Gaussian Distribution Of Error
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Distribution A normal distribution in a variate with mean and variance is a statistic distribution with probability density function
Normal Distribution Pdf
(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,
Normal Distribution Statistics
social scientists refer to 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 multivariate gaussian distribution 2003, p.157). The normal distribution is implemented 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,
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 normal distribution standard deviation quantities using the model. These intervals give the range of plausible values standard normal distribution for the process parameters based on the data and the underlying assumptions about the process. Because of the normal distribution probability statistical nature of 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 http://mathworld.wolfram.com/NormalDistribution.html a probabilistic interpretation that 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 http://www.itl.nist.gov/div898/handbook/pmd/section2/pmd214.htm data. Of course the random errors 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 n
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings http://stats.stackexchange.com/questions/23479/why-do-we-assume-that-the-error-is-normally-distributed 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 normal distribution 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 Why do we assume that the error is normally distributed? up vote 13 down vote favorite 5 I wonder why do we use the Gaussian assumption when modelling the error. In Stanford's ML course, gaussian distribution of Prof. Ng describes it basically in two manners: It is mathematically convenient. (It's related to Least Squares fitting and easy to solve with pseudoinverse) Due to the Central Limit Theorem, we may assume that there are lots of underlying facts affecting the process and the sum of these individual errors will tend to behave like in a zero mean normal distribution. In practice, it seems to be so. I'm interested in the second part actually. The Central Limit Theorem works for iid samples as far as I know, but we can not guarantee the underlying samples to be iid. Do you have any ideas about the Gaussian assumption of the error? regression normality share|improve this question edited Feb 22 '12 at 22:16 chl♦ 37.5k6125243 asked Feb 9 '12 at 13:34 petrichor 8501915 What setting are you talking about? Classification, regression, or something more general? –tdc Feb 9 '12 at 14:24 I asked the question for the general case. Most of the stories start with Gaussian error assumption. But, personally, my own interest is matrix factorizations and linear model solutio
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