Transcript N - Erwin Sitompul

Probability and Statistics
Lecture 1
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
2 0 1 3
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Probability and Statistics
Textbook and Syllabus
Textbook:
“Probability and Statistics for Engineers &
Scientists”, 9th Edition, Ronald E. Walpole, et.
al., Pearson, 2010.
Syllabus:
 Chapter 1: Introduction
 Chapter 2: Probability
 Chapter 3: Random Variables and
Probability Distributions
 Chapter 4: Mathematical Expectation
 Chapter 5: Some Discrete Probability Distributions
 Chapter 6: Some Continuous Probability Distributions
 Chapter 8: Fundamental Sampling Distributions and
Data Descriptions
 Chapter 9: One- and Two-Sample Estimation Problems
 Chapter 10: One- and Two-Sample Tests of Hypotheses
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Probability and Statistics
Grade Policy
Final Grade = 10% Homework + 20% Quizzes +
30% Midterm Exam + 40% Final Exam +
Extra Points
 Homeworks will be given in fairly regular basis. The average of
homework grades contributes 10% of final grade.
 Homeworks are to be submitted on A4 papers, otherwise they
will not be graded.
Probability and Statistics
Homework 2
R. Suhendra
009202100008
21 March 2023
No. 1. Answer: . . . . . . . .
 Heading of Homework Papers (Required)
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Probability and Statistics
Grade Policy
 Homeworks must be submitted on time, one day before the
schedule of the lecture. Late submission will be penalized by point
deduction of –10·n, where n is the number of lateness made.
 There will be 3 quizzes. Only the best 2 will be counted. The
average of quiz grades contributes 20% of the final grade.
 Make up for quizzes must be requested within one week after the
date of the respective quizzes.
 Mid exam and final exam follow the schedule released by AAB
(Academic Administration Bureau).
 Make up for mid exam and final exam must be requested directly
to AAB.
 In order to maintain the integrity, the score of a make up quiz or
exam, upon discretion, can be multiplied by 0.9 (i.e., the
maximum score for a make up is then 90).
 Extra points will be given if you raise a question or solve a
problem in front of the class. You will earn 1, 2, or 3 points.
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Probability and Statistics
Grade Policy
 You are responsible to read and understand the lecture slides.
I am responsible to answer your questions.
 Lecture slides can be copied during class session. It also will be
available on internet around 1 days after class. Please check the
course homepage regularly.
http://zitompul.wordpress.com
 The use of internet for any purpose during class sessions is strictly
forbidden.
 You are expected to write a note along the lectures to record your
own conclusions or materials which are not covered by the lecture
slides.
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Chapter 1
Introduction
Chapter 1
Introduction
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Chapter 1
Introduction
What is Probability?
 Probability is the measure of the likeliness that a random event will
occur, or the knowledge upon an underlying model in figuring out
the chance that different outcomes will occur.
 By definition, probability values are between 0 and 1.
 Probability theory is the study of the mathematical rules that
govern random events.
 If we flip a fair coin 3 times, what is the probability of obtaining 3
heads?
 If we throw a dice 2 times, what is the probability that the sum of
the faces is 10?
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Chapter 1
Introduction
What is Statistics?
 Statistics is a tool to get information from data.
Data
Probability
Statistics
Information
• Knowledge about the
population concerning
some particular facts
• Facts (mostly numerical),
collected from a certain
population
 Statistics is used because the underlying model that governs a
certain experiments is not known.
 All that available is a sample of some outcomes of the experiment.
 The sample is used to make inference about the probability model
that governs the experiment.
 So, a thorough understanding of probability is essential to
understand statistics.
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Chapter 1
Introduction
Branches of Statistics
 Descriptive statistics, is the branch of statistics that involves the
organization, summarization, and display of data when the
population can be enumerated completely.
 Inferential statistics, is the branch of statistics that involves
using a sample of a population to draw conclusions about the
whole population. A basic tool in the study of inferential statistics
is probability.
 Descriptive statistics: There are 45 students in the Probability
and Statistics class. Twenty are younger than 24 years old. 16 are
older than 36 years old. What can be concluded?
 Inferential statistics: As many as 860 people in a Jakarta were
questioned. People who drives bicycle daily have average age of 31
years old. For people who drives motorcycle, the average age is
21. What can be concluded?
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Chapter 1
Introduction
Steps in Inferential Statistics
 Design the experiments and collect the data.
 Organize and arrange the data to aid understanding.
 Analyze the data and draw general conclusions from data.
 Estimate the present and predict the future.
 In conducing the steps mentioned above, Statistics use the
support of Probability, which can model chance mathematically
and enables calculations of chance in complicated cases.
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Chapter 1
Introduction
Why do ITs need Probability and Statistics?
 Security Design
How safe is your password?
 Risk Analysis
As IT has become increasingly important to the competitive position of
firms, managers have grown more sensitive to potential losses incurred by
companies because of problems with their sophisticated IT systems.
 Data Mining
An application of various statistical methods to huge databases. The goal
is to filter available records to produce association, rules, or pattern, that
may be used to the benefit of the user. <Ex: online store, software
testing, stock exchange>
 Randomized Algorithm
Some algorithms benefit from using random steps
deterministic steps. <Ex: optimization, numerical method>
rather
 Queuing and Reliability
than
The mathematical study of waiting time. Based on statistical data, the
optimum resource can be allocated to satisfy optimum number customers.
<Ex: supermarket, computers, public transports>
!And many more....
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Chapter 2
Probability
Chapter 2
Probability
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Chapter 2.1
Sample Space
Some Terminologies
 Data: result of observation that consists of information, in the
form of counts, measurements, or responses.
 Parameter: numerical description of a population characteristics.
 Statistic: numerical description of a sample characteristics.
 Population: the collection of all outcomes, counts,
measurements, or responses that are of interest.
 Sample: a subset of a population.
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Chapter 2.1
Sample Space
Sample Space
 Experiment: any process that generates a set of data.
 Sample space: the set of all possible outcomes of a statistical
experiment. It is represented by the symbol S.
 Element or member: each outcome in a sample space.
Sometimes simply called a sample point.
The sample space S, of possible outcomes when a coin is tossed may
be written as
S  H , T 
where H and T correspond to “heads” and “tails”, respectively.
The sample space can be written according to the point of interest.
Consider the experiment of tossing a die. The sample space can be
S1  1, 2, 3, 4, 5, 6
S2  even, odd
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Chapter 2.1
Sample Space
Sample Space
Suppose that three items are selected at random from a
manufacturing process. Each item is inspected and classified
defective, D, or nondefective, N. As we proceed along each possible
outcome, we see that the sample space is
S  DDD, DDN , DND, DNN , NDD, NDN , NND, NNN 
 Sample spaces with a large or infinite number of sample points are
best described by a statement or rule.
For example, if the possible outcomes of an experiment are the set
of cities in the world with a population over million, the sample space
is written
S   x x is a city with a population over 1 million
If S is the set of all points (x, y) on the boundary or the interior of a
circle of radius 2 with center at the origin, we write


S  ( x, y ) x 2  y 2  4
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Chapter 2.2
Events
Events
 A set is a collection of unique objects.
 A set A is a subset of another set B if every element of A is also
an element of B. We denote this as A B.
 Event: a subset of a sample space. We are interested in
probabilities of events.
The event A that the outcome when a die is tossed is divisible by 3 is
the subset of the sample space S1, and can be expressed as
A  3, 6
The event B that the number of defectives is greater than 1 in the
example on the previous slide can be written as
B  DDD, DDN , DND, NDD 
Given the sample space S = {t | t ≥ 0}, where t is the life in years of
a certain electronic components, then the event A that the
component fails before the end of the fifth year is the subset
A ={t|0 ≤ t < 5}.
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Chapter 2.2
Events
Events
 Null set: a subset that contains no elements at all. It is denoted
by the symbol .
B   x x is an even factor of 7  
 The complement of an event A with respect to S is the subset of
all elements of S that are not in A. We denote the complement of A
by the symbol A’.
Let R be the event that a red card is selected from an ordinary deck
of 52 playing cards, and let S be the entire deck. Then R’ is the
event that the card selected from the deck is not a red but a black
card.
S
A
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Chapter 2.2
Events
Events
 The intersection of two events A and B, denoted by A  B, is the
event containing all elements that are common to A and B.
S
A
B
A B
 Two events A and B are mutually exclusive, or disjoint if A  B
= , that is, if A and B have no elements in common.
S
A
B
A B = 
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Chapter 2.2
Events
Events
 The union of two events A and B, denoted by A  B, is the event
containing all elements that belong to A or B or both.
S
A
B
A B
Let A = {a, b, c} and B = {b, c, d, e}; then
A  B = {b, c}
A  B = {a, b, c, d, e}
If M = {x |3 < x < 9} and N = {y | 5 < y < 12}; then
M  N = {z | 3 < z <12}
MN =?
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Chapter 2.2
Events
Events
If S = {x | 0 < x < 12}, A = {x | 1 ≤ x < 9}, and B = {x | 0 < x < 5},
determine
(a) A  B
(b) A  B
(c) A’  B’
(a) A  B = {x | 0 < x < 9}
(b) A  B = {x | 1 ≤ x < 5}
(c) A’  B’ = (A  B)’
= {x | 0 < x < 1, 5 ≤ x <12}
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Chapter 2.2
Events
Venn Diagram
 Like already seen previously, the relationship between events and
the corresponding sample space can be illustrated graphically by
means of Venn diagrams.
S
A
B
2
7
4
1
6
3
5
C
AB
BC
AC
B’  A
A B  C
(A B) C’
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= {1, 2}
= {1, 3}
= {1, 2, 3, 4, 5, 7}
= {4, 7}
= {1}
= {2, 6, 7}
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Chapter 2.3
Counting Sample Points
Counting Sample Points
 Goal: to count the number of points in the sample space without
actually enumerating each element.
 |Multiplication Rule| If an operation can be performed in n1
ways, and if for each of these a second operation can be
performed in n2 ways, then the two operations can be performed
together in n1·n2 ways.
How may sample points are in the sample space when a pair of dice
is thrown once?
n1  6, n2  6
 n1  n2  36 possible ways
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Chapter 2.3
Counting Sample Points
Counting Sample Points
Sam is going to assemble a computer by himself. He has the choice
of ordering chips from two brands, a hard drive from four, memory
from three and an accessory bundle from five local stores. How
many different ways can Sam order the parts?
Since n1 = 2, n2 = 4, n3 = 3, and n4 = 5,
there are n1·n2·n3·n4 = 2·4·3·5 = 120 different ways to
order the parts
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Chapter 2.3
Counting Sample Points
Counting Sample Points
How many even four-digit numbers can be formed from the digits 0,
1, 2, 5, 6, and 9 if each number can be used only once?
For even numbers, there are n1 = 3 choices for units position.
However, the thousands position cannot be 0.
If units position is 0, n1 = 1, then we have n2 = 5 choices for
thousands position, n3 = 4 for hundreds position, and n4 = 3 for tens
position. In this case, totally n1·n2·n3·n4 = 1·5·4·3 = 60 numbers.
If units position is not 0, n1 = 2, then we have n2 = 4, n3 = 4, and
n4 = 3. In this case, totally n1·n2·n3·n4 = 2·4·4·3 = 96 numbers.
The total number of even four-digit numbers can be calculated by
60 + 96 = 156.
?
How if each number can be
used more than once?
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Chapter 2.3
Counting Sample Points
Permutation
 A permutation is an arrangement of all or part of a set of objects.
Consider the three letters a, b, and c. There are 6 distinct
arrangements of them: abc, acb, bac, bca, cab, and cba.
There are n1 = 3 choices for the first position, then n2 = 2 for the
second, and n3 = 1 choice for the last position, giving a total
n1·n2·n3 = 3·2·1 = 6 permutations.
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Chapter 2.3
Counting Sample Points
Permutation
 In general, n distinct objects can be arranged in
n(n–1)(n–2) · · · (3)(2)(1) ways.
 This product is represented by the symbol n!, which is read “n
factorial.”
 The number of permutations of n distinct objects is n!
 The number of permutations of n distinct objects taken r at a time
is
n Pr 
n!
(n  r )!
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Chapter 2.3
Counting Sample Points
Permutation
Consider the four letters a, b, c, and d. Now consider the number of
permutations that are possible by taking 2 letters out of 4 at a time.
The possible permutations are ab, ac, ad, ba, bc, bd, ca, cb, cd, da,
db, and dc.
d
There are n1 = 4 choices for the first position, and n2 = 3 for the
second, giving a total n1·n2 = 4·3 = 12 permutations.
Another way, by using formula,
4!
 12
4 P2 
(4  2)!
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Chapter 2.3
Counting Sample Points
Permutation
Three awards (research, teaching and service) will be given one year
for a class of 25 graduate students in a statistics department. If each
student can receive at most one award, how many possible
selections are there?
25 P3 
25!
25!
25  24  23  22!
 13,800


(25  3)!
22!
22!
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Chapter 2.3
Counting Sample Points
Permutation
A president and a treasurer are to be chosen from a student club
consisting of 50 people. How many different choices of officers are
possible if
(a) There are no restrictions
(b) A will serve only if he is president
(c) B and C will serve together or not at all
(d) D and E will not serve together
(a) 50P2
(b) 49P1 +
(c) 2P2 +
49P2
48P2
(d) 50P2 – 2
or
{2·2P1·48P1 + 48P2}
?
For detailed explanation read
the e-book.
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Chapter 2.3
Counting Sample Points
Permutation
 The number of distinct permutations of n things of which n1 are of
one kind, n2 of a second kind, …, nk of a kth kind is
n!
n1 !n2 ! nk !
How many distinct permutations can be made from the letters a, a,
b, b, c, and c?
6!
 90
2!2!2!
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Chapter 2.3
Counting Sample Points
Permutation
In a college football training session, the defensive coordinator needs
to have 10 players standing in a row. Among these 10 players, there
are 1 freshman, 2 sophomore, 4 juniors, and 3 seniors, respectively.
How many different ways can they be arranged in a row if only their
class level will be distinguished?
10!
 12, 600
1!2!4!3!
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Probability and Statistics
Homework 1A
1. Disk of polycarbonate plastic from a
supplier are analyzed for scratch
and shock resistance. The result
from 100 disks are summarized as
follows.
Let A denote the event that a disk
has high shock resistance, and let B
denote the event that a disk has
high scratch resistance.
(a) Determine the number of disks in A B, A’, and A  B.
(Mo.2.26)
(b) Construct a Venn Diagram that represents the analysis result above.
Can you indicate all the events mentioned in (a)?
2. Two balls are “randomly drawn” from a bowl containing 6 white and 5
black balls. What is the probability that one of the drawn balls is white
and the other black?
(Ro.E3.5a)
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Probability and Statistics
Homework 1B
1. You are given two boxes with balls numbered 1 to 5. One box contains
balls 1, 3, and 5. The other box contains balls 2 and 4. You first pick a
box at random, then pick a ball from that box at random. What is the
probability that you pick the ball number 2?
(Utah.L2)
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