where  = 1.
We would like to maximize over the choice of hyperplanes the minimal distance from t
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 6 Notes: Rate Changes
Last time we proved the Pessimistic VC inequality:
P
n
C
1
n
sup
i P
i=1
I(X
C)
(C)
C
sup
V
Hence, the rate is
1
2
nt
e
,
8
2en
log 4 + V log
n
i
n
V
4
8
t=
P
2en
t
which can be rewritten with
as
V
P
n
nlog 4 + Vlog
8
i=1
V
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 3 Notes: Average Generalization Error
Assume we have samples z1 = (x1, y1), . . . , zn = (xn, yn) as well as a new sample zn+1. The classifier trained
on the data z1, . . . , zn is fz1,.,zn .
The error of this classifier is
Error(z , . . . , z ) =
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 5 Notes: Bennetts Profit Theorem
2
2
Last time we proved Bennetts inequality: EX = 0, EX = , X < M = const, X1, , Xn independent copies of
X, and t 0. Then
P
n
Xi t
2
exp
M
i=1
tM
n
2
2
n
,
where (x) = (1 + x) log(1 + x) x.
2
x
2
2
2
x
) x = x
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 10 Notes: Uniform Economics
In a classification setup, we are given cfw_(xi, yi) : xi X , yi cfw_1, +1i=1,n, and are required to construct a
classifier y = sign(f (x) with minimum testing error. For any x, the term y f(x) is called margin can be
c
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
Lecture 8 Notes: Variable Changes
j=1 R(j (f ) j1(f ). We first show how to control R on the links. Assume
Hoedings
Lj1,j . Then by
inequality
P
n
i=1
1
i i
exp
t
1 2
n 2 i
2
n
2
t
nt2
i=1 i2
2 n1
n
= ex p
2
exp
nt
222j+4
Note that
2
cardLj1,j cardFj1 c
Introduction to Microeconomic Concepts and Formulas
ECON 115

Winter 2006
We all started as strangers at the beginning of May, entering this classroom not knowing what
to expect. Unsure of ourselves we all had one goal in mind; we wanted to help others. So our
CNA journey began in the classroom, it was a lot of material to cove