1.a theoretical distribution with finite mean and variance
1.(MeSH)Continuous frequency distribution of infinite range. Its properties are as follows: 1, continuous, symmetrical distribution with both tails extending to infinity; 2, arithmetic mean, mode, and median identical; and 3, shape completely determined by the mean and standard deviation.
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Statistical Distributions[Hyper.]
Normal Distribution (n.) [MeSH]↕
calculation; counting; count; ciphering; computation; figuring; reckoning[Classe]
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instance[Domaine]
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emp : plur (fr)[Syntagme]
arrangement, organisation, organization, system  applied math, applied mathematics[Hyper.]
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statistics[Domaine]
normal distribution (n.)↕
Probability density function The red curve is the standard normal distribution 

Cumulative distribution function 

Notation  

Parameters  μ ∈ R — mean (location) σ^{2} > 0 — variance (squared scale) 
Support  x ∈ R 
CDF  
Mean  μ 
Median  μ 
Mode  μ 
Variance  
Skewness  0 
Ex. kurtosis  0 
Entropy  
MGF  
CF  
Fisher information 
In probability theory, the normal (or Gaussian) distribution is a continuous probability distribution that has a bellshaped probability density function, known as the Gaussian function or informally the bell curve:^{[nb 1]}
The parameter μ is the mean or expectation (location of the peak) and σ^{ 2} is the variance. σ is known as the standard deviation. The distribution with μ = 0 and σ^{ 2} = 1 is called the standard normal distribution or the unit normal distribution. A normal distribution is often used as a first approximation to describe realvalued random variables that cluster around a single mean value.
The normal distribution is considered the most prominent probability distribution in statistics. There are several reasons for this:^{[1]} First, the normal distribution arises from the central limit theorem, which states that under mild conditions the sum of a large number of random variables drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution. This gives it exceptionally wide application in, for example, sampling. Secondly, the normal distribution is very tractable analytically, that is, a large number of results involving this distribution can be derived in explicit form.
For these reasons, the normal distribution is commonly encountered in practice, and is used throughout statistics, natural sciences, and social sciences^{[2]} as a simple model for complex phenomena. For example, the observational error in an experiment is usually assumed to follow a normal distribution, and the propagation of uncertainty is computed using this assumption. Note that a normallydistributed variable has a symmetric distribution about its mean. Quantities that grow exponentially, such as prices, incomes or populations, are often skewed to the right, and hence may be better described by other distributions, such as the lognormal distribution or Pareto distribution. In addition, the probability of seeing a normallydistributed value that is far (i.e. more than a few standard deviations) from the mean drops off extremely rapidly. As a result, statistical inference using a normal distribution is not robust to the presence of outliers (data that is unexpectedly far from the mean, due to exceptional circumstances, observational error, etc.). When outliers are expected, data may be better described using a heavytailed distribution such as the Student's tdistribution.
From a technical perspective, alternative characterizations are possible, for example:
The normal distributions are a subclass of the elliptical distributions.
The simplest case of a normal distribution is known as the standard normal distribution, described by the probability density function
The factor in this expression ensures that the total area under the curve ϕ(x) is equal to one^{[proof]}, and 12 in the exponent makes the "width" of the curve (measured as half the distance between the inflection points) also equal to one. It is traditional in statistics to denote this function with the Greek letter ϕ (phi), whereas density functions for all other distributions are usually denoted with letters f or p.^{[5]} The alternative glyph φ is also used quite often, however within this article "φ" is reserved to denote characteristic functions.
Every normal distribution is the result of exponentiating a quadratic function (just as an exponential distribution results from exponentiating a linear function):
This yields the classic "bell curve" shape, provided that a < 0 so that the quadratic function is concave. f(x) > 0 everywhere. One can adjust a to control the "width" of the bell, then adjust b to move the central peak of the bell along the xaxis, and finally one must choose c such that (which is only possible when a < 0).
Rather than using a, b, and c, it is far more common to describe a normal distribution by its mean μ = − b2a and variance σ^{2} = − 12a. Changing to these new parameters allows one to rewrite the probability density function in a convenient standard form,
For a standard normal distribution, μ = 0 and σ^{2} = 1. The last part of the equation above shows that any other normal distribution can be regarded as a version of the standard normal distribution that has been stretched horizontally by a factor σ and then translated rightward by a distance μ. Thus, μ specifies the position of the bell curve's central peak, and σ specifies the "width" of the bell curve.
The parameter μ is at the same time the mean, the median and the mode of the normal distribution. The parameter σ^{2} is called the variance; as for any random variable, it describes how concentrated the distribution is around its mean. The square root of σ^{2} is called the standard deviation and is the width of the density function.
The normal distribution is usually denoted by N(μ, σ^{2}).^{[6]} Thus when a random variable X is distributed normally with mean μ and variance σ^{2}, we write
Some authors advocate using the precision instead of the variance. The precision is normally defined as the reciprocal of the variance (τ = σ^{−2}), although it is occasionally defined as the reciprocal of the standard deviation (τ = σ^{−1}).^{[7]} This parametrization has an advantage in numerical applications where σ^{2} is very close to zero and is more convenient to work with in analysis as τ is a natural parameter of the normal distribution. This parametrization is common in Bayesian statistics, as it simplifies the Bayesian analysis of the normal distribution. Another advantage of using this parametrization is in the study of conditional distributions in the multivariate normal case. The form of the normal distribution with the more common definition τ = σ^{−2} is as follows:
The question of which normal distribution should be called the "standard" one is also answered differently by various authors. Starting from the works of Gauss the standard normal was considered to be the one with variance σ^{2} = 12 :
Stigler (1982) goes even further and insists the standard normal to be with the variance σ^{2} = 12π :
According to the author, this formulation is advantageous because of a much simpler and easiertoremember formula, the fact that the pdf has unit height at zero, and simple approximate formulas for the quantiles of the distribution.
In the previous section the normal distribution was defined by specifying its probability density function. However there are other ways to characterize a probability distribution. They include: the cumulative distribution function, the moments, the cumulants, the characteristic function, the momentgenerating function, etc.
The probability density function (pdf) of a random variable describes the relative frequencies of different values for that random variable. The pdf of the normal distribution is given by the formula explained in detail in the previous section:
This is a proper function only when the variance σ^{2} is not equal to zero. In that case this is a continuous smooth function, defined on the entire real line, and which is called the "Gaussian function".
Properties:
When σ^{2} = 0, the density function doesn't exist. However a generalized function that defines a measure on the real line, and it can be used to calculate, for example, expected value is
where δ(x) is the Dirac delta function which is equal to infinity at x = μ and is zero elsewhere.
The cumulative distribution function (CDF) describes probability of a random variable falling in the interval (−∞, x].
The CDF of the standard normal distribution is denoted with the capital Greek letter Φ (phi), and can be computed as an integral of the probability density function:
This integral cannot be expressed in terms of elementary functions, so is simply called a transformation of the error function, or erf, a special function. Numerical methods for calculation of the standard normal CDF are discussed below. For a generic normal random variable with mean μ and variance σ^{2} > 0 the CDF will be equal to
The complement of the standard normal CDF, Q(x) = 1 − Φ(x), is referred to as the Qfunction, especially in engineering texts.^{[11]}^{[12]} This represents the upper tail probability of the Gaussian distribution: that is, the probability that a standard normal random variable X is greater than the number x. Other definitions of the Qfunction, all of which are simple transformations of Φ, are also used occasionally.^{[13]}
Properties:
For a normal distribution with zero variance, the CDF is the Heaviside step function (with H(0) = 1 convention):
The quantile function of a distribution is the inverse of the CDF. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:
Quantiles of the standard normal distribution are commonly denoted as z_{p}. The quantile z_{p} represents such a value that a standard normal random variable X has the probability of exactly p to fall inside the (−∞, z_{p}] interval. The quantiles are used in hypothesis testing, construction of confidence intervals and QQ plots. The most "famous" normal quantile is 1.96 = z_{0.975}. A standard normal random variable is greater than 1.96 in absolute value in 5% of cases.
For a normal random variable with mean μ and variance σ^{2}, the quantile function is
The characteristic function φ_{X}(t) of a random variable X is defined as the expected value of e^{itX}, where i is the imaginary unit, and t ∈ R is the argument of the characteristic function. Thus the characteristic function is the Fourier transform of the density ϕ(x). For a normally distributed X with mean μ and variance σ^{2}, the characteristic function is ^{[14]}
The characteristic function can be analytically extended to the entire complex plane: one defines φ(z) = e^{iμz − 1/2σ2z2} for all z ∈ C.^{[15]}
The moment generating function is defined as the expected value of e^{tX}. For a normal distribution, the moment generating function exists and is equal to
The cumulant generating function is the logarithm of the moment generating function:
Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.
The normal distribution has moments of all orders. That is, for a normally distributed X with mean μ and variance σ^{ 2}, the expectation E[X^{p}] exists and is finite for all p such that Re[p] > −1. Usually we are interested only in moments of integer orders: p = 1, 2, 3, ….
Order  Raw moment  Central moment  Cumulant 

1  μ  0  μ 
2  μ^{2} + σ^{2}  σ^{ 2}  σ^{ 2} 
3  μ^{3} + 3μσ^{2}  0  0 
4  μ^{4} + 6μ^{2}σ^{2} + 3σ^{4}  3σ^{ 4}  0 
5  μ^{5} + 10μ^{3}σ^{2} + 15μσ^{4}  0  0 
6  μ^{6} + 15μ^{4}σ^{2} + 45μ^{2}σ^{4} + 15σ^{6}  15σ^{ 6}  0 
7  μ^{7} + 21μ^{5}σ^{2} + 105μ^{3}σ^{4} + 105μσ^{6}  0  0 
8  μ^{8} + 28μ^{6}σ^{2} + 210μ^{4}σ^{4} + 420μ^{2}σ^{6} + 105σ^{8}  105σ^{ 8}  0 
Because the normal distribution is a locationscale family, it is possible to relate all normal random variables to the standard normal. For example if X is normal with mean μ and variance σ^{2}, then
has mean zero and unit variance, that is Z has the standard normal distribution. Conversely, having a standard normal random variable Z we can always construct another normal random variable with specific mean μ and variance σ^{2}:
This "standardizing" transformation is convenient as it allows one to compute the PDF and especially the CDF of a normal distribution having the table of PDF and CDF values for the standard normal. They will be related via
About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 689599.7 rule, or the empirical rule, or the 3sigma rule. To be more precise, the area under the bell curve between μ − nσ and μ + nσ is given by
where erf is the error function. To 12 decimal places, the values for the 1, 2, up to 6sigma points are:^{[16]}
i.e. 1 minus ...  or 1 in ...  

1  0.682689492137  0.317310507863  3.15148718753 
2  0.954499736104  0.045500263896  21.9778945080 
3  0.997300203937  0.002699796063  370.398347345 
4  0.999936657516  0.000063342484  15,787.1927673 
5  0.999999426697  0.000000573303  1,744,277.89362 
6  0.999999998027  0.000000001973  506,797,345.897 
The next table gives the reverse relation of sigma multiples corresponding to a few often used values for the area under the bell curve. These values are useful to determine (asymptotic) confidence intervals of the specified levels based on normally distributed (or asymptotically normal) estimators:^{[17]}
n  n  

0.80  1.281551565545  0.999  3.290526731492  
0.90  1.644853626951  0.9999  3.890591886413  
0.95  1.959963984540  0.99999  4.417173413469  
0.98  2.326347874041  0.999999  4.891638475699  
0.99  2.575829303549  0.9999999  5.326723886384  
0.995  2.807033768344  0.99999999  5.730728868236  
0.998  3.090232306168  0.999999999  6.109410204869 
where the value on the left of the table is the proportion of values that will fall within a given interval and n is a multiple of the standard deviation that specifies the width of the interval.
The theorem states that under certain (fairly common) conditions, the sum of a large number of random variables will have an approximately normal distribution. For example if (x_{1}, …, x_{n}) is a sequence of iid random variables, each having mean μ and variance σ^{2}, then the central limit theorem states that
The theorem will hold even if the summands x_{i} are not iid, although some constraints on the degree of dependence and the growth rate of moments still have to be imposed.
The importance of the central limit theorem cannot be overemphasized. A great number of test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, even more estimators can be represented as sums of random variables through the use of influence functions — all of these quantities are governed by the central limit theorem and will have asymptotically normal distribution as a result.
Another practical consequence of the central limit theorem is that certain other distributions can be approximated by the normal distribution, for example:
Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.
A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.
If X is distributed normally with mean μ and variance σ^{2}, then
If X_{1} and X_{2} are two independent standard normal random variables, then
The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is onedimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.
One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
Normality tests assess the likelihood that the given data set {x_{1}, …, x_{n}} comes from a normal distribution. Typically the null hypothesis H_{0} is that the observations are distributed normally with unspecified mean μ and variance σ^{2}, versus the alternative H_{a} that the distribution is arbitrary. A great number of tests (over 40) have been devised for this problem, the more prominent of them are outlined below:
It is often the case that we don't know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample (x_{1}, …, x_{n}) from a normal N(μ, σ^{2}) population we would like to learn the approximate values of parameters μ and σ^{2}. The standard approach to this problem is the maximum likelihood method, which requires maximization of the loglikelihood function:
Taking derivatives with respect to μ and σ^{2} and solving the resulting system of first order conditions yields the maximum likelihood estimates:
Estimator is called the sample mean, since it is the arithmetic mean of all observations. The statistic is complete and sufficient for μ, and therefore by the Lehmann–Scheffé theorem, is the uniformly minimum variance unbiased (UMVU) estimator.^{[29]} In finite samples it is distributed normally:
The variance of this estimator is equal to the μμelement of the inverse Fisher information matrix . This implies that the estimator is finitesample efficient. Of practical importance is the fact that the standard error of is proportional to , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.
From the standpoint of the asymptotic theory, is consistent, that is, it converges in probability to μ as n → ∞. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
The estimator is called the sample variance, since it is the variance of the sample (x_{1}, …, x_{n}). In practice, another estimator is often used instead of the . This other estimator is denoted s^{2}, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root s is called the sample standard deviation. The estimator s^{2} differs from by having (n − 1) instead of n in the denominator (the so called Bessel's correction):
The difference between s^{2} and becomes negligibly small for large n's. In finite samples however, the motivation behind the use of s^{2} is that it is an unbiased estimator of the underlying parameter σ^{2}, whereas is biased. Also, by the Lehmann–Scheffé theorem the estimator s^{2} is uniformly minimum variance unbiased (UMVU),^{[29]} which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator is "better" than the s^{2} in terms of the mean squared error (MSE) criterion. In finite samples both s^{2} and have scaled chisquared distribution with (n − 1) degrees of freedom:
The first of these expressions shows that the variance of s^{2} is equal to 2σ^{4}/(n−1), which is slightly greater than the σσelement of the inverse Fisher information matrix . Thus, s^{2} is not an efficient estimator for σ^{2}, and moreover, since s^{2} is UMVU, we can conclude that the finitesample efficient estimator for σ^{2} does not exist.
Applying the asymptotic theory, both estimators s^{2} and are consistent, that is they converge in probability to σ^{2} as the sample size n → ∞. The two estimators are also both asymptotically normal:
In particular, both estimators are asymptotically efficient for σ^{2}.
By Cochran's theorem, for normal distribution the sample mean and the sample variance s^{2} are independent, which means there can be no gain in considering their joint distribution. There is also a reverse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between and s can be employed to construct the socalled tstatistic:
This quantity t has the Student's tdistribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this tstatistics will allow us to construct the confidence interval for μ;^{[30]} similarly, inverting the χ^{2} distribution of the statistic s^{2} will give us the confidence interval for σ^{2}:^{[31]}
where t_{k,p} and χ 2
k,p are the p^{th} quantiles of the t and χ^{2}distributions respectively. These confidence intervals are of the level 1 − α, meaning that the true values μ and σ^{2} fall outside of these intervals with probability α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The approximate formulas in the display above were derived from the asymptotic distributions of and s^{2}. The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles z_{α/2} do not depend on n. In particular, the most popular value of α = 5%, results in z_{0.025} = 1.96.
Bayesian analysis of normallydistributed data is complicated by the many different possibilities that may be considered:
The formulas for the nonlinearregression cases are summarized in the conjugate prior article.
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.
This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:
A similar formula can be written for the sum of two vector quadratics: If are vectors of length , and and are symmetric, invertible matrices of size , then
where
Note that the form is called a quadratic form and is a scalar:
In other words, it sums up all possible combinations of products of pairs of elements from , with a separate coefficient for each. In addition, since , only the sum matters for any offdiagonal elements of , and there is no loss of generality in assuming that is symmetric. Furthermore, if is symmetric, then the form .
Another useful formula is as follows:
where
For a set of i.i.d. normallydistributed data points X of size n where each individual point x follows with known variance σ^{2}, the conjugate prior distribution is also normallydistributed.
This can be shown more easily by rewriting the variance as the precision, i.e. using Then if and we proceed as follows.
First, the likelihood function is (using the formula above for the sum of differences from the mean):
Then, we proceed as follows:
In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving . The result is the kernel of a normal distribution, with mean and precision , i.e.
This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:
That is, to combine data points with total precision of (or equivalently, total variance of ) and mean of values , derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precisionweighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)
The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precisionweighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas
For a set of i.i.d. normallydistributed data points X of size n where each individual point x follows with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chisquared distribution. The two are equivalent except for having different parameterizations. The use of the inverse gamma is more common, but the scaled inverse chisquared is more convenient, so we use it in the following derivation. The prior for σ^{2} is as follows:
The likelihood function from above, written in terms of the variance, is:
where
Then:
This is also a scaled inverse chisquared distribution, where
or equivalently
Reparameterizing in terms of an inverse gamma distribution, the result is:
For a set of i.i.d. normallydistributed data points X of size n where each individual point x follows with unknown mean μ and variance σ^{2}, the a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normalinversegamma distribution. Logically, this originates as follows:
The priors are normally defined as follows:
The update equations can be derived, and look as follows:
The respective numbers of pseudoobservations just add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new "interaction term" needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.
Proof is as follows.
The prior distributions are
Therefore, the joint prior is
The likelihood function from the section above with known variance, and writing it in terms of variance rather than precision, is:
where
Therefore, the posterior is (dropping the hyperparameters as conditioning factors):
In other words, the posterior distribution has the form of a product of a normal distribution over times an inverse gamma distribution over , with parameters that are the same as the update equations above.
The occurrence of normal distribution in practical problems can be loosely classified into three categories:
Certain quantities in physics are distributed normally, as was first demonstrated by James Clerk Maxwell. Examples of such quantities are:
Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by a large number of small effects acting additively and independently, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence which has a considerably larger magnitude than the rest of the effects.
“  I can only recognize the occurrence of the normal curve — the Laplacian curve of errors — as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations. — Pearson (1901)  ” 
There are statistical methods to empirically test that assumption, see the above Normality tests section.
In computer simulations, especially in applications of the MonteCarlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a N(μ, σ2
) can be generated as X = μ + σZ, where Z is standard normal. All these algorithms rely on the availability of a random number generator U capable of producing uniform random variates.
The standard normal CDF is widely used in scientific and statistical computing. The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Different approximations are used depending on the desired level of accuracy.
Some authors^{[38]}^{[39]} attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738 ^{[nb 3]} published in the second edition of his "The Doctrine of Chances" the study of the coefficients in the binomial expansion of (a + b)^{n}. De Moivre proved that the middle term in this expansion has the approximate magnitude of , and that "If m or ½n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ℓ, has to the middle Term, is ."^{[40]} Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.^{[41]}
In 1809 Gauss published his monograph "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, M′, M′′, … to denote the measurements of some unknown quantity V, and sought the "most probable" estimator: the one which maximizes the probability φ(M−V) · φ(M′−V) · φ(M′′−V) · … of obtaining the observed experimental results. In his notation φΔ is the probability law of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the wellknown answer: the arithmetic mean of the measured values.^{[nb 4]} Starting from these principles, Gauss demonstrates that the only law which rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:^{[42]}
where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the nonlinear weighted least squares (NWLS) method.^{[43]}
Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.^{[nb 5]} It was Laplace who first posed the problem of aggregating several observations in 1774,^{[44]} although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the integral ∫ e^{−t ²}dt = √π in 1782, providing the normalization constant for the normal distribution.^{[45]} Finally, it was Laplace who in 1810 proved and presented to the Academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.^{[46]}
It is of interest to note that in 1809 an American mathematician Adrain published two derivations of the normal probability law, simultaneously and independently from Gauss.^{[47]} His works remained largely unnoticed by the scientific community, until in 1871 they were "rediscovered" by Abbe.^{[48]}
In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:^{[49]} "The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is
Since its introduction, the normal distribution has been known by many different names: the law of error, the law of facility of errors, Laplace's second law, Gaussian law, etc. Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than "usual"^{[50]}. However, by the end of the 19th century some authors^{[nb 6]} had started using the name normal distribution, where the word "normal" was used as an adjective — the term now being seen as a reflection of the fact that this distribution was seen as typical, common  and thus "normal". Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under certain circumstances."^{[51]} Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.^{[52]}
“  Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'. — Pearson (1920)  ” 
Also, it was Pearson who first wrote the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
The term "standard normal" which denotes the normal distribution with zero mean and unit variance came into general use around 1950s, appearing in the popular textbooks by P.G. Hoel (1947) "Introduction to mathematical statistics" and A.M. Mood (1950) "Introduction to the theory of statistics".^{[53]}
When the name is used, the "Gaussian distribution" was named after Carl Friedrich Gauss, who introduced the distribution in 1809 as a way of rationalizing the method of least squares as outlined above. The related work of Laplace, also outlined above has led to the normal distribution being sometimes called Laplacian,^{[citation needed]} especially in Frenchspeaking countries. Among English speakers, both "normal distribution" and "Gaussian distribution" are in common use, with different terms preferred by different communities.
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Lettris
Lettris is a curious tetrisclone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
boggle
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
English dictionary
Main references
Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).
Copyrights
The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.
Translation
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
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