Normal Error Distribution
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is Inference After fitting a model to the data and validating it, scientific or engineering questions about the process are usually answered by computing statistical intervals for relevant process quantities using normal distribution standard deviation the model. These intervals give the range of plausible values for the multivariate gaussian distribution process parameters based on the data and the underlying assumptions about the process. Because of the statistical nature of
Standard Normal Distribution
the process, however, the intervals cannot always be guaranteed to include the true process parameters and still be narrow enough to be useful. Instead the intervals have a probabilistic interpretation that http://onlinelibrary.wiley.com/doi/10.1002/9780470121498.app4/pdf guarantees coverage of the true process parameters a specified proportion of the time. In order for these intervals to truly have their specified probabilistic interpretations, the form of the distribution of the random errors must be known. Although the form of the probability distribution must be known, the parameters of the distribution can be estimated from the data. Of course the random errors http://www.itl.nist.gov/div898/handbook/pmd/section2/pmd214.htm from different types of processes could be described by any one of a wide range of different probability distributions in general, including the uniform, triangular, double exponential, binomial and Poisson distributions. With most process modeling methods, however, inferences about the process are based on the idea that the random errors are drawn from a normal distribution. One reason this is done is because the normal distribution often describes the actual distribution of the random errors in real-world processes reasonably well. The normal distribution is also used because the mathematical theory behind it is well-developed and supports a broad array of inferences on functions of the data relevant to different types of questions about the process. Non-Normal Random Errors May Result in Incorrect Inferences Of course, if it turns out that the random errors in the process are not normally distributed, then any inferences made about the process may be incorrect. If the true distribution of the random errors is such that the scatter in the data is less than it would be under a normal distribution, it is possible that the intervals used to capture th
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Wed Oct 19 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Statistical http://mathworld.wolfram.com/NormalDistribution.html Distributions>Continuous Distributions> History and Terminology>Wolfram Language Commands> Interactive Entries>Interactive Demonstrations> Normal Distribution A http://stats.stackexchange.com/questions/148803/how-does-linear-regression-use-the-normal-distribution normal distribution in a variate with mean and variance is a statistic distribution with probability density function (1) on the domain . While statisticians and mathematicians uniformly use the term "normal distribution" for this distribution, physicists sometimes call it a Gaussian distribution and, because of its curved flaring shape, social scientists refer to normal distribution it as the "bell curve." Feller (1968) uses the symbol for in the above equation, but then switches to in Feller (1971). de Moivre developed the normal distribution as an approximation to the binomial distribution, and it was subsequently used by Laplace in 1783 to study measurement errors and by Gauss in 1809 in the analysis of astronomical data (Havil 2003, p.157). The normal distribution is implemented normal error distribution in the Wolfram Language as NormalDistribution[mu, sigma]. The so-called "standard normal distribution" is given by taking and in a general normal distribution. An arbitrary normal distribution can be converted to a standard normal distribution by changing variables to , so , yielding (2) The Fisher-Behrens problem is the determination of a test for the equality of means for two normal distributions with different variances. The normal distribution function gives the probability that a standard normal variate assumes a value in the interval , (3) (4) where erf is a function sometimes called the error function. Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically or otherwise approximated. The normal distribution is the limiting case of a discrete binomial distribution as the sample size becomes large, in which case is normal with mean and variance (5) (6) with . The distribution is properly normalized since (7) The cumulative distribution function, which gives the probability that a variate will assume a value , is then the integral of the normal distribution, (8) (9) (10) where erf is the so-called error function. Normal distributions hav
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top How does linear regression use the normal distribution? up vote 6 down vote favorite 6 In linear regression, each predicted value is assumed to have been picked from a normal distribution of possible values. See below. But why is each predicted value assumed to have come from a normal distribution? How does linear regression use this assumption? What if possible values are not normally distributed? regression probability distributions normal-distribution modeling share|improve this question asked Apr 29 '15 at 7:15 luciano 3,05154070 Only the errors follow a normal distribution (which implies the conditional probability of Y given X is normal too). This is probably traditional because of reasons relating to the central limit theorem. But you can replace normal with any symmetric probability distribution and get the same estimates of coefficients via least squares. What differs though would be the residual standard error, goodness of fit and the way you validate the assumptions. –user2432701 Apr 29 '15 at 9:15 2 Normal assumptions mainly come into inference -- hypothesis testing, CIs, PIs. If you make different assumptions, those will be different, at least in small samples.