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Exchangeable Random Variables
In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences : X_1, X_2, X_3, X_4, X_5, X_6 \quad \text \quad X_3, X_6, X_1, X_5, X_2, X_4 both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling. Definition Formally, an exchangeable sequence of random variables is a finite or infinite sequence ''X''1, ''X''2, ''X''3, ... of random variables such that for any finite permutation σ of the indices 1, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Large Numbers
In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law of large numbers states that given a sample of independent and identically distributed values, the sample mean converges to the true mean. The law of large numbers is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a stre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Urn Model
In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. A number of important variations are described below. An urn model is either a set of probabilities that describe events within an urn problem, or it is a probability distribution, or a family of such distributions, of random variables associated with urn problems. History In ''Ars Conjectandi'' (1713), Jacob Bernoulli considered the problem of determining, given a number of pebbles drawn from an urn, the proportions of different colored pebbles within the urn. This problem was known as the ''inverse probability'' problem, and was a topic of research in the eighteenth century, attracting the attention of Abraham d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixture Distribution
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution. In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Combination
In convex geometry and Vector space, vector algebra, a convex combination is a linear combination of point (geometry), points (which can be vector (geometric), vectors, scalar (mathematics), scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the ''count'' of the weights as in a standard weighted average. Formal definition More formally, given a finite number of points x_1, x_2, \dots, x_n in a real vector space, a convex combination of these points is a point of the form : \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where the real numbers \alpha_i satisfy \alpha_i\ge 0 and \alpha_1+\alpha_2+\cdots+\alpha_n=1. As a particular example, every convex combination of two points lies on the line segment between the points. A set is convex set, convex if it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Correlation
In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways. If ''X'' and ''Y'' are two random variables, with means (expected values) ''μX'' and ''μY'' and standard deviations ''σX'' and ''σY'', respectively, then their covariance and correlation are as follows: ; covariance :\text_ = \sigma_ = E X-\mu_X)\,(Y-\mu_Y)/math> ; correlation :\text_ = \rho_ = E X-\mu_X)\,(Y-\mu_Y)(\sigma_X \sigma_Y)\,, so that \rho_ = \sigma_ / (\sigma_X \sigma_Y) where ''E'' is the expected value operator. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. If ''Y'' always takes on the same values as ''X'', we have the covariance of a variable with itself (i.e. \sigma_), which is called the variance and is more commonly denoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Limit
In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\infty, and complex numbers \alpha: # \phi(\alpha x+y) = \alpha\phi(x) + \phi(y) (linearity); # if x_n\geq 0 for all n \in \mathbb, then \phi(x) \geq 0 (positivity); # \phi(x) = \phi(Sx), where S is the shift operator defined by (Sx)_n=x_ (shift-invariance); # if x is a convergent sequence, then \phi(x) = \lim x . Hence, \phi is an extension of the continuous functional \lim: c \to \mathbb C where c \subset\ell^\infty is the complex vector space of all sequences which converge to a (usual) limit in \mathbb C. In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Distribution Function
In statistics, an empirical distribution function ( an empirical cumulative distribution function, eCDF) is the Cumulative distribution function, distribution function associated with the empirical measure of a Sampling (statistics), sample. This cumulative distribution function is a step function that jumps up by at each of the data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. The empirical distribution function is an Estimator, estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of Convergence of random variables#Convergence in distribution, convergence of the empirical distribution function to the underlying cumulative distribution function. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequentist Statistics
Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or proportion of findings in the data. Frequentist inference underlies frequentist statistics, in which the well-established methodologies of statistical hypothesis testing and confidence intervals are founded. History of frequentist statistics Frequentism is based on the presumption that statistics represent probabilistic frequencies. This view was primarily developed by Ronald Fisher and the team of Jerzy Neyman and Egon Pearson. Ronald Fisher contributed to frequentist statistics by developing the frequentist concept of "significance testing", which is the study of the significance of a measure of a statistic when compared to the hypothesis. Neyman-Pearson extended Fisher's ideas to apply to multiple hypotheses. They posed that the ratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
