It is important to consider ways in which **probability** distributions are determined in practice. One way is by symmetry. For the case of the toss of a coin, we do not see any physical difference between the two sides of a coin that should affect the chance of one side or the other turning up. Similarly, with an ordinary die there is no essential difference between any two sides of the die, and so by symmetry we assign the same **probability** for any possible outcome. In general, considerations of symmetry often suggest the uniform distribution function. Care must be used here. We should not always assume that, just because we do not know any reason to suggest that one outcome is more likely than another, it is appropriate to assign equal probabilities. For example, consider the experiment of guessing the sex of a newborn child. It has been observed that the proportion of newborn children who are boys is about .513. Thus, it is more appropriate to assign a distribution function which assigns **probability** .513 to the outcome boy and **probability** .487 to the outcome girl than to assign **probability** 1/2 to each outcome. This is an example where we use statistical observations to determine probabilities. Note that these probabilities may change with new studies and may vary from country to country. Genetic engineering might even allow an individual to influence this **probability** for a particular case.

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2.12 In the UEFA Euro 2004 playoﬀs draw 10 national football teams were matched in pairs. A lot of people complained that “the draw was not fair,” because each strong team had been matched with a weak team (this is commercially the most interesting). It was claimed that such a matching is extremely unlikely. We will compute the **probability** of this “dream draw” in this exercise. In the spirit of the three-envelope example of Section 2.1 we put the names of the 5 strong teams in envelopes labeled 1, 2, 3, 4, and 5 and of the 5 weak teams in envelopes labeled 6, 7, 8, 9, and 10. We shuﬄe the 10 envelopes and then match the envelope on top with the next envelope, the third envelope with the fourth envelope, and so on. One particular way a “dream draw” occurs is when the ﬁve envelopes labeled 1, 2, 3, 4, 5 are in the odd numbered positions (in any order!) and the others are in the even numbered positions. This way corresponds to the situation where the ﬁrst match of each strong team is a home match. Since for each pair there are two possibilities for the home match, the total number of possibilities for the “dream draw” is 2 5 = 32 times as large.

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COURSE OVERVIEW: The objective of this course is to enable the student to identify and perform calculations involving the equations to the various conic sections; recognize and work with concepts in calculus (limits, differentiation, and integration); and recognize the elements of introductory **probability** theory.

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A more general situation arises when the method of choosing is not “fair” or “uniform.” Suppose U = { H, T } is a set of two letters, H and T . We select either H or T by taking a coin and flipping it. If “heads” comes up, we choose H, otherwise we choose T . The coin, typically, will be dirty, have scratches in it, etc., so the “chance” of H being chosen might be different from the chance of T being chosen. If we wanted to do a bit of work, we could flip the coin 1000 times and keep some records. Interpreting these records might be a bit tricky in general, but if we came out with 400 heads and 600 tails, we might suspect that tails was more likely. It is possible to be very precise about these sort of experiments (the subject of statistics is all about this sort of thing). But for now, let’s just suppose that the “**probability**” of choosing H is 0.4 and the **probability** of choosing T is 0.6. Intuitively, we mean by this that if you toss the coin a large number N of times, about 0.4N will be heads and 0.6N will be tails. The function P with domain U = { H, T } and values P (H) = 0.4 and P (T ) = 0.6 is an example of a “**probability** function” on a sample space U .

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This final common interpretation of **probability** is somewhat controversial, but does not suffer from the problems that the other interpretations do. It suggests that the association of probabilities to events is a personal (subjective) process, relating to your degree of belief in the likelihood of the event occurring. It is controversial because it accepts that different people will assign different probabilities to the same event. Whilst in some sense it gives up on an objective notion of prob- ability, it is in no sense arbitrary. It can be defined in a precise way, from which the axioms of **probability** may be derived as requirements of self-consistency.

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Example 3.10 Given a group of k people (2 ¸ k ¸ 365), what is the **probability** that at least 2 people in the group have the same birthday? To simplify the problem, assume that birthdays are unrelated (there are no twins) and that each of the 365 days of the year are equally likely to be the birthday of any person. The sample space S then consists of 365 k possible outcomes. The number of outcomes in S for which

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In the second edition of **Probability** and Statistics, which appeared in 2000, the guiding principle was to make changes in the first edition only where necessary to bring the work in line with the emphasis on topics in con- temporary texts. In addition to refinements throughout the text, a chapter on nonparametric statistics was added to extend the applicability of the text without raising its level. This theme is continued in the third edition in which the book has been reformatted and a chapter on Bayesian methods has been added. In recent years, the Bayesian paradigm has come to enjoy increased popularity and impact in such areas as economics, environmental science, medicine, and finance. Since Bayesian statistical analysis is highly computational, it is gaining even wider ac- ceptance with advances in computer technology. We feel that an **introduction** to the basic principles of Bayesian data analysis is therefore in order and is consistent with Professor Murray R. Spiegel’s main purpose in writing the original text—to present a modern **introduction** to **probability** and statistics using a background of calculus. J. S CHILLER

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The theory of SLE is a major piece of contemporary mathematics which promises to explain phase transitions in an important class of two-dimen- sional disordered systems, and to help bridge the gap between **probability** theory and conformal field theory. It plays a key role in the study of critical percolation (see Chapter 5), and also of the critical random-cluster and Ising models, [224, 225]. In addition, it has provided complete explanations of conjectures made by mathematicians and physicists concerning the intersec- tion exponents and fractionality of frontier of two-dimensional Brownian motion, [160, 161, 162]. The purposes of the current section are to give a brief non-technical **introduction** to SLE, and to indicate its relevance to the scaling limits of LERW and UST.

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The fact that conditioning is problematic if one conditions on something not equal to a partition has in fact been known for a long time, see e.g. Shafer (1985) for the first landmark reference. Our point is simply to show that the issue fits in well with the safety concept. There is an obvious analogy here with the Borel-Kolmogorov paradox (Schweder and Hjort, 1996) which presumably could also be recast in terms of safety. As Kolmogorov (1933) writes, “ The concept of a conditional **probability** with regard to an isolated hypothesis whose **probability** equals 0 is inadmissible.” Safe **probability** suggests something more radical: standard conditional probabilities with regard to an isolated hypothesis (event) are never admissible — if one does not know whether the alternatives form a partition, setting ˜ P to be the standard conditional distribution is inherently unsafe.

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In this paper, a new routing protocol named as PRoWait is designed to overcome some of the shortcomings of the existing protocols in Oppnet. As the area of routing in Oppnets is still under research and much work is to be done in different aspects, the main focus in this work is mainly to increase the packet delivery ratio and lower delay and hop count. The PRoWait is designed to overcome some of the shortcomings of the existing protocols in Oppnet. Through simulation analysis the performance of PRoWait is evaluated and compared with Epidemic, Spray and Wait, and Prophet protocols in terms of delivery **probability**, overhead ratio, and average hop count performance metrics. It has been observed that PRoWait outperforms these protocols on the basis of aforementioned performance metrics.

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P(event) = number of favorable outcomes total # of possible outcomes.. An favorable outcome is the outcome (or[r]

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Predict how this player’s score will change the median, mode, range, and mean of the data and explain your reasoning.. Then compute each of these measures to check your predic[r]

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• A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values). – thickness of an item[r]

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What is the probability that a person’s primary news source is the internet and they are a college graduate?. What is the probability that a college graduate’s primary news source is t[r]

This report will deal with many issues that DNA evidence raises. First, DNA cases currently in the news will be discussed. Next, the prosecutor’s fallacy will be explained, and it will be determined if jury members really do understand the **probability** behind DNA matches. The actual composition of DNA will then be explained to help comprehend the number of possible DNA permutations involved. Following this, the **probability** behind DNA will be discussed with the help of Bayes’ theorem. Also, the concept of compounding evidence will be applied to the trial of the century, the O.J. Simpson trial. Finally, a simple analogy of what a consistent DNA match indicates will be presented.

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Definition 11.1 The continuous random variables X and Y are said to have a joint probability density function f x, y if the joint cumulative probability distribution function PX :Sa, Y :[r]

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Deﬁnitions Suppose a random variable X can take any real number in an interval. Of course, the number that we record is often rounded to some appropriate number of decimal places, so we don’t actually observe X but Y = X rounded to the nearest /2 units. So, for example, the **probability** that we record the number Y = y is the **probability** that X falls in the interval y −/2 < X ≤ y +/2. If F (x) is the cumulative distribution function of X, this **probability** is P [Y = y ] = F (y + /2) − F (y − /2). Suppose now that is very small and that the cumulative distribution function is piecewise continuously differentiable with a derivative given in an interval by

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Series A, Statistics in society 0964-1998 Journal of the Royal Statistical Society. Methodological 0035-9246 Journal of the Royal Statistical Society[r]

probabilities for chance nodes. Learning probabilities from data is a good approach but is useless in case of no available data set. In this paper, we propose a method to elicit probabilities from a group of domain experts. The theory of interval **probability** is adapted to handle the fuzziness of experts’ knowledge. Using a software tool, the experts assign interval probabilities to chance nodes, and then their judgments are combined. Finally, by using maximum entropy principle, the interval probabilities are converted into point-value probabilities. With this me- thod, experts feel more comfortable and confident to make judgments. We believe this method is useful in in- fluence diagrams modeling as well as in other **probability** elicitation situations.

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adopting sequential **probability** ratio test instead of using current test procedure for testing mean of a normally distributed variate and found that the sequential test results in an average saving of at least 47 percent in the necessary number of observations as compared with the current test.

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