...Chapter 5 Discrete Probability Distributions Learning Objectives 1. Understand the concepts of a random variable and a probability distribution. 2. Be able to distinguish between discrete and continuous random variables. 3. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable. 4. Be able to compute and work with probabilities involving a binomial probability distribution. 5. Be able to compute and work with probabilities involving a Poisson probability distribution. A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution n Poisson Distribution [pic] A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals. Example: JSL Appliances n Discrete random variable with a finite number of...
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...Answer Key Business Statistics Chapter I: 1.Which scale of measurement can be either numeric or nonnumeric? a. | Nominal | b. | Ratio | c. | Interval | d. | None of these alternatives is correct. | ANS: A PTS: 1 TOP: Descriptive Statistics 2.Which of the following can be classified as quantitative data? a. | interval and ordinal | b. | ratio and ordinal | c. | nominal and ordinal | d. | interval and ratio | ANS: D PTS: 1 TOP: Descriptive Statistics 3.A population is a. | the same as a sample | b. | the selection of a random sample | c. | the collection of all items of interest in a particular study | d. | always the same size as the sample | ANS: C PTS: 1 TOP: Descriptive Statistics 4.On a street, the houses are numbered from 300 to 450. The house numbers are examples of a. | categorical data | b. | quantitative data | c. | both quantitative and categorical data | d. | neither quantitative nor categorical data | ANS: A PTS: 1 TOP: Descriptive Statistics Chapter II: Exhibit 2-1 ========================================================================== The numbers of hours worked (per week) by 400 statistics students are shown below. Number of hours | Frequency | 0 - 9 | 20 | 10 - 19 | 80 | 20 - 29 | 200 | 30 - 39 | 100 | =========================================================================== 5.Refer to Exhibit 2-1. The class width for this distribution a. | is 9 | b. | is 10 | ...
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...Chapter 1 Discrete Probability Distributions 1.1 Simulation of Discrete Probabilities Probability In this chapter, we shall first consider chance experiments with a finite number of possible outcomes ω1 , ω2 , . . . , ωn . For example, we roll a die and the possible outcomes are 1, 2, 3, 4, 5, 6 corresponding to the side that turns up. We toss a coin with possible outcomes H (heads) and T (tails). It is frequently useful to be able to refer to an outcome of an experiment. For example, we might want to write the mathematical expression which gives the sum of four rolls of a die. To do this, we could let Xi , i = 1, 2, 3, 4, represent the values of the outcomes of the four rolls, and then we could write the expression X 1 + X 2 + X 3 + X4 for the sum of the four rolls. The Xi ’s are called random variables. A random variable is simply an expression whose value is the outcome of a particular experiment. Just as in the case of other types of variables in mathematics, random variables can take on different values. Let X be the random variable which represents the roll of one die. We shall assign probabilities to the possible outcomes of this experiment. We do this by assigning to each outcome ωj a nonnegative number m(ωj ) in such a way that m(ω1 ) + m(ω2 ) + · · · + m(ω6 ) = 1 . The function m(ωj ) is called the distribution function of the random variable X. For the case of the roll of the die we would assign equal probabilities or probabilities 1/6 to each of the outcomes....
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...Part I (Chapters 1 – 11) MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. A. Review of Basic Statistics (Chapters 1-11) Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain events. We use statistical methods and statistical analysis to make decisions in uncertain environment. Population: Sample: A population is the complete set of all items in which an investigator is interested. A sample is a subset of population values. & Example: Population - High school students - Households in the U.S. Sample - A sample of 30 students - A Gallup poll of 1,000 consumers - Nielson Survey of TV rating Random Sample: A random sample of n data values is one selected from the population in such a way that every different sample of size n has an equal chance of selection. & Example: Random Selection - Lotto numbers - Random numbers Random Variable: A variable takes different possible values for a given subject of study. Numerical Variable: A numerical variable takes some countable finite numbers or infinite numbers. Categorical Variable: A categorical variable takes values that belong to groups or categories. Data: Data are measured values of the variable. There are two types of data: quantitative data and qualitative data. 1 Part I (Chapters 1 – 11) Quantitative Data: Qualitative Data: & Example: 1. 2. 3. 3. 4. 5. 6. 7. 8. Statistics: Quantitative data are data measured on a numerical scale. Qualitative data are non-numerical...
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...215 CHAPTER Discrete Probability Distributions CONTENTS STATISTICS IN PRACTICE: CITIBANK 5.1 RANDOM VARIABLES Discrete Random Variables Continuous Random Variables 5.2 DEVELOPING DISCRETE PROBABILITY DISTRIBUTIONS 5.3 EXPECTED VALUE AND VARIANCE Expected Value Variance 5.4 BIVARIATE DISTRIBUTIONS, COVARIANCE, AND FINANCIAL PORTFOLIOS A Bivariate Empirical Discrete Probability Distribution Financial Applications Summary 5.5 BINOMIAL PROBABILITY DISTRIBUTION A Binomial Experiment Martin Clothing Store Problem Using Tables of Binomial Probabilities Expected Value and Variance for the Binomial Distribution POISSON PROBABILITY DISTRIBUTION An Example Involving Time Intervals An Example Involving Length or Distance Intervals HYPERGEOMETRIC PROBABILITY DISTRIBUTION 5 5.6 5.7 74537_05_ch05_p215-264.qxd 10/8/12 4:05 PM Page 219 5.1 Random Variables 219 Exercises Methods SELF test 1. Consider the experiment of tossing a coin twice. a. List the experimental outcomes. b. Define a random variable that represents the number of heads occurring on the two tosses. c. Show what value the random variable would assume for each of the experimental outcomes. d. Is this random variable discrete or continuous? 2. Consider the experiment of a worker assembling a product. a. Define a random variable that represents the time in minutes required to assemble the product. b. What values may the random variable assume? c. Is the random variable discrete or continuous...
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...Concepts 2 Expectations, Moments and Descriptive Statistics 3 Bivariate Distributions 4 Estimation 5 The Normal and Related Distributions and Interval Estimation 6 Hypothesis Testing These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also consult the reading as given in the unit outline and the lectures. 1 10 19 26 35 43 Chapter 1 Some Basic Concepts 1.1 Probability It is customary to begin courses in statistics with a discussion of probability and then go on to derive certain propositions in probability theory. The problem with this approach is that probability is a difficult and potentially confusing subject. The foundations of probability theory are not well established. There are at least three different ways of thinking about probability. (a) (b) (c) Relative Frequency Subjective belief Mathematical theory The relative frequency idea of probability is the oldest concept and dates back to the eighteenth century (and probably before) when European mathematicians become interested in analysing games of chance. The basic idea is that the probability of an event occurring is the relative number of times the event occurs in a given number of trials. This introduces the important idea that the probability of the event A occurring lies between 0 and 1, or 0 ≤ P(A) ≤ 1 (1.1) It also means that the probability of a series of outcomes from a gambling game can be written as a simple...
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...Statistics Chapter 5 - Review - A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula. A discrete random variable has either a finite number of values or a countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they can be associated with a counting process, so that the number of values is 0 or 1 or 2 or 3, etc. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions. Examples: 1.) Discrete: x = the number of eggs that a hen lays in a day. This is a discrete random variable because its only possible values are 0, or 1, or 2, and so on. No hen can lay 2.343115 eggs, which would have been possible if the data had come from a continuous scale. 2.) Continuous: x = the amount of milk a cow produces in one day. This is a continuous random variable because it can have any value over a continuous span. During a single day, a cow might yield an amount of milk that can be any value between 0 and 5 gallons. It would be possible to get 4.123456 gallons, because the cow is not restricted to the discrete amounts of 0,1,2,3,4, or 5 gallons. Requirements...
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... [pic] Discrete and Continuous Probability Distributions Business Statistics With Canadian Applications Hummelbrunner Rak Gray Third Edition Week6 Pages 261-263 chapter 8 Pages 288-314, 320-325 chapter 9 Arranged by: Neiloufar Aminneia Probability distribution A probability distribution is a list of all events of an experiment together with the probability associated with each event in a tabular form. It is used for business and economic problems. We learned frequency distribution to classify data, relating to actual observations and experiments but probability distribution describes how outcomes are expected to vary. Probability distribution for rolling a true die x P(x) Events Frequencies Probability 1 1 1/6 2 1 1/6 3 1 1/6 4 1 1/6 5 1 1/6 6 1 1/6 Total: 1 [pic] Probability distribution for tossing 3 coins x P(x) Events(# of Heads) Frequencies Probability 0 1 (TTT)) 1/8 = 0.125 ...
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...Lesson 2: Review of basic concepts of probability theory Coverage: Basic probability rules Random variables and associate concepts Normal distributions Reading: Chapter 2 (1-7), Chapter 3 (1-5, 10-11) and Chapter 4 (1-8) (1 8) Homework: Replicate and complete all the classroom exercises. Print answers of 2.1 (b-c-d), 2.3 (b-c-d) and 2.4(b-c-d) in one (1) page. 1 Business Statistics Lesson 2 - Page 2 Objectives At the end of the lesson, you should be able to: Define and apply the basic probability rules Describe the basic concepts related to random variables D ib th b i t l t dt d i bl Describe and use the properties of means and variances Recognize and understand the most commonly used probability distributions Use the basic data manipulation and descriptive statistical features of SPSS and transfer between SPSS and Excel SPSS, 1 Business Statistics • • • • Review of probability concepts Lesson 2 - Page 3 Probability: is defined on random events (occurrences), takes values between 0 and 1, and can be interpreted as limit of relative frequency (objective probability) Note: In everyday usage, probability might mean the extent of our belief in the occurrence of the event (subjective probability). However, statistics mostly deals with objective interpretation based on relative frequency. j p q y Basic probability rules: P( Sure event) = 1 and P( Impossible event) = 0 P(A or B) = P(A) + P(B) – P(A and B) Consequences: P( not A) = 1 - P(A)...
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...(Prerequisite: MAT 104) COURSE DESCRIPTION This course examines the principles of probability and of descriptive and inferential statistics. Topics include probability concepts, measures of central tendency, normal distributions, and sampling techniques. The application of these principles to simple hypothesis testing methods and to confidence intervals is also covered. The application of these topics in solving problems encountered in personal and professional settings is also discussed. INSTRUCTIONAL MATERIALS Required Resources ALEKS Access Code (bundled with course text when purchased from the Strayer Bookstore) Bluman, A. G. (2013). Elementary statistics: a brief version (6th ed.). New York, NY: McGraw-Hill. Note: Course materials for this class must be purchased from the Strayer Bookstore at http://www.strayerbookstore.com Supplemental Resources Hand, D. J. (2008). Statistics: a very short introduction. Oxford, UK: Oxford University Press. Rumsey, D. (2011). Statistics for dummies (2nd ed.). Hoboken, NJ: Wiley Publishing. Standard Normal Distribution Table. (2012). Retrieved from http://www.mathsisfun.com/data/standard-normal-distribution-table.html Statistics Calculator Free App for your Smartphone, created by Christian Gollner. Retrieved from https://play.google.com/store/apps/details?id=com.cgollner&hl=en COURSE LEARNING OUTCOMES 1. Describe the differences between the various types of data. 2. Apply various descriptive graphical techniques...
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...Six hours (you can take the Exam only at one stretch) The test will consist of 17 True/False type (4 points each), 16 Multiple/Choices (7 points each) and 8 Essay Type (15 points each) with a total of 300 points. The Chapters covered are 1, 2, 3, 5, 6, and 7. The topics to focus in these Chapters are listed below following the general guidelines. General Guidelines for Preparation for the Test At the outset I want you to realize that this is an open book test, but full preparation is absolutely necessary. If you are not prepared enough and need to turn the pages of the book or power point slides or my Instructions at every step, then it would not be possible for you to finish all questions in time. Also remember that the questions are based on the Text Book chapter topics and my Instructions but are not directly from the book (as was also true in your bi-weekly assignments). Review my Instructions on the six chapters covered in the Midterm. Besides, also review the Two Tri-Weekly assignment solutions. As you have already seen, I give the solution to the Tri-Weekly Assignments with necessary explanations and details immediately after I finish grading (usually within 48 hours from the deadline for assignment submission). And I provide ppt slides for all chapters covered in this course. Moreover, I provide hints and explanations in my responses to all your emails everyday of the week and to questions in the Virtual Office. So you have enough resources available to prepare for the...
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...DECISION TREES AND SIMULATIONS In the last chapter, we examined ways in which we can adjust the value of a risky asset for its risk. Notwithstanding their popularity, all of the approaches share a common theme. The riskiness of an asset is encapsulated in one number – a higher discount rate, lower cash flows or a discount to the value – and the computation almost always requires us to make assumptions (often unrealistic) about the nature of risk. In this chapter, we consider a different and potentially more informative way of assessing and presenting the risk in an investment. Rather than compute an expected value for an asset that that tries to reflect the different possible outcomes, we could provide information on what the value of the asset will be under each outcome or at least a subset of outcomes. We will begin this section by looking at the simplest version which is an analysis of an asset’s value under three scenarios – a best case, most likely case and worse case – and then extend the discussion to look at scenario analysis more generally. We will move on to examine the use of decision trees, a more complete approach to dealing with discrete risk. We will close the chapter by evaluating Monte Carlo simulations, the most complete approach of assessing risk across the spectrum. Scenario Analysis The expected cash flows that we use to value risky assets can be estimated in one or two ways. They can represent a probability-weighted average of cash flows under all possible...
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...Introduction to Discrete-Event System Simulation 1 Introduction to Simulation A simulation is the imitation of the operation of a real-world process or system over time. Whether done by hand or on a computer, simulation involves the generation of an artificial history of a system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system. The behavior of a system as it evolves over time is studied by developing a simulation model. This model usually takes the form of a set of assumptions concerning the operation of the system. These assumptions are expressed in mathematical, logical, and symbolic relationships between the entities, or objects of interest, of the system. Once developed and validated, a model can be used to investigate a wide variety of “what-if” questions about the realworld system. Potential changes to the system can first be simulated in order to predict their impact on system performance. Simulation can also be used to study systems in the design stage, before such systems are built. Thus, simulation modeling can be used both as an analysis tool for predicting the effect of changes to existing systems, and as a design tool to predict the performance of new systems under varying sets of circumstances. In some instances, a model can be developed which is simple enough to be “solved” by mathematical methods. Such solutions may be found by the use of differential calculus, probability theory, algebraic...
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...collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: • Exploring Data: Describing patterns and departures from patterns • Sampling and Experimentation: Planning and conducting a study • Anticipating Patterns: Exploring random phenomena using probability and simulation • Statistical Inference: Estimating population parameters and testing hypotheses Students who successfully complete the course and examination may receive credit and/or advanced placement for a one-semester introductory college statistics course. Textbook: The Practice of Statistics, 3rd ed. (2008) by Yates, Moore and Starnes (Freeman Publishers) Calculator needed: TI-83 Graphing Calculator (Rentals Available) TI-83+, TI-84, TI-84+ are acceptable calculators as well Note: Any other calculator may/may not have statistical capabilities, and the instructor shall assist whenever possible, but in these instances, the student shall have sole responsibility for the calculator’s use and application in this course. AP STATISTICS Textbook: The Practice of Statistics, 3rd edition by Yates, Moore and Starnes Preliminary Chapter – What Is Statistics? (2 Days) A. Where Do Data Come From? 1. Explain why we should not draw conclusions based on personal experiences. 2. Recognize whether a study is an experiment, a survey, or an observational study that is not a survey. 3. Determine the best method...
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...While, this unit provides opportunities for students to deal with situations involving probability, descriptive and inferential statistics, regression analysis and other basic mathematical tools used by business analysts at their workplaces, it intends to further expand student’s ability and interest into the area of economic research. Though the unit may seem mathematical at first there are certain characteristics that distinguish it from any mathematics unit. The unit provides students with the necessary tools to interpret results rationally. Finally, it provides the basis for a unit in econometrics at the higher level. The student shall be able to: • • • • • • • • • • • Differentiate the two general bodies of methods that together constitute the subject called statistics: descriptive and inferential statistics Understand the usage of graphical descriptive methods to summarise and describe sets of data understand the usage of numerical descriptive measures to summarise and describe sets of data Manage the data collection and sampling process Understand the basic concepts behind the rules and techniques of probability Acquire knowledge on the characteristics of probability distributions including binomial and other discrete probability distributions. Understand the importance of normal distribution and its practical usage. Link numerical descriptive statistics and probability distributions to statistical inference Understand the relevance of estimating population parameters and...
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