...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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...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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...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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... 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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...= {-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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...1.M/G/ Queue a. Show that Let A(t) : Number of arrivals between time (0, t] “ n should be equal to or great than k” since if n is less than k (n<k), Pk(t)=0 Let’s think some customer C, Let’s find P{C arrived at time x and in service at time t | x=(0,t)] } P{C arrives in (x, x+dx] | C arrives in (0, t] }P{C is in service | C arrives at x, and x = (0,t] } Since theorem of Poisson Process, The theorem is that Given that N(t) =n, the n arrival times S1, S2, …Sn have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on the interval (0, t) Thus, P{C is in service | C arrives between time (0, t] } Since let y=t-x, x=0 → y=t, x=t →y=o, dy=-dx Therefore, In conclusion, ------ (1) 1-a Solution Since b. let 1-b Solution ------------------------------------------------- 2. notation Page 147 in “Fundamentals of Queuing Theory –Third Edition- , Donald Gross Carl M. Harris a. b. ------------------------------------------------- ------------------------------------------------- ------------------------------------------------- 3. a. let X=service time (Random variable) and XT=total service time (Random variable) X2=X+X, X3=X+X+X, ….. f2(x2)...
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...Probability & Mathematical Statistics | “The frequency concept of Probability” | [Type the author name] | What is probability & Mathematical Statistics? It is the mathematical machinery necessary to answer questions about uncertain events. Where scientists, engineers and so forth need to make results and findings to these uncertain events precise... Random experiment “A random experiment is an experiment, trial, or observation that can be repeated numerous times under the same conditions... It must in no way be affected by any previous outcome and cannot be predicted with certainty.” i.e. it is uncertain (we don’t know ahead of time what the answer will be) and repeatable (ideally).The sample space is the set containing all possible outcomes from a random experiment. Often called S. (In set theory this is usually called U, but it’s the same thing) Discrete probability Finite Probability This is where there are only finitely many possible outcomes. Moreover, many of these outcomes will mostly be where all the outcomes are equally likely, that is, uniform finite probability. An example of such a thing is where a fair cubical die is tossed. It will come up with one of the six outcomes 1, 2, 3, 4, 5, or 6, and each with the same probability. Another example is where a fair coin is flipped. It will come up with one of the two outcomes H or T. Terminology and notation. We’ll call the tossing of a die a trial or an experiment. Where we...
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...Model Answers for Chapter 4: Evaluating Classification and Predictive Performance Answer to 4.3.a: Leftmost bar: If we take the 10% "most probable 1’s(frauds)” (as ranked by the model), it will yield 6.5 times as many 1’s (frauds), as would a random selection of 10% of the records. 2nd bar from left: If we take the second highest decile (10%) of records that are ranked by the model as “the most probable 1’s (frauds ” it will yield 2.7 times as many 1’s (frauds), as would a random selection of 10 % of the records. Answer to 4.3.b: Consider a tax authority that wants to allocate their resources for investigating firms that are most likely to submit fraudulent tax returns. Suppose that there are resources for auditing only 10% of firms. Rather than taking a random sample, they can select the top 10% of firms that are predicted to be most likely to report fraudulently (according to the decile chart). Or, to preserve the principle that anyone might be audited, they can establish differential probabilities for being sampled -- those in the top deciles being much more likely to be audited. . Answer to 4.3.c: Classification Confusion Matrix Predicted Class 1 (Fraudulent) Actual Class 1 (Fraudulent) 0 (Non-fraudulent) Error rate = 0 (Non-fraudulent) 30 58 32 920 n0,1 + n1,0 32 + 58 = = 0.0865 = 8.65% n 1040 Our classification confusion matrix becomes Classification Confusion Matrix Predicted Class 1 (Fraudulent) ...
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...Memorandum To: CC: From: Date: Re: The Cincinnati Enquirer Kristen DelGuzzi Ashley N. Ear; September 16, 2007 Data Analysis of Hamilton County Judges Probabilities used to assist with Ranking of Hamilton County Judges After the current statistics were gathered to produce data analysis regarding Hamilton County Judges, we can come to a conclusion and rank judges appropriately by their probability to be appealed, reversed and a combination of the both. With the provided data analysis, I have included statistics to all probabilities including: total cases disposed, appealed cases, reversed cases, probability of appeal, rank by probability of appeal, probability of reversal, rank by probability of reversal, conditional probability of reversal given appeal, rank by conditional probability of reversal given appeal and overall sum of ranks. The judges that rank the highest (i.e. 1st, 2nd, 3rd) have the lowest probability to have appealed cases, reversed cases and lowest conditional probability of reversed cases given appeal. In my opinion, by ranking the judges as such, we can see how often their ruling is upheld, which is ultimately desirable when concerning the credibility of a judge. I have provided rankings for all three different courts including: Common Pleas Court, Domestic Relations Court and Municipal Court. These overall rankings are gathered by summing up all of the rankings by the three probability variables. I have also provided data analysis which interprets who...
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...2 Broad Study in Statistics Descriptive (mô tả) : provides simple summaries about the data collected & about the preliminary observations that have beed made. Such summaries maybe either quantitative (numerical measures) or visual (e.g. simple-to-understand graphs) e.g. Present a summary report of this year business result to management Inferential (suy luận) : are systems of procedures that can be used to draw conclusions from datasets arising from systems affected by random variation. The type of inferential statistical procedure used depends upon the type of data collected as well as the distribution of the data. The procedures are usually used to test hypotheses and establish probability. e.g. Estimate the IQ score of Kaplan students by observing a small group of students Population : e.g. A population is a collection of all individuals, objects, or measurement of interest Sample : e.g. A sample is a portion or part of the population of interst MCQ 1. The process of using sample statistics to draw conclusions about true 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...
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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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...determine whether a member of the population is infected. The medical test for the infection is not completely reliable: for someone infected with the B3F1 virus, there is a probability of 0.98 that the test will prove positive; whereas for someone not infected with the B3F1 virus, there is a probability of 0.01 that the test will prove positive. One person is randomly chosen from the population and tested. Let I be the event that the person is infected and T be the event that the person is tested positive. Determine the least value of x such that the probability that a person is infected with the B3F1 virus given that he is tested positive is at least 0.95. [3] (Hint : Draw a tree diagram.) 2. [RJC/2009/Promo] (a) [ A' denotes the event ‘A does not occur’.] 11 8 Given that P ( A ∪ B ) = and P ( A '∩ B ) = , determine P ( B ) if 15 45 (i) A and B are two mutually exclusive events, (ii) A and B are two independent events. [4] (Hint for (ii) : Find P(A) first. Then use the fact that since A and B are two independent events, A' and B are independent as well.) (Hint for both parts : Draw Venn diagram.) (b) In a class of 20 students, there are 7 boys and 13 girls. Of the 7 boys, 3 of them read General Paper and of the 13 girls, 8 of them read General Paper. The rest of the students read Knowledge and Inquiry. Three students are chosen at random from the class. Find the probability that (i) there is at least one student who reads Knowledge...
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