...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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...implies a higher market value for the property. The expiration of some of the rental leases also present an uncertainty as it is not clear if the company will be able to find a new tenant before May 2011. The company will want to minimize the vacancies in order to attract the best bid. It is reasonable to assume that $22.5 million is the best bid that the company can obtain. Using this assumption, we can estimate the likelihood that the company will be able to obtain a higher bid than $22.5 million. The company expects to earn a 10% hurdle rate on its investment. Therefore, the future expected value of the company will have to be discounted at 10% before comparing it with the present bid. Due to the uncertainties associated with the future value of the property, it is impossible to estimate the exact value of the property. However, we can estimate the distribution of the possible future property values. The distribution can be constructed by generating a random distribution of the tenant mix after one year. The distribution is based on the estimated probabilities of the lease renewal. The capitalization rate after one year can also be assumed to randomly vary between 8% and 10%. After one thousand iterations of this probability weighted scenario analysis, we can arrive at a range of the possible property prices in the future. The distribution of the present value of these property prices is shown in the histogram below. The distribution of the histogram can be assumed approximately...
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...papers are of a preliminary nature, it may be useful to contact the author of a particular working paper about results or caveats before referring to, or quoting, a paper. Any comments on working papers should be sent directly to the author. Markov or Not Markov – This Should Be a Question Abstract: Although it is well known that Markov process theory, frequently applied in the literature on income convergence, imposes some very restrictive assumptions upon the data generating process, these assumptions have generally been taken for granted so far. The present paper proposes, resp. recalls chi-square tests of the Markov property, of spatial independence, and of homogeneity across time and space to assess the reliability of estimated Markov transition matrices. As an illustration we show that the evolution of the income distribution across the 48 coterminous U.S. states from 1929 to 2000 clearly has not followed a Markov process. Keywords: Convergence, Markov process, chi-square tests, U.S. regional growth JEL classification: C12, O40, R11 Frank Bickenbach Kiel Institute of World Economics 24100 Kiel, Germany Telephone: +49/431/8814-274 Fax: +49/431/8814-500 E-mail: fbickenbach@ifw.unikiel.de Eckhardt Bode Kiel Institute of World Economics 24100 Kiel, Germany Telephone: +49/431/8814-462 Fax: +49/431/8814-500 E-mail: ebode@ifw.uni-kiel.de 3 1. Introduction Since the late 1980s the issue of convergence or divergence of per-capita income ...
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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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...Institute of Management Sciences Peshawar Bachelors in Business Studies Course Plan Course Title: Statistics for Business Instructor: Shahid Ali Contact Email shahid.ali@imsciences.edu.pk Semester/Duration: 16 Weeks Course objectives : To introduce students to the concepts of statistics and to equip them with analytical tools to be used in business decision making. The course is intended to polish the numeric ability of the students to identify business problems, describe them numerically and to provide intelligible solutions by data collection and inferential principles. Course pre-requisites Intermediate statistics Attendance Policy: Late arrivals are highly discouraged. Any student coming late to a class late by 5 minutes after the scheduled start time will be marked as absent for the day. The teacher reserves discretion, however, to allow or disallow any student, to sit in the class in case of late arrivals. Attendance is not be entertained once the attendance register is closed. Class Project Students will be divided in groups for a class project. Each group will have to nominate a group leader. The details of the project will be made available to the group leader. Class Presentations Each student will have to make at least one individual presentation and one group presentation in the class. The group presentation will be on the project explained earlier. The individual presentations will...
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...CHAPTER 6: THE NORMAL DISTRIBUTION AND OTHER CONTINUOUS DISTRIBUTIONS 1. In its standardized form, the normal distribution a) has a mean of 0 and a standard deviation of 1. b) has a mean of 1 and a variance of 0. c) has an area equal to 0.5. d) cannot be used to approximate discrete probability distributions. ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: standardized normal distribution, properties 2. Which of the following about the normal distribution is NOT true? a) Theoretically, the mean, median, and mode are the same. b) About 2/3 of the observations fall within 1 standard deviation from the mean. c) It is a discrete probability distribution. d) Its parameters are the mean, , and standard deviation, . ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 3. If a particular batch of data is approximately normally distributed, we would find that approximately a) 2 of every 3 observations would fall between 1 standard deviation around the mean. b) 4 of every 5 observations would fall between 1.28 standard deviations around the mean. c) 19 of every 20 observations would fall between 2 standard deviations around the mean. d) all of the above ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 4. For some positive value of Z, the probability that a standardized normal variable is between 0 and Z is 0.3770. The value of Z is a) 0.18. b) 0.81. c) 1.16. d) 1...
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...In addition it will provide necessary statistical knowledge and wide rage of ways to analyze data, which will improve the students statistical analytical and decision making skills. Session Lecture Outline Learning Objectives 01 Basic Probability and Discrete Probability Distributions Simple Probability To develop an understanding of basic probability concepts To introduce conditional probability To use Bayes’ Theorem to revise probabilities in light of new information To provide an understanding of the basic concepts of discrete probability distributions and their characteristics To develop the concept of mathematical expectation for a discrete random variable To introduce the covariance and illustrate its application in finance To present applications of the binomial distribution in business To present applications of the Poisson distribution in business 02 Counting Techniques 03 Continued 04 Conditional Probability 05 Discrete probability distribution. 06 Mathematical Expectation of Discrete Random Variable 07 Properties of mean and variance of a discrete random Variable. 08 Binomial Distribution. 09 Poisson Distribution. 10. The Normal Distribution and Sampling Distribution Introduction Mathematical Models of Continuous Random Variables To...
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...Introduction to Probability Randomness and statistical regularity There are many instances in nature for which we cannot predict the possible event that may happen. We can say in these cases that the occurrence of the event is random. However for whatever reason that the event cannot be predicted we can make a definite average pattern of results can be seen in these situations leading to random occurrences when the situation that led to the event is repeated a number of times. The simplest example for this would be if a coin is flipped many times, for roughly half of the flips it shall turn up heads. Probability measures the likelihood of an event occurring. Statistical regularity is the tendency of repeated experiments for the pattern...
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...504 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 Stochastic Modeling of an Automated Guided Vehicle System With One Vehicle and a Closed-Loop Path Aykut F. Kahraman, Abhijit Gosavi, Member, IEEE, and Karla J. Oty Abstract—The use of automated guided vehicles (AGVs) in material-handling processes of manufacturing facilities and warehouses is becoming increasingly common. A critical drawback of an AGV is its prohibitively high cost. Cost considerations dictate an economic design of AGV systems. This paper presents an analytical model that uses a Markov chain approximation approach to evaluate the performance of the system with respect to costs and the risk associated with it. This model also allows the analytic optimization of the capacity of an AGV in a closed-loop multimachine stochastic system. We present numerical results with the Markov chain model which indicate that our model produces results comparable to a simulation model, but does so in a fraction of the computational time needed by the latter. This advantage of the analytical model becomes more pronounced in the context of optimization of the AGV’s capacity which without an analytical approach would require numerous simulation runs at each point in the capacity space. Note to Practitioners—This paper presents a model for determining the optimal capacity of an automated guided vehicle (AGV) to be purchased by a manufacturer. This paper was motivated by work...
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...chancy in case of an individual is predictable and uniform in the case of a large group. * This law forms the basis for the expectation of probable-loss upon which insurance premium rates are computed. Also called law of averages. Law of Large Numbers Observe a random variable X very many times. In the long run, the proportion of outcomes taking any value gets close to the probability of that value. The Law of Large Numbers says that the average of the observed values gets close to the mean μ X of X. 4.slide ; Law of Large Numbers for Discrete Random Variables * The Law of Large Numbers, which is a theorem proved about the mathematical model of probability, shows that this model is consistent with the frequency interpretation of probability. 5.slide ; Chebyshev Inequality * To discuss the Law of Large Numbers, we first need an important inequality called the Chebyshev Inequality. * Chebyshev’s Inequality is a formula in probability theory that relates to the distribution of numbers in a set. * This formula is able to prove with little provided information the probability of outliers existing at a certain interval. 6.slide * Given X is a random variable, A stands for the mean of the set, K is the number of standard deviations, and Y is the value of the standard deviation, the formula reads as follows: *...
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...experiment if the outcome of that experiment is one of the elements of E. 6 Examples • When a fair die is rolled, probability of each outcome is 1/6. • So probability of E: Even numbers is P(E) = 3/6 = 0.5. • But a die does not need to be fair, and outcomes are not always equally likely! • For example, suppose we have a loaded die with probabilities of 1, 2, 3, 4, 5 and 6 resp. 0.25, 0.15, 0.15, 0.15, 0.15 and 0.15. • Then P(E) = 0.45. 7 Basic Concepts For any event A: i) 0 ≤ P(A) ≤ 1. ii) P(Φ) = 0, where Φ is the null event (no outcomes). iii) P(S) = 1, where S is the sample space (all outcomes). iv) If A and B are two events with no common outcome, (i.e. “mutually exclusive”) then P(A B) P(A) P(B) v) If A and B are two mutually exclusive events such that P(A B) 1 then A and B are called complements of each other. We denote B as A or AC. 8 General Formula P(A B) P(A) P( B) P(A B) 9 Example • Consider the fair die being rolled twice • P(sum is even) = • P(product is odd) = • P(sum is even, product is odd) = • Re-compute the probabilities for the loaded die. 10 Example Work out the probabilities i) P(sum is even given the product is odd) ii) P(product is odd given sum is even) for the fair die as well as the loaded die. How to proceed? 11 Conditional Probability • Let A, B be two events such that P(A) > 0. Then P ( A B) P( B | A) ...
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...PROBABILITY 1. ACCORDING TO STATISTICAL DEFINITION OF PROBABILITY P(A) = lim FA/n WHERE FA IS THE NUMBER OF TIMES EVENT A OCCUR AND n IS THE NUMBER OF TIMES THE EXPERIMANT IS REPEATED. 2. IF P(A) = 0, A IS KNOWN TO BE AN IMPOSSIBLE EVENT AND IS P(A) = 1, A IS KNOWN TO BE A SURE EVENT. 3. BINOMIAL DISTRIBUTIONS IS BIPARAMETRIC DISTRIBUTION, WHERE AS POISSION DISTRIBUTION IS UNIPARAMETRIC ONE. 4. THE CONDITIONS FOR THE POISSION MODEL ARE : • THE PROBABILIY OF SUCCESS IN A VERY SMALL INTERAVAL IS CONSTANT. • THE PROBABILITY OF HAVING MORE THAN ONE SUCCESS IN THE ABOVE REFERRED SMALL TIME INTERVAL IS VERY LOW. • THE PROBABILITY OF SUCCESS IS INDEPENDENT OF t FOR THE TIME INTERVAL(t ,t+dt) . 5. Expected Value or Mathematical Expectation of a random variable may be defined as the sum of the products of the different values taken by the random variable and the corresponding probabilities. Hence if a random variable X takes n values X1, X2,………… Xn with corresponding probabilities p1, p2, p3, ………. pn, then expected value of X is given by µ = E (x) = Σ pi xi . Expected value of X2 is given by E ( X2 ) = Σ pi xi2 Variance of x, is given by σ2 = E(x- µ)2 = E(x2)- µ2 Expectation of a constant k is k i.e. E(k) = k fo any constant k. Expectation of sum of two random variables is the sum of their expectations i.e. E(x +y) = E(x) + E(y) for any two...
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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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...PROBABILITY SEDA YILDIRIM 2009421051 DOKUZ EYLUL UNIVERSITY MARITIME BUSINESS ADMINISTRATION CONTENTS Rules of Probability 1 Rule of Multiplication 3 Rule of Addition 3 Classical theory of probability 5 Continuous Probability Distributions 9 Discrete vs. Continuous Variables 11 Binomial Distribution 11 Binomial Probability 12 Poisson Distribution 13 PROBABILITY Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics. There are several competing interpretations of the actual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution. The conditional probability of an event A assuming that B has occurred, denoted ,equals The two faces of probability introduces a central ambiguity which has been around for 350 years and still leads to disagreements about...
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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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