Circular permutation
There are n! different permutations of n di ti t
Th
! diff
t
t ti
f distinct
(
objects. arranged in a line ).
How about other kinds of arrangement?
Say, if the people are randomly arranged in a
circle, how many different ways do we ha
Probability
y
Originate from the study of games of chance and gambling during the
16th century.
Be a measure of the possibility that a random phenomenon occurs,
e.g. the tossing of a dice.
Although we cannot exactly predict what we get in any random
p
p
Math
M th 2411 A li d St ti ti
Applied Statistics
(Spring, 2012)
by
Dr. Yu, Chi Wai
Instructor: Dr. Yu, Chi Wai
Office: Room 3492
Email: macwyu@ust.hk
Course webpage: http:/www.math.ust.hk/~macwyu/
Office Hours:
14:00-14:50 (Mon,
14:00 14:50 (Mon Tue) or
Math2411 (2013, Fall)
Introduction to R
Part II: Statistics
1
Introduction to R
Risafreestatisticalprogramminglanguage.
Downloadat:http:/www.rproject.org.
Or, simply google [R], it should appear as the first hit
Part II: Statistics
2
Part II: Statistics
3
Chapter Five
Hypothesis Testing
(Statistical Inference: Part III Hypothesis Testing)
Part II: Statistics
1
In the previous chapter, we introduced the point and interval
estimation of an unknown parameter(s), say and 2.
In this chapter, we consider scienti
Chapter Six
Regression Models
(Simple linear regression models)
Part II: Statistics
1
Introduction
In the previous chapter, we discussed how to make a statistical
inference about the mean difference, when two samples of random
variables, say X and Y, are
Final
Examination
MATH2411 FORMULA SHEET AND DISTRIBUTION TABLES
Seat Number:_
Do NOT mark or write down anything (including your name) on this formula
sheet or distribution tables. Otherwise, 5 points will be deducted for each
marked page.
Page 1 of 6
Fi
Chapter Four
Confidence Interval
(Statistical Inference: Part II Interval estimation)
Part II: Statistics
1
Problem with point estimation
The point estimator can only produce a single estimated
value of the unknown parameter. Indeed, we can seldom
estimat
What is Statistics?
It is a science that involves data summarization, experimental
design, data collection, etc.
Recently, it has been considered to encompass the science of
basing inferences on data and the entire problem of making
decisions in the fac
Chapter Four
Parameter Estimation
(Statistical Inference: Part I Point estimation)
Part II: Statistics
1
What is Statistical Inference
Use a statistical approach to make an inference about
the distribution which generates (a sample of) data we collect.
W
Independence
Irrelevant information
Based on a new information, we can update the sample space and
events to be new ones, so that we can calculate a conditional probability
of the updated event.
However, sometimes, the new information may be irrelevant.
T
Random Variables and
Probability Distributions
On page 3 of lecture note 2
Question 1: Flip a fair coin three times. List all the possible
outcomes, i.e. find the sample space of this experiment.
S = cfw_ HHH, HHT HTH, THH TTH THT, HTT , TTT .
,
,
,
Varia
MATH 2411 - L2&3
Week 8, Fall 2012
TA: Willis
SOLUTIONS TO EXAMPLES
Example 1
In this question, we are given that the r.v. X follows N(50,
). We need to be careful that the
conventional notation of normal distribution is to write out the mean
question we
MATH2411 - L2 & L3
Extra Notes on Hypothesis Testing
TA: Willis
yp
g
Hypothesis Testing Process
Step 1: Define Hypothesis
The Null Hypothesis, H0
The Alternative Hypothesis, H1
Always contains the = sign
Never contains the = sign
Step 2: Collect Data and
MATH2411 - L2 & L3
Extra Notes on Hypothesis Testing
Page 1 of 6
TA: Willis
MATH2411 - L2 & L3
Extra Notes on Hypothesis Testing
Page 2 of 6
TA: Willis
MATH2411 - L2 & L3
Extra Notes on Hypothesis Testing
Page 3 of 6
TA: Willis
MATH2411 - L2 & L3
Extra No
MATH2411 - L2&3
Week 8, Fall 2012
TA: Willis
Example 1
Given that the r.v. X follows a normal distribution with = 50 and = 10. Find
the probability of X taking value between 45 and 62.
Example 2
Suppose that a particular company has a history of making er
Summary for Math2411 (Spring 2012)
1. Numerical and graphical tools to describe data
cfw_xi , i = 1, 2, ., n is a set of data with n data points, and cfw_x(i) , i = 1, 2, ., n is its corresponding ordered data
set, with x(i) as the ith smallest data point
FB2200 Management Sciences I
Topic 6: One-Sample Test Extra Exercise (t-test)
Q1.
You are the manager of a restaurant for a fast-food franchise. Last month, the mean waiting
time at the drive-through window for branches in your geographical region, as mea
Def. 7.3 Unbiasedness
Unbiasedness of Point Estimator
The sample statistic is an unbiased estimator of
the population parameter if E( )= , where E( )
is the expected value of the sample statistic .
Note: The reason for using n-1 instead of n in the
denomi
Counting sample points
What is the number of different ways to order distinct objects?
For example, suppose that there are 5 people, say
A B C D E
Now,
Now we want to arrange them in a row randomly
randomly,
then how many ways are possible?
Consider the f
Review
Probability
obab ty
be a measure of the possibility that a random phenomenon occurs,
t find th regularity of these random phenomena.
to fi d the
l it f th
d
h
Basics of probability theory
p
y
y
Sample space
Complement
Element (or sample point)
Permutation
Permutation
Theorem
The number of permutations of n distinct objects is n!.
In general, the number of permutations of n distinct objects
taken r n at a time is
n!
n(n 1)(n 2) (n r 1)
(n r )!
There are r terms
denoted by
n
r
P
(Linear) Permuta
Conditional probability
Conditional probability
So far, all (unconditional) probabilities were calculated with respect
to the sample space S.
However, in many situations, we can obtain some new information
such that we can update the sample space.
More sp
Discrete Random Variables
Part I: Probability
1
Probability mass function
Consider a discrete random variable X and its range
If a function given by p(x) = P(X = x) for each
0 p(x) 1, for all
.
x satisfies
x , and p(x)=0 if x
p( x) 1
x
Then we call p(x
Probability
Table entry for z is the area under the standard normal curve to the left of z .
z
TABLE A Standard normal probabilities
z 3.4 3.3 3.2 3.1 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5
MATH144: Applied Statistics (L2) - Fall 09/10
Exercise 4 (Random Variables and Probability Distributions)
1. Determine the value c so that each of the following functions can serve as a probability distribution of
the discrete random variable X:
a) f (x)
Solutions for Exercise 5
1.
a) Responses of all people in Richmond who have telephones.
b) Outcomes for a large or innite number of tosses of a coin.
c) Length of life of such tennis shoes when worn on the professional tour.
d) All possible time intervals