Normal Distribution Of Error
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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 normally distributed error term relevant process quantities using the model. These intervals give the range of normally distributed errors regression plausible values for the process parameters based on the data and the underlying assumptions about the process. Because
Gaussian Distribution Error Function
of the 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
Normal Distribution Curve
the intervals have 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 normal distribution examples can be estimated from the 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 t
For other uses, see Bell curve (disambiguation). Normal distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation N ( μ , σ 2 ) {\displaystyle {\mathcal σ
Normal Distribution Standard Deviation
4}(\mu ,\,\sigma ^ σ 3)} Parameters μ ∈ R — mean (location) σ2 normal distribution of errors experiment > 0 — variance (squared scale) Support x ∈ R PDF 1 2 σ 2 π e − ( standard error normal distribution x − μ ) 2 2 σ 2 {\displaystyle {\frac σ 0{\sqrt − 9\pi }}}\,e^{-{\frac {(x-\mu )^ − 8} − 7}}}} CDF 1 2 [ 1 + erf ( x − http://www.itl.nist.gov/div898/handbook/pmd/section2/pmd214.htm μ σ 2 ) ] {\displaystyle {\frac − 2 − 1}\left[1+\operatorname − 0 \left({\frac 9{\sigma {\sqrt 8}}}\right)\right]} Quantile μ + σ 2 erf − 1 ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt 2}\operatorname 1 ^{-1}(2F-1)} Mean μ Median μ Mode μ Variance σ 2 {\displaystyle \sigma ^ − 8\,} Skewness 0 Ex. kurtosis 0 Entropy 1 https://en.wikipedia.org/wiki/Normal_distribution 2 ln ( 2 σ 2 π e ) {\displaystyle {\tfrac − 6 − 5}\ln(2\sigma ^ − 4\pi \,e\,)} MGF exp { μ t + 1 2 σ 2 t 2 } {\displaystyle \exp\{\mu t+{\frac − 0 σ 9}\sigma ^ σ 8t^ σ 7\}} CF exp { i μ t − 1 2 σ 2 t 2 } {\displaystyle \exp\ σ 2 σ 1}\sigma ^ σ 0t^ μ 9\}} Fisher information ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) {\displaystyle {\begin μ 41/\sigma ^ μ 3&0\\0&1/(2\sigma ^ μ 2)\end μ 1}} In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the nu
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 http://stats.stackexchange.com/questions/130775/why-do-we-care-so-much-about-normally-distributed-error-terms-and-homoskedastic 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 normal distribution top Why do we care so much about normally distributed error terms (and homoskedasticity) in linear regression when we don't have to? up vote 43 down vote favorite 24 I suppose I get frustrated every time I hear someone say that non-normality of residuals and /or heteroskedasticity violates OLS assumptions. To estimate parameters in an OLS model neither of these assumptions are necessary by the Gauss-Markov theorem. I see how this matters in Hypothesis Testing normal distribution of for the OLS model, because assuming these things give us neat formulas for t-tests, F-tests, and more general Wald statistics. But it is not too hard to do hypothesis testing without them. If we drop just homoskedasticity we can calculate robust standard errors and clustered standard errors easily. If we drop normality altogether we can use bootstrapping and, given another parametric specification for the error terms, likelihood ratio, and Lagrange multiplier tests. It's just a shame that we teach it this way, because I see a lot of people struggling with assumptions they do not have to meet in the first place. Why is it that we stress these assumptions so heavily when we have the ability to easily apply more robust techniques? Am I missing something important? regression assumptions robust teaching share|improve this question edited Dec 30 '14 at 23:28 conjugateprior 13.3k12761 asked Dec 30 '14 at 22:22 Zachary Blumenfeld 2,560514 1 Seems to be a disciplinary thing. In my experience, at the extremes, Econometrics texts almost always cover what inferences each assumption buys and Psychology texts never seem to mention anything about the topic. –conjugateprior Dec 30 '14 at 23:35 10 Homoscedasticity is necessary for OLS to be BLUE though. –Momo Dec 30 '14 at 23:39 3 I think you are right, those assumptions receive undue attention.