...CHAPTER 6 RANDOM VARIABLES PART 1 – Discrete and Continuous Random Variables OBJECTIVE(S): • Students will learn how to use a probability distribution to answer questions about possible values of a random variable. • Students will learn how to calculate the mean and standard deviation of a discrete random variable. • Students will learn how to interpret the mean and standard deviation of a random variable. Random Variable – Probability Distribution - Discrete Random Variable - The probabilities of a probability distribution must satisfy two requirements: a. b. Mean (expected value) of a discrete random variable [pic]= E(X) = = 1. In 2010, there were 1319 games played in the National Hockey League’s regular season. Imagine selecting one of these games at random and then randomly selecting one of the two teams that played in the game. Define the random variable X = number of goals scored by a randomly selected team in a randomly selected game. The table below gives the probability distribution of X: Goals: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001 a. Show that the probability distribution for X is legitimate. b. Make a histogram of the probability distribution. Describe what you see. 0.25 0.20 0.15 0.10 ...
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...Additional information, including supplemental material and rights and permission policies, is available at http://ite.pubs.informs.org. Vol. 9, No. 1, September 2008, pp. 1–9 issn 1532-0545 08 0901 0001 informs ® doi 10.1287/ited.1080.0014 © 2008 INFORMS INFORMS Transactions on Education Using Simulation to Model Customer Behavior in the Context of Customer Lifetime Value Estimation Shahid Ansari, Alfred J. Nanni Accounting and Law Division, Babson College, Wellesley, Massachusetts 02457 {sansari@babson.edu, nanni@babson.edu} Dessislava A. Pachamanova, David P. Kopcso Mathematics and Science Division, Babson College, Wellesley, Massachusetts 02457 {dpachamanova@babson.edu, kopcso@babson.edu} T his article illustrates how simulation can be used in the classroom for modeling customer behavior in the context of customer lifetime value estimation. Operations research instructors could use this exercise to introduce multiperiod spreadsheet simulation models in a business setting that is of great importance in practice, and the simulation approach to teaching this subject could be of interest also to marketing and accounting instructors. At Babson College, the spreadsheet simulation exercise is part of an integrated one-case teaching day of the marketing, accounting, and operations research disciplines in the full-time MBA program, but the exercise is directly transferable to stand-alone courses as well. In our experience, students have...
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...Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #07 Random Variables So, far we were discussing the laws of probability so, in the laws of the probability we have a random experiment, as a consequence of that we have a sample space, we consider a subset of the, we consider a class of subsets of the sample space which we call our event space or the events and then we define a probability function on that. Now, we consider various types of problems for example, calculating the probability of occurrence of a certain number in throwing of a die, probability of occurrence of certain card in a drain probability of various kinds of events. However, in most of the practical situations we may not be interested in the full physical description of the sample space or the events; rather we may be interested in certain numerical characteristic of the event, consider suppose I have ten instruments and they are operating for a certain amount of time, now after amount after working for a certain amount of time, we may like to know that, how many of them are actually working in a proper way and how many of them are not working properly. Now, if there are ten instruments, it may happen that seven of them are working properly and three of them are not working properly, at this stage we may not be interested in knowing the positions, suppose we are saying one instrument, two instruments and so, on tenth...
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...Course Outline: Probability and Statistics Week Tue-Sat 1 2 3 4 Topics/Sub-Topics Introduction to Statistics What is Statistics, Definition of Statistics, Types of Statistics, Application of Statistics in Real life, Variable and its types, Constant and its types. Definition of Data, Primary & Secondary Data, Frequency and Frequency Distribution, Class Limit & Boundary. Organizing and Graphing Data Organizing and Graphing of Qualitative (Simple Bar Chart, Multiple Bar Chart, Percentage Pie Chart) data. Organizing and Graphing of Quantitative data (Stem and Leaf Plot, Histogram, Frequency Polygon, Ogive) Numerical descriptive Measures Measure of central tendency for Ungrouped and Grouped Data (Mean, Median and Mode), Measure of Dispersion and its Types. Cox-Box Plot Variance and use of Standard Deviation, Co-efficient of Variation. Introduction to moment. Week Tue-Sat 5 6 7 8 9 Quizzes/ Assignments Topics/Sub-Topics Moment about origin and Central Moments for Frequency Distribution, Moment Ratios and its interpretation. Introduction to Probability Counting Principle, Probability and its Approaches, Deterministic and non-deterministic Experiment, Sample Space and Events, Outcome, Permutation and Combination. Types of Events Mutually Exclusive Events, Collectively Exhaustive events, Complementary events, Addition Laws of Probability. MID TERM EXAMINATION Assignment 1 Quiz 1 Quizzes/ Assignments Assignment...
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...CHAPTER 1 : INTRODUCTION 1.1. BACKGROUND OF STUDY Life at the turn of the 21st century are exceptionally testing and not the us effectively we anticipated. Besides, the monetary emergency that is hitting the world these days is no special case for Malaysia likewise influences somewhat by ordinary life in life. As an aftereffect of this, of numerous who wander into the business to oblige the minimal present as a consequence of the present downturn now including understudies. This study is to see the effect of the college understudy working low maintenance on the execution of learning. The purpose of this study was to examine the work while the impression towards academic achievements. Percentage shows between 55% to 80% of students will work while learning (Miller, 1997; King, 1998). This high percentage is also causing some researchers to believe that the students who will work towards the achievement of academic decline (Steinberg, Dornbusch, & Fegley 1993). At the same time also, there are a discovered that work while learning provides a positive impact if they follow the correct percentage (In & Hoyt, 1981). Inquiry about "impression of working part time on academic Achievement" is mixed. Along these lines, the study will endeavour to give more proof of a much clearer and point by point to comprehend the impression of working low maintenance towards scholastic accomplishments in North Malaysia College understudies particularly. 1.2. STATEMENTS OF PROBLEM...
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...Chapter 6 Statistical Process Control 6.0 Introduction One of the axioms or truisms in law of nature is “No two items of any category at any instant in the universe are the same”. Manufacturing process is no exception to it. It means that variability is part of life and is an inherent property of any process. Measuring, monitoring and managing are rather engineers’ primary job in the global competition. A typical manufacturing scenario can be viewed as shown in the Figure 6.1. That is if one measures the quality characteristic of the output, he will come to know that no two measured characteristics assume same value. This way the variablility conforms one of the axioms or truisms of law of nature; no two items in the universe under any category at any instant will be exactly the same. In maunufacturing scenario, this variability is due to the factors (Random variables) acting upon the input during the process of adding value. Thus the process which is nothing but value adding activity is bound ot experience variability as it is inherent and integral part of the process. Quality had been defined in many ways. Quality is fitness for use is the most common way of looking at it. This fitness for use is governed by the variability. In a maufacturing scenario, despite the fact that a machine operator uses the same precision methods and machines and endeavours to produce identical parts, but the finished products will show a definite variablity. The variability of a product...
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... 7 2 Graphs and Displays 2.1 9 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Medians, Modes, and Means Revisited . . . . . . . . . . . 10 2.1.3 z-Scores and Percentile Ranks Revisited . . . . . . . . . . 11 2.2 Stem and Leaf Displays . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Five Number Summaries and Box and Whisker Displays . . . . . 12 3 Probability 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2.2 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.3 Variance and Standard Deviation . . . . . . . . . . . . . . 17 3.2.4 “Shortcuts” for Binomial Random Variables . . . . . . . . 18 1 4 Probability Distributions 19 4.1 Binomial Distributions . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Poisson Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.2 As an Approximation to the Binomial . . . . . . . . . . . 22 Normal Distributions . ....
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...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. With this assignment of probabilities, one could write P (X ≤ 4) = 1 2 3 2 CHAPTER 1. DISCRETE PROBABILITY DISTRIBUTIONS to mean that the probability is 2/3 that a roll of a die will have a value which does not exceed 4. Let Y be the random variable which represents the toss of a coin. In this case, there are two possible outcomes, which we can label as H and T. Unless we have reason...
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...can begin to use probabilistic ideas in statistical inference and modelling, and the study of stochastic processes. Probability axioms. Conditional probability and independence. Discrete random variables and their distributions. Continuous distributions. Joint distributions. Independence. Expectations. Mean, variance, covariance, correlation. Limiting distributions. The syllabus is as follows: 1. Basic notions of probability. Sample spaces, events, relative frequency, probability axioms. 2. Finite sample spaces. Methods of enumeration. Combinatorial probability. 3. Conditional probability. Theorem of total probability. Bayes theorem. 4. Independence of two events. Mutual independence of n events. Sampling with and without replacement. 5. Random variables. Univariate distributions - discrete, continuous, mixed. Standard distributions - hypergeometric, binomial, geometric, Poisson, uniform, normal, exponential. Probability mass function, density function, distribution function. Probabilities of events in terms of random variables. 6. Transformations of a single random variable. Mean, variance, median, quantiles. 7. Joint distribution of two random variables. Marginal and conditional distributions. Independence. iii iv 8. Covariance, correlation. Means and variances of linear functions of random variables. 9. Limiting distributions in the Binomial case. These course notes explain the naterial in the syllabus. They have been “fieldtested” on the class of 2000. Many of the examples...
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...little space to motivate this. Suppose that the price of a stock or other asset at time 0 is known to be S(0) and we want to model its future price S(10) at time 10—note that some texts use the notation S0 and S10 instead. Let’s break the time interval from 0 to 10 into 10,000 pieces of length 0.001, and let’s let Sk stand for S(0.001k), the price at time 0.001k. I know the price S0 = S(0) and want to model the price S10000 = S(10). I can write (1.1) S(10) = S10000 = S2 S1 S10000 S9999 ··· S0 . S9999 S9998 S1 S0 Now suppose that the ratios Rk = SSk that appear in Equation 1.1 that represent the growth factors k−1 in price over each interval of length 0.001 are random variables, and—to get a simple model—are all independent of one another. Then Equation 1.1 writes S(10) as a product of a large number of independent random...
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...times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. Quiz: __________________________________ is the chance or likelihood of some event occurring. The _________________________ of rolling 1 on a 12 sided dice is 1/12. __________________________ is the result of a trial of a probability experiment. I pulled a red marble out of a bag that had 5 red marbles and 5 blue marbles the _____________________ is that I pulled a red marble out. Find the probability of selecting a letter at random from the word MISSISSIPPI. Write the probability as a fraction. P(M) P(I)...
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...COURSE SYLLABUS BMGT 230 - BUSINESS STATISTICS Summer Session 0301 - 2014 Instructor Information Professor: Frank B. Alt (falt@rhsmith.umd.edu ) Office: 4323 Van Munching Hall (VMH) Office Hours: After all teaching days (2:00-3:00 p.m.) and by appointment Office Phone: 301-405-2231 Course Assistant Mr. Daniel Klein Office Hours: After all class days (except 6/19) from 3:00pm – 4:30pm Office: 4308 Email: dklein99@terpmail.umd.edu Class Information Classroom: Van Munching Hall, Room 1330 Meeting Times: 10:00 a.m. - 1:10 p.m. Meeting Dates: June 2 - 5 (Monday – Thursday) June 9 - 12 (Monday – Thursday) June 16 – 19 (Monday – Thursday) Information regarding official university closings and delays can be found at the campus website or by calling the weather emergency phone line (301-405-7669). If a class is cancelled, the dates on the Course Outline will be changed to reflect this. Students will be notified of such changes by an email from me. Please refer to the inclement weather policy on page 3. Required Course Materials Text: Basic Statistical Ideas for Managers, 2nd ed, D. Hildebrand, R. Ott and J. Gray, Duxbury Press (Thompson-Brooks/Cole), 2005, ISBN 0-534-37805-6. The text comes with a CD-ROM containing an Excel Add-in and Data Sets. If your text does not have the CD, that is okay since I can post the data sets and we will not...
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...Syllabus Cambridge International A Level Further Mathematics Syllabus code 9231 For examination in June and November 2013 Contents Cambridge A Level Further Mathematics Syllabus code 9231 1. Introduction ..................................................................................... 2 1.1 1.2 1.3 1.4 Why choose Cambridge? Why choose Cambridge International A Level Further Mathematics? Cambridge Advanced International Certificate of Education (AICE) How can I find out more? 2. Assessment at a glance .................................................................. 5 3. Syllabus aims and objectives ........................................................... 7 4. Curriculum content .......................................................................... 8 4.1 Paper 1 4.2 Paper 2 5. Mathematical notation................................................................... 17 6. Resource list .................................................................................. 22 7 Additional information.................................................................... 26 . 7 .1 7 .2 7 .3 7 .4 7 .5 7 .6 Guided learning hours Recommended prior learning Progression Component codes Grading and reporting Resources Cambridge A Level Further Mathematics 9231. Examination in June and November 2013. © UCLES 2010 1. Introduction 1.1 Why choose Cambridge? University of Cambridge International Examinations (CIE) is the world’s largest provider of international...
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...rushing attempt, both of which rank No.1 in FBS; The Spartans lead the FBS in time of possession (34:56 per game).(SpartyOn.com) In Michigan, from 2009 to 2013 there were 88.9 percent of people age of 25 above have education of high school graduate or higher which is higher than United State average of 86.0 percent, and also have 25.9 percent people have Bachelor’s degree or higher. In addition, Michigan has an average salary of jobs of $58,000. (Census.gov) Where these numbers came from? It is Statistics. “People cannot make research without statistics and analysis.”(Zeleke) Statistics is important and critical, it can solve for society problem. Nowadays, we have a lot of technological advancement, mobile technology and data in our daily life. The world becomes more and more complex. We need people to be proficient to understand all information out there. How do we process? What is important? How do we analyze the information and make the right decision? Now, there are lots of problem that we are facing: human, race, climate change, economics hardships, and political instability etc. For solving these problems, we need statistics and modeling. Statistics become more and more important; it is not only for statistician but also for everyone. Statistics not only help us organize the complicate data, but also help society. In other words, Statistics is also one of the strongest evidence; it can help people decide more things. Such as the research "Single statistic can strengthen...
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...DESCRIPTIVE STATISTICS & PROBABILITY THEORY 1. Consider the following data: 1, 7, 3, 3, 6, 4 the mean and median for this data are a. 4 and 3 b. 4.8 and 3 c. 4.8 and 3 1/2 d. 4 and 3 1/2 e. 4 and 3 1/3 2. A distribution of 6 scores has a median of 21. If the highest score increases 3 points, the median will become __. a. 21 b. 21.5 c. 24 d. Cannot be determined without additional information. e. none of these 3. If you are told a population has a mean of 25 and a variance of 0, what must you conclude? a. Someone has made a mistake. b. There is only one element in the population. c. There are no elements in the population. d. All the elements in the population are 25. e. None of the above. 4. Which of the following measures of central tendency tends to a. be most influenced by an extreme score? b. median c. mode d. mean 5. The mean is a measure of: a. variability. b. position. c. skewness. d. central tendency. e. symmetry. 6. Suppose the manager of a plant is concerned with the total number of man-hours lost due to accidents for the past 12 months. The company statistician has reported the mean number of man-hours lost per month but did not keep a record of the total sum. Should the manager order the study repeated to obtain the desired information? Explain...
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