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Continuity Correction
In probability theory, a continuity correction is an adjustment that is made when a discrete distribution is approximated by a continuous distribution. Examples Binomial If a random variable ''X'' has a binomial distribution with parameters ''n'' and ''p'', i.e., ''X'' is distributed as the number of "successes" in ''n'' independent Bernoulli trials with probability ''p'' of success on each trial, then :P(X\leq x) = P(X |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Software
Statistical software are specialized computer programs for analysis in statistics and econometrics. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a software suite for non-linear statistical modeling based on C++ which uses automatic differentiation * Chronux – for neurobiological time series data * DAP – free replacement for SAS * Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) a software framework for developing data mining algorithms in Java * Epi Info – statistical software for epidemiology developed by Centers for Disease Control and Prevention (CDC). Apache 2 licensed * Fityk – nonlinear regression software (GUI and command line) * GNU Octave – programming language very similar to MATLAB with statistical features * gretl – gnu regression, econometrics and time-series library * intrinsic Noise Analyzer (iNA) – For analyzing intrinsic fluctua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Probability Distributions
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be scientific, belong to a non-scientific discipline, or no discipline at all. Depending on the context, a theory's assertions might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and compre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Proportion Confidence Interval
In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion confidence interval is an interval estimate of a success probability ''p'' when only the number of experiments ''n'' and the number of successes ''nS'' are known. There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. In general, a binomial distribution applies when an experiment is repeated a fixed number of times, each trial of the experiment has two possible outcomes (success and failure), the probability of success is the same for each trial, and the trials are statistically independent. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approximations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yates's Correction For Continuity
In statistics, Yates's correction for continuity (or Yates's chi-squared test) is used in certain situations when testing for independence (probability theory), independence in a contingency table. It aims at correcting the error introduced by assuming that the discrete probabilities of frequencies in the table can be approximated by a continuous distribution (chi-squared distribution, chi-squared). In some cases, Yates's correction may adjust too far, and so its current use is limited. Correction for approximation error Using the chi-squared distribution to interpret Pearson's chi-squared test, Pearson's chi-squared statistic requires one to assume that the Discrete probability distribution, discrete probability of observed binomial distribution, binomial frequencies in the table can be approximated by the continuous chi-squared distribution. This assumption is not quite correct, and introduces some error. To reduce the error in approximation, Frank Yates, an England, English stati ... [...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] |
Binomial Test
In statistics, the binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories using sample data. Usage The binomial test is useful to test hypotheses about the probability (\pi) of success: : H_0:\pi=\pi_0 where \pi_0 is a user-defined value between 0 and 1. If in a sample of size n there are k successes, while we expect n\pi_0, the formula of the binomial distribution gives the probability of finding this value: : \Pr(X=k)=\binomp^k(1-p)^ If the null hypothesis H_0 were correct, then the expected number of successes would be n\pi_0. We find our p-value for this test by considering the probability of seeing an outcome as, or more, extreme. For a one-tailed test, this is straightforward to compute. Suppose that we want to test if \pi\pi_0 using the summation of the range from k to n instead. Calculating a p-value for a two-tailed test is slightly more complicated, since a bin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Hypothesis Test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. History Early use While hypothesis testing was popularized early in the 20th century, early forms were used in the 1700s. The first use is credited to John Arbuthnot (1710), followed by Pierre-Simon Laplace (1770s), in analyzing the human sex ratio at birth; see . Modern origins and early controversy Modern significance testing is largely the product of Karl Pearson ( ''p''-value, Pearson's chi-squared test), William Sealy Gosset (Student's t-distribution), and Ronald Fisher ("null hypothesis", analysis of variance, "significance test"), while hypothesis testing was developed by Jerzy Neyman and Egon Pearson (son of Karl). Ronald Fisher began his life in statistics as a Bayesian (Zabell 1992), but Fisher soon grew disenchanted with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and Statistical independence, independently of the time since the last event. It is named after France, French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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