Transcript B - Erwin Sitompul

Probability and Statistics
Lecture 1
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
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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 7: Functions of Random Variables
 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 =
5% Homework + 30% Quizzes +
30% Midterm Exam + 40% Final Exam +
Extra Points
 Homeworks will be given in fairly regular basis. The average of
homework grades contributes 5% of final grade.
 Homeworks are to be submitted on A4 papers, otherwise they
will not be graded.
 Homeworks must be submitted on Tuesday evening in my office
(4th floor) at the latest. Lateness will cause deduction of –40
points
 There will be 3 quizzes. Only the best 2 will be counted. The
average of quiz grades contributes 30% of the final grade.
 Midterm and final exam schedule will be announced in time.
 Make up of quizzes and exams will be held one week after the
schedule of the respective quizzes and exams.
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Probability and Statistics
Grade Policy
 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.
 You are responsible to read and understand the lecture slides. I
am responsible to answer your questions.
Probability and Statistics
Homework 2
R. Suhendra
009202100008
21 March 2022
No. 1. Answer: . . . . . . . .
 Heading of Homework Papers (Required)
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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.
 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 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
 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 1
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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