President UniversityErwin SitompulPBST 1/1 Dr.-Ing. Erwin Sitompul President University Lecture 1...

President University Erwin Sitompul PBST 1/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 1

Probability and Statistics

http://zitompul.wordpress.com


Textbook:“Probability and Statistics for Engineers & Scientists”, 9th Edition, Ronald E. Walpole, et. al., Pearson, 2010.

Textbook and SyllabusProbability and Statistics

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


Grade PolicyProbability and Statistics

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.


Grade PolicyProbability and Statistics

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.

Heading of Homework Papers (Required)

Probability and StatisticsHomework 2

R. Suhendra009202100008

21 March 2022

No. 1. Answer: . . . . . . . .


Chapter 1

Introduction


What is Probability?Chapter 1 Introduction

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?


What is Statistics?Chapter 1 Introduction

Statistics is a tool to get information from data.

Data

ProbabilityStatistics Information

• Facts (mostly numerical), collected from a certain population

• Knowledge about the population concerning some particular facts

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.


Branches of StatisticsChapter 1 Introduction

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?


Steps in Inferential StatisticsChapter 1 Introduction

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.


Chapter 2

Probability

Chapter 2 Probability


Some TerminologiesChapter 2.1 Sample Space

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.


Sample SpaceChapter 2.1 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.

,S H T

The sample space S, of possible outcomes when a coin is tossed may be written as

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

1 1, 2, 3, 4, 5, 6S

2 even, oddS


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

Sample SpaceChapter 2.1 Sample Space

, , , , , , ,S DDD DDN DND DNN NDD NDN NND NNN

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 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

2 2( , ) 4S x y x y

Sample spaces with a large or infinite number of sample points are best described by a statement or rule.


EventsChapter 2.2 Events

Event: a subset of a sample space. We are interested in probabilities of events.

3, 6A

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

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}.


Events Null set: a subset that contains no elements at all. It is denoted by

the symbol Æ.

is an even factor of 7 B x x

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.

A

S

A’

Chapter 2.2 Events


Events The intersection of two events A and B, denoted by A Ç B, is the

event containing all elements that are common to A and B.

Two events A and B are mutually exclusive, or disjoint if A Ç B = Æ, that is, if A and B have no elements in common.

Chapter 2.2 Events

A

S

B

A Ç B

A

S

B

A Ç B = Æ


Events The union of two events A and B, denoted by A È B, is the event

containing all elements that belong to A or B or both.

Chapter 2.2 Events

S

B

A È B

A

Let A = {a, b, c} and B = {b, c, d, e}; thenA Ç B = {b, c}A È B = {a, b, c, d, e}

If M = {x |3 < x < 9} and N = {y | 5 < y < 12}; then M È N = {z | 3 < z <12}M Ç N = ?


EventsChapter 2.2 Events

If S = {x | 0 < x < 12}, A = {x | 1 ≤ x < 9}, and B = {x | 0 < x < 5}, determine(a) A È B(b) A Ç B(c) A’ È B’

(a) A È B =

(b) A Ç B =

(c) A’ È B’ =

{x | 0 < x < 9}

{x | 1 ≤ x < 5}

(A Ç B)’

= {x | 0 < x < 1, 5 ≤ x <12}


Venn Diagram Like already seen previously, the relationship between events and

the corresponding sample space can be illustrated graphically by means of Venn diagrams.

Chapter 2.2 Events

SBA

C

17

26

4 3

5

A Ç B = B Ç C = A È C =

B’ Ç A = A Ç B Ç C =

(A È B) Ç C’ =

{1, 2}{1, 3}{1, 2, 3, 4, 5, 7}{4, 7}{1}{2, 6, 7}


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.

Chapter 2.3 Counting Sample Points

How may sample points are in the sample space when a pair of dice is thrown once?

1 2

1 2

6, 636 possibleways

n nn n


Counting Sample PointsChapter 2.3 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


Counting Sample PointsChapter 2.3 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?


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.


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 r

nP

n r


PermutationChapter 2.3 Counting Sample Points

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.

4 2

4!

(4 2)!P

12

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,


13,800


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 3

25!

(25 3)!P

25!

22! 25 24 23 22!

22!


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 + 49P2

(c) 2P2 + 48P2

(d) 50P2 – 2 or {2·2P1·48P1 + 48P2}


?For detailed explanation read the e-book.


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


1 2

!

! ! !k

n

n n n

How many distinct permutations can be made from the letters a, a, b, b, c, and c?

906!

2!2!2!


12,600


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!

1!2!4!3!


Homework 11. 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 Ç B, A’, and A È B.

(Mo.2.26)

(b) Construct a Venn Diagram that represents the analysis result above. Can you indicate all the events mentioned in (a)?

Probability and Statistics

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)

President UniversityErwin SitompulPBST 1/1 Dr.-Ing. Erwin Sitompul President University Lecture 1...

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