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Pearson Distribution
The Pearson distribution is a family of continuous probability distribution, continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. History The Pearson system was originally devised in an effort to model visibly skewness, skewed observations. It was well known at the time how to adjust a theoretical model to fit the first two cumulants or moment (mathematics), moments of observed data: Any probability distribution can be extended straightforwardly to form a location-scale family. Except in pathological (mathematics), pathological cases, a location-scale family can be made to fit the observed mean (mathematics), mean (first cumulant) and variance (second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which the skewness (standardized third cumulant) and kurtosis (standardized fourth cumulant) could be adjuste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pearson System
Pearson may refer to: Organizations Education *Lester B. Pearson College, Victoria, British Columbia, Canada *Pearson College (UK), London, owned by Pearson PLC *Lester B. Pearson High School (other) Companies *Pearson PLC, a UK-based international media conglomerate, best known as a book publisher **Pearson Education, the textbook division of Pearson PLC ***Pearson-Longman, an imprint of Pearson Education *Pearson Yachts Places *Pearson, California (other) *Pearson, Georgia, a US city *Pearson, Texas, an unincorporated community in the US *Pearson, Victoria, a ghost town in Australia *Pearson, Wisconsin, an unincorporated community in the US *Toronto Pearson International Airport, in Toronto, Ontario, Canada *Pearson Field, in Vancouver, Washington, US *Pearson Island, an island in South Australia which is part of the Pearson Isles *Pearson Isles, an island group in South Australia Other uses *Pearson (surname) *Pearson correlation coefficient, a statistical mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standardized Moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments. Standard normalization Let ''X'' be a random variable with a probability distribution ''P'' and mean value \mu = \mathrm /math> (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of ''X''. Then the standardized moment of degree ''k'' is \frac, that is, the ratio of the ''k''th moment about the mean : \mu_k = \operatorname \left ( X - \mu )^k \right = \int_^ (x - \mu)^k P(x)\,dx, to the ''k''th power of the standard deviation, :\sigma^k = \left(\sqrt\right)^k. The power of ''k'' is because moments scale as x^k, meaning that \mu_k(\lambda X) = \lambda^k \mu_k(X): they are homogeneous functions of de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Mass Function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. A probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability. The value of the random variable having the largest probability mass is called the mode. Formal definition Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function p: \R \to ,1/math> defined by for -\inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Prime Distribution
In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kindJohnson et al (1995), p 248) is an absolutely continuous probability distribution. Definitions Beta prime distribution is defined for x > 0 with two parameters ''α'' and ''β'', having the probability density function: : f(x) = \frac where ''B'' is the Beta function. The cumulative distribution function is : F(x; \alpha,\beta)=I_\left(\alpha, \beta \right) , where ''I'' is the regularized incomplete beta function. The expected value, variance, and other details of the distribution are given in the sidebox; for \beta>4, the excess kurtosis is :\gamma_2 = 6\frac. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Sealy Gosset
William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He pioneered small sample experimental design and analysis with an economic approach to the logic of uncertainty. Gosset published under the pen name Student and developed most famously Student's t-distribution – originally called Student's "z" – and "Student's test of statistical significance". Life and career Born in Canterbury, England the eldest son of Agnes Sealy Vidal and Colonel Frederic Gosset, R.E. Royal Engineers, Gosset attended Winchester College before matriculating as Winchester Scholar in natural sciences and mathematics at New College, Oxford. Upon graduating in 1899, he joined the brewery of Arthur Guinness & Son in Dublin, Ireland; he spent the rest of his 38-year career at Guinness. The site cites ''Dictionary of Scientific Biograph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Student's T-distribution
In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student". The ''t''-distribution plays a role in a number of widely used statistical analyses, including Student's ''t''-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. Student's ''t''-distribution also arises in the Bayesian analysis of data from a normal family. If we take a sample of n observations from a normal distribution, then the ''t''-distribution with \nu=n-1 degrees of freedom can be de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use: #With a shape parameter k and a scale parameter \theta. #With a shape parameter \alpha = k and an inverse scale parameter \beta = 1/ \theta , called a rate parameter. In each of these forms, both parameters are positive real numbers. The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a 1/x base measure) for a random variable X for which E 'X''= ''kθ'' = ''α''/''β'' is fixed and greater than zero, and E n(''X'')= ''ψ''(''k'') + ln(''θ'') = ''ψ''(''α'') − ln(''β'') is fixed (''ψ'' is the digamma function). Definitions The parameterization with ''k'' and ''θ'' appears to be more common in econo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Probability
In probability theory, inverse probability is an obsolete term for the probability distribution of an unobserved variable. Today, the problem of determining an unobserved variable (by whatever method) is called inferential statistics, the method of inverse probability (assigning a probability distribution to an unobserved variable) is called Bayesian probability, the "distribution" of data given the unobserved variable is rather the likelihood function (which is not a probability distribution), and the distribution of an unobserved variable, given both data and a prior distribution, is the posterior distribution. The development of the field and terminology from "inverse probability" to "Bayesian probability" is described by . The term "inverse probability" appears in an 1837 paper of Augustus De Morgan, De Morgan, in reference to Laplace's method of probability (developed in a 1774 paper, which independently discovered and popularized Bayesian methods, and a 1812 book), though th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/ yes/true/ one with probability ''p'' and failure/no/ false/zero with probability ''q''. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and ''p'' would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and ''p'' would be the probability of tails). In particular, unfair coins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Distribution
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
