1. President UniversityErwin SitompulPBST 1/1 Dr.-Ing. Erwin Sitompul President University Lecture 1 Probability and Statistics http://zitompul.wordpress.com

2. President UniversityErwin SitompulPBST 1/2 Textbook: “Probability and Statistics for Engineers & Scientists”, 9 th Edition, Ronald E. Walpole, et. al., Pearson, 2010. Textbook and Syllabus Probability 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

3. President UniversityErwin SitompulPBST 1/3 Grade Policy Probability 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 (4 th 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.

4. President UniversityErwin SitompulPBST 1/4 Grade Policy Probability 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 Statistics Homework 2 R. Suhendra 009202100008 21 March 2022 No. 1. Answer:........

5. President UniversityErwin SitompulPBST 1/5 Chapter 1 Introduction

6. President UniversityErwin SitompulPBST 1/6 What is Probability? Chapter 1Introduction 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?

7. President UniversityErwin SitompulPBST 1/7 What is Statistics? Chapter 1Introduction Statistics is a tool to get information from data. Data Probability Statistics 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.

8. President UniversityErwin SitompulPBST 1/8 Branches of Statistics Chapter 1Introduction 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?

9. President UniversityErwin SitompulPBST 1/9 Steps in Inferential Statistics Chapter 1Introduction 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.

10. President UniversityErwin SitompulPBST 1/10 Chapter 2 Probability Chapter 2Probability

11. President UniversityErwin SitompulPBST 1/11 Some Terminologies Chapter 2.1Sample 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.

12. President UniversityErwin SitompulPBST 1/12 Sample Space Chapter 2.1Sample 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 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

13. President UniversityErwin SitompulPBST 1/13 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 Space Chapter 2.1Sample 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 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 Sample spaces with a large or infinite number of sample points are best described by a statement or rule.

14. President UniversityErwin SitompulPBST 1/14 Events Chapter 2.2Events 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 S 1, 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 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}.

15. President UniversityErwin SitompulPBST 1/15 Events Null set: a subset that contains no elements at all. It is denoted by the symbol  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’A’ Chapter 2.2Events

16. President UniversityErwin SitompulPBST 1/16 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. Two events A and B are mutually exclusive, or disjoint if A B = , that is, if A and B have no elements in common. Chapter 2.2Events A S B A B A S B A B = 

17. President UniversityErwin SitompulPBST 1/17 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. Chapter 2.2Events S B A B A 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 = ?

18. President UniversityErwin SitompulPBST 1/18 Events Chapter 2.2Events 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 = (b) A  B = (c) A’  B’ = {x | 0 < x < 9} {x | 1 ≤ x < 5} (A  B)’ ={x | 0 < x < 1, 5 ≤ x <12}

19. President UniversityErwin SitompulPBST 1/19 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 2.2Events S B A C 1 7 2 6 4 3 5 A  B=A  B= B  C=B  C= A  C=A  C= B’  A= A B  C= (A B) C’= {1, 2} {1, 3} {1, 2, 3, 4, 5, 7} {4, 7} {1} {2, 6, 7}

20. President UniversityErwin SitompulPBST 1/20 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 n 1 ways, and if for each of these a second operation can be performed in n 2 ways, then the two operations can be performed together in n 1 ·n 2 ways. Chapter 2.3Counting Sample Points How may sample points are in the sample space when a pair of dice is thrown once?

21. President UniversityErwin SitompulPBST 1/21 Counting Sample Points Chapter 2.3Counting 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 n 1 = 2, n 2 = 4, n 3 = 3, and n 4 = 5, there are n 1 ·n 2 ·n 3 ·n 4 = 2·4·3·5 = 120different ways to order the parts

22. President UniversityErwin SitompulPBST 1/22 Counting Sample Points Chapter 2.3Counting 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 n 1 = 3 choices for units position. However, the thousands position cannot be 0. If units position is 0, n 1 = 1, then we have n 2 = 5 choices for thousands position, n 3 = 4 for hundreds position, and n 4 = 3 for tens position. In this case, totally n 1 ·n 2 ·n 3 ·n 4 = 1·5·4·3 = 60 numbers. If units position is not 0, n 1 = 2, then we have n 2 = 4, n 3 = 4, and n 4 = 3. In this case, totally n 1 ·n 2 ·n 3 ·n 4 = 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?

23. President UniversityErwin SitompulPBST 1/23 Permutation A permutation is an arrangement of all or part of a set of objects. Chapter 2.3Counting Sample Points Consider the three letters a, b, and c. There are 6 distinct arrangements of them: abc, acb, bac, bca, cab, and cba. There are n 1 = 3 choices for the first position, then n 2 = 2 for the second, and n 3 = 1 choice for the last position, giving a total n 1 ·n 2 ·n 3 = 3·2·1 = 6 permutations.

24. President UniversityErwin SitompulPBST 1/24 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.” Chapter 2.3Counting Sample Points The number of permutations of n distinct objects is n! The number of permutations of n distinct objects taken r at a time is

25. President UniversityErwin SitompulPBST 1/25 Permutation Chapter 2.3Counting 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. d There are n 1 = 4 choices for the first position, and n 2 = 3 for the second, giving a total n 1 ·n 2 = 4·3 = 12 permutations. Another way, by using formula,

26. President UniversityErwin SitompulPBST 1/26 Permutation Chapter 2.3Counting Sample Points 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?

27. President UniversityErwin SitompulPBST 1/27 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) 50 P 2 (b) 49 P 1 + 49 P 2 (c) 2 P 2 + 48 P 2 (d) 50 P 2 – 2 or {2· 2 P 1 · 48 P 1 + 48 P 2 } Chapter 2.3Counting Sample Points ? For detailed explanation read the e-book.

28. President UniversityErwin SitompulPBST 1/28 Permutation The number of distinct permutations of n things of which n 1 are of one kind, n 2 of a second kind, …, n k of a k th kind is Chapter 2.3Counting Sample Points How many distinct permutations can be made from the letters a, a, b, b, c, and c?

29. President UniversityErwin SitompulPBST 1/29 Permutation Chapter 2.3Counting Sample Points 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?

30. President UniversityErwin SitompulPBST 1/30 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)? 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)