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Of practical importance is the fact that the standard error of μ ^ {\displaystyle \scriptstyle {\hat {\mu }}} is proportional to 1 / n {\displaystyle \scriptstyle 1/{\sqrt − 6}} , that Show Gaussian curve IndexDistribution functionsApplied statistics concepts HyperPhysics*****HyperMath *****Algebra Go Back Gaussian Distribution Function The full width of the gaussian curve at half the maximum is Show IndexApplied statistics concepts The independence between μ ^ {\displaystyle \scriptstyle {\hat {\mu }}} and s can be employed to construct the so-called t-statistic: t = μ ^ − μ s / n = x 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 .

Haskell: An erf package[18] exists that provides a typeclass for the error function and implementations for the native (real) floating point types. Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The Poisson distribution with parameter λ is approximately normal with mean λ and variance λ, for large values of λ.[21] The chi-squared distribution χ2(k) is approximately normal with mean k and

Contents 1 Definition 1.1 Standard normal distribution 1.2 General normal distribution 1.3 Notation 1.4 Alternative parameterizations 2 Properties 2.1 Symmetries and derivatives 2.1.1 Differential equation 2.2 Moments 2.3 Fourier transform and 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 At maximum entropy, a small variation δf(x) about f(x) will produce a variation δL about L which is equal to zero: 0 = δ L = ∫ − ∞ ∞ δ Confidence intervals[edit] See also: Studentization By Cochran's theorem, for normal distributions the sample mean μ ^ {\displaystyle \scriptstyle {\hat {\mu }}} and the sample variance s2 are independent, which means there

An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression Furthermore, if A is symmetric, then the form x ′ A y = y ′ A x . {\displaystyle \mathbf μ 2 '\mathbf μ 1 \mathbf μ 0 =\mathbf σ 9 These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:[20] F(μ + nσ) − F(μ − nσ) n F(μ Bayesian analysis of the normal distribution[edit] Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered: Either the mean, or the variance, or neither,

Please try the request again. LCCN65-12253. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR2723248 External links[edit] MathWorld – Erf Authority control NDL: 00562553 Retrieved from Generated Sat, 15 Oct 2016 15:11:52 GMT by s_ac4 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection

This other estimator is denoted s2, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root s is called the sample standard deviation. New York: Dover, pp.297-309, 1972. Online Integral Calculator» Solve integrals with Wolfram|Alpha. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.

If L is sufficiently far from the mean, i.e. μ − L ≥ σ ln ⁡ k {\displaystyle \mu -L\geq \sigma {\sqrt {\ln {k}}}} , then: Pr [ X ≤ L and Watson, G.N. IDL: provides both erf and erfc for real and complex arguments. Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as

The mean value is a=np where n is the number of events and p the probability of any integer value of x (this expression carries over from the binomial distribution ). Integrals and Series, Vol.2: Special Functions. Intermediate levels of Re(ƒ)=constant are shown with thin red lines for negative values and with thin blue lines for positive values. Fortran 77 implementations are available in SLATEC.

and Oldham, K.B. "The Error Function and Its Complement ." Ch.40 in An Atlas of Functions. The same family is flat with respect to the (±1)-connections ∇(e) and ∇(m).[36] Related distributions[edit] Operations on a single random variable[edit] If X is distributed normally with mean μ and variance New York: Chelsea, 1948. The system returned: (22) Invalid argument The remote host or network may be down.

Normal distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation N ( μ , σ 2 ) {\displaystyle {\mathcal σ 4}(\mu ,\,\sigma ^ σ Differential equation[edit] It satisfies the differential equation σ 2 f ′ ( x ) + f ( x ) ( x − μ ) = 0 , f ( 0 ) In particular, the quantile z0.975 is 1.96; therefore a normal random variable will lie outside the interval μ ± 1.96σ in only 5% of cases. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x)=−erf(−x).

If Z is a standard normal deviate, then X = Zσ+μ will have a normal distribution with expected value μ and standard deviationσ. The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e−z2 into its Maclaurin series and integrating term by term, one obtains The Generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors. Mathematics of Statistics, Pt.2, 2nd ed.

For , (5) where is the incomplete gamma function. However it can be shown that the biased estimator σ ^ 2 {\displaystyle \scriptstyle {\hat {\sigma }}^ σ 4} is "better" than the s2 in terms of the mean squared error Your cache administrator is webmaster. Applications[edit] When the results of a series of measurements are described by a normal distribution with standard deviation σ {\displaystyle \textstyle \sigma } and expected value 0, then erf ( a

In particular, the most popular value of α = 5%, results in |z0.025| = 1.96. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. The mean, variance and third central moment of this distribution have been determined[41] E ( x ) = μ + 2 π ( σ 2 − σ 1 ) {\displaystyle E(x)=\mu A Course in Modern Analysis, 4th ed.

Referenced on Wolfram|Alpha: Erf CITE THIS AS: Weisstein, Eric W. "Erf." From MathWorld--A Wolfram Web Resource. The absolute value of normalized residuals, |X - μ|/σ, has chi distribution with one degree of freedom: |X - μ|/σ ~ χ1(|X - μ|/σ). This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule. Generated Sat, 15 Oct 2016 15:11:52 GMT by s_ac4 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = Amer., p.16, 1990.