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Fair Coin
In probability theory and statistics, a sequence of Independence (probability theory), 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 (British coin), 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 met ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coin Toss (3635981474)
Coin flipping, coin tossing, or heads or tails is using the thumb to make a coin go up while spinning in the air and checking obverse and reverse, which side is showing when it is down onto a surface, in order to randomly choose between two alternatives. It is a form of sortition which inherently has two possible outcomes. 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 Roman Emperor, 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 an unpredictable number of 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; caught and inverted; or allowed to land on the g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...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 sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Entropy Function
Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that takes two arguments * Binary relation, a relation involving two elements * Finger binary, a system for counting in binary numbers on the fingers of human hands Computing * Binary code, the representation of text and data using only the digits 1 and 0 * Bit, or binary digit, the basic unit of information in computers * Binary file, composed of something other than human-readable text ** Executable, a type of binary file that contains machine code for the computer to execute * Binary tree, a computer tree data structure in which each node has at most two children * Binary-coded decimal, a method for encoding for decimal digits in binary sequences Astronomy * Binary star, a star system with two stars in it * Binary planet, t ... [...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 property of conditional expectation, 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. The conditional expected value \operatorname( X \mid Y ), with Y a random variable, is not a simple number; it is a random variable whose value depends on the value of Y. That is, the conditional expected value of X given the ''event'' Y = y is a number and it is a function of y. If we write g(y) for the value of \operatorname ( X \mid Y = y) then the random variable \operatorname( X \mid Y ) is g( Y ) . One special case states that if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Of Zero
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formalizati ... [...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 family of urn models that can be used to interpret many commonly used statistical models. The model represents objects of interest (such as atoms, people, cars, etc.) as colored balls in an urn. In the basic Pólya urn model, the experimenter puts ''x'' white and ''y'' black balls into an urn. At each step, one ball is drawn uniformly at random 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. If by random chance, more black balls are drawn than white balls in the initial few draws, it would make it more likely for more black balls to be drawn later. Similarly for the white balls. Thus the urn has a self-reinforcing property (" the rich get richer"). It is the opposite of sampling without replacement, where every time a particular value is observed, it is less likely to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malleable
Ductility refers to the ability of a material to sustain significant plastic deformation before fracture. Plastic deformation is the permanent distortion of a material under applied stress, as opposed to elastic deformation, which is reversible upon removing the stress. Ductility is a critical mechanical performance indicator, particularly in applications that require materials to bend, stretch, or deform in other ways without breaking. The extent of ductility can be quantitatively assessed using the percent elongation at break, given by the equation: \% \mathrm= \left ( \frac \right )\times100 where l_ is the length of the material after fracture and l_0 is the original length before testing. This formula helps in quantifying how much a material can stretch under tensile stress before failure, providing key insights into its ductile behavior. Ductility is an important consideration in engineering and manufacturing. It defines a material's suitability for certain manufactur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, integrating Basic research, pure and Applied science#Applied research, applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including Cellular automaton, cellular automata, the Von Neumann universal constructor, universal constructor and the Computer, digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA. During World War II, von Neumann worked on the Manhattan Project. He developed the mathematical models behind the explosive lense ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Process
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables ''X''''i'' are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable ''X''''i'' in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme. The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of checking whether a coin is fair. Definition A ''Bernoulli process'' is ... [...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 ''forecastin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
