You have been given the
task to find the correlation
from the sample given in
the next two columsn
X
86
56
70
48
66
56
97
52
54
94
44
71
49
46
95
41
96
99
81
46
41
Y
5
7
4
8
8
7
6
7
7
5
7
7
5
4
5
4
5
8
6
4
4
STEP I
Find the Sum and
Averages of Both the
Va

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Page 1
Permutations and Combinations
BUILDING ON
listing outcomes of probability experiments
solving equations
BIG IDEAS
Counting str

Mathematics
Learning CentreEmAlC
COUNTING TECHNIQUES
Counting Techniques
Sue Gordon
Mathematics Learning Centre
University of Sydney
NSW 2006
c 1994
University of Sydney
Acknowledgements
I gratefully acknowledge the many ideas and suggestions provided by

PROBABILITY
SIMPLE PROBABILITY
SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a
number between 0 and 1.
There are two categories of simple probabilities.
THEORETICAL PROBABILITY is calculated probability. If every ev

Permutations &
Combinations
Extension 1 Mathematics
HSC Revision
Multiplication Rule
If one event can occur in m ways, a second
event in n ways and a third event in r, then
the three events can occur in m n r ways.
Example Erin has 5 tops, 6 skirts and 4

Stat110 Lecture Notes
1
1.1
Diana Cai
Probability and Counting
Naive Denition of Probability
A sample space is the set of all possible outcomes of an experiment. An event is a subset
of the sample space.
Denition (Naive denition of probability). Let A be

UCLA STAT 100A
Introduction to Probability Theory
Instructor:
Ivo Dinov,
Asst. Prof. In Statistics and Neurology
Teaching
Assistant:
Romeo Maciuca,
UCLA Statistics
University of California, Los Angeles, Fall 2002
http:/www.stat.ucla.edu/~dinov/
Stat100A,U

PROBABILITY
#24
PROBABILITY is the likelihood that a specific outcome will occur, represented by a number
between 0 and 1.
There are two categories of probability.
THEORETICAL PROBABILITY is calculated probability. If every outcome is equally
likely, it i

C H A P T E R
1
Combinatorial Analysis
1.1
1.2
1.3
1.4
1.5
1.6
INTRODUCTION
THE BASIC PRINCIPLE OF COUNTING
PERMUTATIONS
COMBINATIONS
MULTINOMIAL COEFFICIENTS
THE NUMBER OF INTEGER SOLUTIONS OF EQUATIONS
1.1
INTRODUCTION
Here is a typical problem of inter

MATH 2: Axioms of probability
In the course of solving various counting problems we have discovered (and developed)
a framework that seems to be common to all these problems. The main concepts are:
Sample space = the set W of all possible outcomes of a r

Exercise 1: Prove the De Morgans Law
(A B)c = Ac B c
Solution:
x (A B)c x A B x A and x B x Ac and x B c x Ac B c
/
/
/
Exercise 2: Let (, F, P ) be a probability space. Show that for all sets A1 , A2 , . . . F
we have that
n
n
Ai
P
P [Ai ]
i=1
i=1
Solut

Axiomatic Probability
The objective of probability is to assign to each event A a number
P(A), called the probability of the event A, which will give a precise
measure of the chance thtat A will occur.
Probability Axioms:
AXIOM 1 For any event A, P(A) 0.

h
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Qu i z
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3 Co
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d i r Bo
& D a t a A n a l s is
y
n a l ! x c c t a l 1o n
h l a r ko
p
T im
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e
t r u c t io n s
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P le a s e
re a l t t B
1c
q u e s t io
n s c a lm
ly
an
N
St a t e M a r k o
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,

Chapter 9
Two-Sample Tests
Paired t Test (Correlated Groups t Test)
Effect Sizes and Power
Paired t Test Calculation
Summary
Independent t Test
Chapter 9 Homework
Power and Two-Sample Tests: Paired
Versus Independent Designs
I
f you have any interest in

L
,
W O RE
U N IV E Rsi
E C2 3 0 St a t i
s t ic s & D
Qu i z 1 P e r m u t a i
N
WAGEI v1ENT S c i E N C E S
A n a lysi s
d B a sic C o
u n tin
g P r in c i p l e
a ta
t o ns a
n
T im e : 45
In str u c tio
T y OF
n u te s
P le a s e
n s :
rea
d t h e u e

CHAPTER 5 Sampling Distributions
Sections: 5.1 & 5.2
Introduction
In this chapter we focus our attention on the sampling distribution of the sample mean and the
sampling distribution of the sample proportion. These sampling distributions will give us the

Chapter 6
The t-test and Basic Inference
Principles
The t-test is used as an example of the basic principles of statistical inference.
One of the simplest situations for which we might design an experiment is
the case of a nominal two-level explanatory va

2
Elements of probability theory
Probability theory provides mathematical models for random phenomena, that
is, phenomena which under repeated observations yield dierent outcomes that
cannot be predicted with certainty.
2.1
SAMPLE SPACES
A situation whose

STATISTICS, Lesson 1.
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1. Permutations. How many dierent orders of n objects exist?
Without repetition (there are n dierent objects): n!
With repetition (there are n objects of which n1 , . . . , nr are alike):
n!
n1 !.nr !
2. Vari

Chapter 10
Classical Business Cycle
Analysis: Market-Clearing
Macroeconomics
I.
Business Cycles in the Classical Model
(Sec. 10.1)
1. Two key questions about business
cycles
2. Any business cycle theory has two
components
A)
The real business cycle theo

Chapter 13
Open Economy Macroeconomics
Introduction
Our
previous model has assumed a single
country exists in isolation, with no trade or
financial flows with any other country.
This chapter relaxes the single country
assumption, and revises our theory

C H A P T E R
2
Axioms of Probability
2.1
2.2
2.3
2.4
2.5
2.6
2.7
INTRODUCTION
SAMPLE SPACE AND EVENTS
AXIOMS OF PROBABILITY
SOME SIMPLE PROPOSITIONS
SAMPLE SPACES HAVING EQUALLY LIKELY OUTCOMES
PROBABILITY AS A CONTINUOUS SET FUNCTION
PROBABILITY AS A ME

6.041/6.341 Quiz I Review (Fall 08)
I. We will begin our quiz review by summarizing the important
concepts and formulae from the rst part of this class. We will
go over:
Sample space and probability,
Discrete random variables, and
Introduction to conti

STAT 514
Probability Quiz
Name:
Question 1: According to the axiom of countable additivity for probability measures, if
E1 , E2 , E3 , . . . is a sequence of disjoint events,
P
Ei
=
.
i=1
Question 2: Suppose that X is a Poisson random variable and 10P (X