... Probability – the chance that an uncertain event will occur (always between 0 and 1) Impossible Event – an event that has no chance of occurring (probability = 0) Certain Event – an event that is sure to occur (probability = 1) Assessing Probability probability of occurrence= probability of occurrence based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation Events Simple event An event described by a single characteristic Joint event An event described by two or more characteristics Complement of an event A , All events that are not part of event A The Sample Space is the collection of all possible events Simple Probability refers to the probability of a simple event. Joint Probability refers to the probability of an occurrence of two or more events. ex. P(Jan. and Wed.) Mutually exclusive events is the Events that cannot occur simultaneously Example: Randomly choosing a day from 2010 A = day in January; B = day in February Events A and B are mutually exclusive Collectively exhaustive events One of the events must occur the set of events covers the entire sample space Computing Joint and Marginal Probabilities The probability of a joint event, A and B: Computing a marginal (or simple) probability: Probability is the numerical measure of the likelihood that an event will occur The probability of any event must be between 0 and 1, inclusively The sum of the...
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...Unit 2 DB Subjective Probability “ A probability derived from an individual's personal judgment about whether a specific outcome is likely to occur. Subjective probabilities contain no formal calculations and only reflect the subject's opinions and past experience.” (investopedia.com, 2013) There are three elements of a probability which combine to equal a result. There is the experiment ,the sample space and the event (Editorial board, 2012). In this case the class is the experiment because the process of attempting it will result in a grade which could vary from an A to F. The different grades that can be achieved in the class are the sample space. The event or outcome is the grade that will be received at the end of the experiment. I would like to achieve an “A” in this class but due to my lack of experience in statistical analysis, my hesitation towards advanced mathematics, and the length of time it takes for me to complete my course work a C in this class may be my best result. I have a 1/9 chance or probability to receive an “A” in the data range presented to me which is (A,A-,B,B-,C,C-,D,D- AND F). By the grades that have been posted I would say that the other students have a much better chance of receiving a better grade than mine. I have personally use subjective probability in my security guard business in bidding on contracts based on the clients involved , the rates that I charge versus the rates other companies charge and the amount of work involved...
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...the stage where one 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...
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...population parameters is called Statistical inference. Keywords: inferential statistics 2. Those methods involving the collection, presentation, and characterization of a set of data in order to properly describe the various features of that set of data are called Descriptive statistics. 3. The collection characteristics of the employees of a particular firm is an example of Descriptive statistics. 4. The estimation of the population average family expenditure on food based on the sample average expenditure of 1,000 families is an example of Inferential statistics. 5. The universe or "totality of items or things" under consideration is called A population. 6. The portion of the universe that has been selected for analysis is called A sample. 7. A summary measure that is computed to describe a...
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...[pic] FACULTY OF LAW AND BUSINESS School of Business North Sydney Campus Semester 2, 2014 STAT102: BUSINESS DATA ANALYSIS Probability Trees A survey of STAT102 students reveals that among the students that attain CR or better grades, 60% of them attended both lectures and tutorials. Among the students that attain PASS or FAILURE, only 20% of them attended both lectures and tutorials. Not attending both lectures and tutorials and with self-study, a student feels that the chance of attaining CR or better grades is quite low, 10%. The student is willing to attend both lectures and tutorials only if the efforts would increase the probability of achieving CR or better grades to 20% or more. Please use the following notations for the various states: C = CR or better grades CC = PASS or FAILURE L = Attend both lectures and tutorials LC = Do not attend both lectures and tutorials Required Should the student attend both lectures and tutorials? Please justify your answer with the posterior probability (prior probabilities are revised after the decision to attend both lectures and tutorials). The prior probability (probabilities determined prior to the decision of attending both lectures and tutorials) of attaining CR or better grades is P(C) = 0.1 P(CC) = 0.9 Complement rule Conditional probabilities are P(L(C) = 0.6 (for a student that have achieved a CR or better grade, the...
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...latest decision tools and techniques Linear Programming Operations management often presents complex problems that can be modelled by linear functions. The mathematical technique of linear programming is instrumental in solving a wide range of operations management problems. Linear programming is used to solve problems in many aspects of business administration including: • product mix planning • distribution networks • truck routing • staff scheduling • financial portfolios • corporate restructuring Decision Trees In many problems chance (or probability) plays an important role. Decision analysis is the general name that is given to techniques for analysing problems containing risk/uncertainty/probabilities. Decision trees are one specific decision analysis technique. Sensitivity analysis Queuing Theory (Waiting Line...
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...A Decision of Uncertainty A Decision of Uncertainty Paper There are decisions that people make where the outcome is presumably known and, there are the decisions that people make where the results are unknown. The latter part of the aforementioned statement is also known as decisions of uncertainty. To make these choices with more confidence, we will explore concepts that will formulate these judgments. We also have to include appropriate probability concepts that will help limit uncertainty in certain decisions. This paper will disclose the decision to reside in the tri-state area with the probability of destructive hurricanes occurring. Next this paper will reveal concepts and the outcome from the statistical analysis that was used to determine the final decision and, the tradeoffs between accuracy and precision required by various probability concepts. As a final point, this paper will demonstrate the effects the decision had on the data provided and the decision that was ultimately made. Probability Concepts and Application Hurricane Sandy hit Atlantic City, New Jersey on October 29, 2012. A 900-mile wide storm, Sandy affected the entire northeastern United States with devastating winds, rain and floods. New Jersey and New York suffered the worst from the super storm leaving thousands of people without power for days. People living in this area were not prepared or expected the storm to devastate the area as it did leaving homes and personal...
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...Computer Science. Most upper level Computer Science courses require probability in some form, especially in analysis of algorithms and data structures, but also in information theory, cryptography, control and systems theory, network design, artificial intelligence, and game theory. Probability also plays a key role in fields such as Physics, Biology, Economics and Medicine. There is a close relationship between Counting/Combinatorics and Probability. In many cases, the probability of an event is simply the fraction of possible outcomes that make up the event. So many of the rules we developed for finding the cardinality of finite sets carry over to Probability Theory. For example, we’ll apply an Inclusion-Exclusion principle for probabilities in some examples below. In principle, probability boils down to a few simple rules, but it remains a tricky subject because these rules often lead unintuitive conclusions. Using “common sense” reasoning about probabilistic questions is notoriously unreliable, as we’ll illustrate with many real-life examples. This reading is longer than usual . To keep things in bounds, several sections with illustrative examples that do not introduce new concepts are marked “[Optional].” You should read these sections selectively, choosing those where you’re unsure about some idea and think another example would be helpful. 2 Modelling Experimental Events One intuition about probability is that we want to predict how likely it is for a given experiment to...
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...Reliability Daniel Straub Engineering Risk Analysis Group TU München Ever increasing amounts of information are available Sensor data Satelite data Spatial measurements on structures Advanced simulation Sources: Frey et al. (in print); Gehlen et al. (2010); Michalski et al (2011); Schuhmacher et al. (2011) 2 1 01.06.2012 1973: 3 Probabilistic Updating of Flaw Information Tang (1973) • Imperfect information through inspection modeled by probability-ofdetection: 4 2 01.06.2012 Probabilistic Updating of Flaw Information Tang (1973) 5 Updating models and reliability computations with (indirect) information • Bayes‘ rule: ∝ 6 3 01.06.2012 How to compute the reliability of a geotechnical site conditional on deformation monitoring outcomes? -> Integrate Bayesian updating in structural reliability methods 7 Prior model in structural reliability • Failure domain: Ω 0 • Probability of failure: Pr ∈Ω d 8 4 01.06.2012 Information in structural reliability • Inequality information: Ω 0 • Conditional probability of failure: Pr | Pr ∩ Pr ∈ Ω ∩Ω ∈Ω d d 9 Information in structural reliability • Equality information: Ω 0 • Conditional probability of failure: Pr | Pr ∩ Pr 0 0 ? 10 5 01.06.2012 In statistics, information is expressed as likelihood function • Likelihood function for information event Z: ∝ Pr | • Example: – Measurement of system...
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...an abnormally high repeat rate whereas “change-of-pace” brands can be defined as brands with an abnormally low repeat rate for a given penetration level. Key notions Here are some basic notions you need be familiar with prior to carrying out this exercise: (i) (ii) (iii) (iv) (v) (vi) (vii) What a “variable” is. What a mean (expected value) is and how to compute it; What the notion of independence between two events means and how to test for it; What a correlation between two variables is and how to measure it; What a market share is and how to interpret it; What a penetration is and how to interpret it; What the duplication between two brands is and how to interpret it. Analysis as a scientific process Analyzing data consists of a three-step procedure: (i) (ii) (iii) Defining expectations: What do you expect to find and why? If you have “no idea” about your expectations, you need to develop these ideas by discussing with colleagues, “experts”, reading textbooks and/or using other supporting material; Running the analysis; Comparing your expectations with the findings. Do they match? Did you obtain any surprising result? If so, what are their implications for the practitioner? In summary, in order to obtain results, one does not (simply) “crunch” a data set. The scientific process that leads to findings comes from an interaction between expectations (“theory”) versus results. Never take an isolated number for granted (i.e., as representing ...
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...THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems ∏ Cătălin Bărboianu INFAROM Publishing Applied Mathematics office@infarom.com http://www.infarom.com http://probability.infarom.ro ISBN 978-973-1991-11-5 Publisher: INFAROM Author: Cătălin Bărboianu Correction Editor: CarolAnn Johnson Copyright © INFAROM 2009 This work is subject to copyright. All rights are reserved, whether the whole work or part of the material is concerned, specifically the rights of translation, reprinting, reuse of formulas and tables, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of Copyright Laws, and permission for use must always be obtained from INFAROM. 2 Contents (of the complete edition) Introduction ...................................................................................... 5 The Rules of Lottery ...................................................................…. 11 Supporting Mathematics ......................................................…....... 15 Probability space ..............................................................…......... 16 Probability properties and formulas used .........................…......... 19 Combinatorics …………………………………………………... 22 Parameters of the lottery matrices …………………………......... 25 Number Combinations .......………….………………………...
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...= {-20, -19, …, -1, 0, 1, …, 19, 20} Number of people arriving at a bank in a day: S = {0, 1, 2, …} Inspection of parts till one defective part is found: S = {d, gd, ggd, gggd, …} Temperature of a place with a knowledge that it ranges between 10 degrees and 50 degrees: S = {any value between 10 to 50} Speed of a train at a given time, with no other additional information: S = {any value between 0 to infinity} 4 Sample Space (cont…) Discrete sample space: One that contains either finite or countable infinite set of outcomes • Out of the previous examples, which ones are discrete sample spaces??? Continuous sample space: One that contains an interval of real numbers. The interval can be either finite or infinite 5 Events A collection of certain sample points A subset of the sample space Denoted by ‘E’ Examples: • Getting an odd number in dice throwing experiment S = {1, 2, 3, 4,...
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...[pic] [pic] Markov Chain [pic] Bonus Malus Model [pic] [pic] This table justifies the matrix above: | | | |Next state | | | |State |Premium |0 Claims |1 Claim |2 Claims |[pic]Claims | |1 | |1 |2 |3 |4 | |2 | |1 |3 |4 |4 | |3 | |2 |4 |4 |4 | |4 | |3 |4 |4 |4 | | | | | | | | |P11 |P12 |P13 |P14 | | | |P21 |P22 |P23 |P24 | | | |P31 |P32 |P33 |P34 | | | |P41 |P42 |P43 |P44 | | | | ...
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...Permutations The word ‘coincidence’ is defined as an event that might have been arranged though it was accidental in actuality. Most of us perceive life as a set of coincidences that lead us to pre-destined conclusions despite believing in a being who is free from the shackles of time and space. The question is that a being, for whom time and space would be nothing more than two more dimensions, wouldn’t it be rather disparaging to throw events out randomly and witness how the history unfolds (as a mere spectator)? Did He really arrange the events such that there is nothing accidental about their occurrence? Or are all the lives of all the living beings merely a result of a set of events that unfolded one after another without there being a chronological order? To arrive at satisfactory answers to above questions we must steer this discourse towards the concept of conditional probability. That is the chance of something to happen given that an event has already happened. Though, the prior event need not to be related to the succeeding one but must be essential for it occurrence. Our minds as I believe are evolved enough to analyze a story and identify the point in time where the story has originated or the set of events that must have happened to ensure the specific conclusion of the story. To simplify the conundrum let us assume a hypothetical scenario where a man just became a pioneer in the field of actuarial science. Imagine him telling us his story in reverse. “I became...
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...presence with probability 0.99. If it is not present, the radar falsely registers an aircraft presence with probability 0.10. We assume that an aircraft is present with probability 0.05. What is the probability of false alarm (a false indication of aircraft presence), and the probability of missed detection (nothing registers, even though an aircraft is present)? A sequential representation of the sample space is appropriate here, as shown in Fig. 1. Figure 1: Sequential description of the sample space for the radar detection problem Solution: Let A and B be the events A={an aircraft is present}, B={the radar registers an aircraft presence}, and consider also their complements Ac={an aircraft is not present}, Bc={the radar does not register an aircraft presence}. The given probabilities are recorded along the corresponding branches of the tree describing the sample space, as shown in Fig. 1. Each event of interest corresponds to a leaf of the tree and its probability is equal to the product of the probabilities associated with the branches in a path from the root to the corresponding leaf. The desired probabilities of false alarm and missed detection are P(false alarm)=P(Ac∩B)=P(Ac)P(B|Ac)=0.95∙0.10=0.095, P(missed detection)=P(A∩Bc)=P(A)P(Bc|A)=0.05∙0.01=0.0005. Application of Bayes` rule in this problem. We are given that P(A)=0.05, P(B|A)=0.99, P(B|Ac)=0.1. Applying Bayes’ rule, with A1=A and A2=Ac, we obtain P(aircraft present | radar registers) =...
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