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Fair Coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000. In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coin Toss (3635981474)
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute between two parties. It is a form of sortition which inherently has two possible outcomes. The party who calls the side that is facing up when the coin lands wins. History Coin flipping was known to the Romans as ''navia aut caput'' ("ship or head"), as some coins had a ship on one side and the head of the emperor on the other. In England, this was referred to as ''cross and pile''. Process During a coin toss, the coin is thrown into the air such that it rotates edge-over-edge several times. Either beforehand or when the coin is in the air, an interested party declares "heads" or "tails", indicating which side of the coin that party is choosing. The other party is assigned the opposite side. Depending on custom, the coin may be caught; cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time-series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements. Time series ''analysis'' comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series ''forecasti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Experiment (probability Theory)
In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be ''random'' if it has more than one possible outcome, and ''deterministic'' if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial. When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis. Experiments and trials Random experiments are often conducted repeatedly, so that the collective results may be subj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Gelman
Andrew Eric Gelman (born February 11, 1965) is an American statistician and professor of statistics and political science at Columbia University. Gelman received bachelor of science degrees in mathematics and in physics from MIT, where he was a National Merit Scholar, in 1986. He then received a master of science in 1987 and a doctor of philosophy in 1990, both in statistics from Harvard University, under the supervision of Donald Rubin. Career Gelman is the Higgins Professor of Statistics and Professor of Political Science and the Director of the Applied Statistics Center at Columbia University. He is a major contributor to statistical philosophy and methods especially in Bayesian statistics and hierarchical models. He is one of the leaders of the development of the statistical programming framework Stan. Perspective on Statistical Inference and Hypothesis Testing Gelman's approach to statistical inference emphasizes studying variation and the associations between dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller's Coin-tossing Constants
Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in ''n'' independent tosses of a fair coin, no run of ''k'' consecutive heads (or, equally, tails) appears. William Feller showed that if this probability is written as ''p''(''n'',''k'') then : \lim_ p(n,k) \alpha_k^=\beta_k where α''k'' is the smallest positive real root of :x^=2^(x-1) and :\beta_k=. Values of the constants For k=2 the constants are related to the golden ratio, \varphi, and Fibonacci numbers; the constants are \sqrt-1=2\varphi-2=2/\varphi and 1+1/\sqrt. The exact probability ''p''(n,2) can be calculated either by using Fibonacci numbers, ''p''(n,2) = \tfrac or by solving a direct recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coin Flipping
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute between two parties. It is a form of sortition which inherently has two possible outcomes. The party who calls the side that is facing up when the coin lands wins. History Coin flipping was known to the Romans as ''navia aut caput'' ("ship or head"), as some coins had a ship on one side and the head of the emperor on the other. In England, this was referred to as ''cross and pile''. Process During a coin toss, the coin is thrown into the air such that it rotates edge-over-edge several times. Either beforehand or when the coin is in the air, an interested party declares "heads" or "tails", indicating which side of the coin that party is choosing. The other party is assigned the opposite side. Depending on custom, the coin may be caught; cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Checking Whether A Coin Is Fair
In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and probability theory can provide guidance on two types of question; specifically those of how many trials to undertake and of the accuracy of an estimate of the probability of turning up heads, derived from a given sample of trials. A fair coin is an idealized randomizing device with two states (usually named "heads" and "tails") which are equally likely to occur. It is based on the coin flip used widely in sports and other situations where it is required to give two parties the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Total Expectation
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected value \operatorname(X) is defined, and Y is any random variable on the same probability space, then :\operatorname (X) = \operatorname ( \operatorname ( X \mid Y)), i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X. One special case states that if _i is a finite or countable partition of the sample space, then :\operatorname (X) = \sum_i. Note: The conditional expected value E(''X'' , ''Z'') is a random variable whose value depend on the value of ''Z''. Note that the conditional expected value of ''X'' given the ''event'' ''Z'' = ''z'' is a function of ''z''. If we write E(''X'' , ''Z'' = ''z'') = ''g''(''z'') then the random variable E(''X'' , ''Z'') is ''g''(''Z''). Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Of Zero
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that hap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pólya Urn Model
In statistics, a Pólya urn model (also known as a Pólya urn scheme or simply as Pólya's urn), named after George Pólya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments. In an urn model, objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. In the basic Pólya urn model, the urn contains ''x'' white and ''y'' black balls; one ball is drawn randomly from the urn and its color observed; it is then returned in the urn, and an additional ball of the same color is added to the urn, and the selection process is repeated. Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out. This endows the urn with a self-reinforcing property sometimes expressed as '' the rich get richer''. Note that in some sense, the Pólya urn model is the "opposite" of the model of sampling without replacement, where every time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malleable
Ductility is a mechanical property commonly described as a material's amenability to drawing (e.g. into wire). In materials science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stress before failure. Ductility is an important consideration in engineering and manufacturing. It defines a material's suitability for certain manufacturing operations (such as cold working) and its capacity to absorb mechanical overload.. Some metals that are generally described as ductile include gold and copper. However, not all metals experience ductile failure as some can be characterized with brittle failure like cast iron. Polymers generally can be viewed as ductile materials as they typically allow for plastic deformation. Malleability, a similar mechanical property, is characterized by a material's ability to deform plastically without failure under compressive stress. Historically, materials were considered malleable if they were am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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