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Gaussian

Definition: Gaussian

Gaussian

Adjective

1. Of or relating to Karl Gauss or his mathematical theories of magnetics or electricity or astronomy or probability; "Gaussian distribution".

Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved.
 

 

Specialty Definition: Gaussian

(From Wikipedia, the free Encyclopedia)

See:

The relationship between these two concepts is that the probability density function of the normal probability distribution is a Gaussian function.

Also see:

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Gaussian function

(From Wikipedia, the free Encyclopedia)

A Gaussian function (the first syllable rhymes with house) is a function of the form

for some real constants a > 0, b, and c. The eponym of these functions is Carl Friedrich Gauss.

Gaussian functions with c2=2 are eigenfunctions of the Fourier transform.

Gaussian functions are among those functions that are "elementary" but lack "elementary antiderivatives", i.e., their antiderivatives are not among the functions ordinarily considered in first-year calculus courses. Nonetheless their definite integrals over the whole real line can be evaluated exactly.

This calculation can be performed by the residue theorem of complex analysis, but there is also a simple and instructive way to do the calculation. Call the value of this integral I. Then,

Note the renaming of the variable of integration from x to y (see dummy variable). We now change to plane polar coordinates

(The substitution u = r2, du = 2r dr was used.)

The density function of the normal probability distribution is a Gaussian function.

See also: Lorentzian function

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Gaussian function."

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Normal distribution

(From Wikipedia, the free Encyclopedia)

The normal distribution is an extremely important probability distribution considered in statistics. Among people whose field is not primarily probability theory or statistics (notably in physics) it is often called the Gaussian distribution. It is actually a family of distributions of the same general form, differing only in their location and scale parameters: the mean and standard deviation. The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. Because the graph of its probability density resembles a bell, it is often called the bell curve.

History

The normal distribution was first introduced by de Moivre in an article in 1733 (reprinted in the second edition of his Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n. His result was extended by Laplace in his book Analytical Theory of Probabilities (1812), and is now called the Theorem of de Moivre-Laplace.

Laplace used the normal distribution in the analysis of errors of experiments. The important method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have used the method since 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors.

The name "bell curve" goes back to Jouffret who used the term "bell surface" in 1872 for a bivariate normal with independent components. The name "normal distribution" was coined independently by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875 [Stigler]. This terminology is unfortunate, since it reflects and encourages the fallacy that "everything is Gaussian". (See the discussion of "occurrence" below).

Specification of the normal distribution

There are various ways to specify a random variable. The most visual is the probability density function (plot at the top), which represents how likely each value of the random variable is. The cumulative density function is a conceptually cleaner way to specify the same information, but to the untrained eye its plot is much less informative (see below). Equivalent ways to specify the normal distribution are: the characteristic function (related to the cumulant generating function) and the moment generating function. These are very useful for theoretical work, but much less intuitive. See probability distribution for a discussion.

Distribution functions

Probability density function

The probability density function of the normal distribution with mean μ and standard deviation σ (or variance σ2) is also known as the Gaussian function

(see exponential function and pi). If a random variable X follows this distribution, we write X ~ N(μ, σ2). If μ = 0 and σ = 1, we talk about the standard normal distribution, with formula

The picture at the top is the graph of the probability density function of the standard normal distribution. The distribution is symmetric about its mean value. About 68% of the area under the curve is within one standard deviation of the mean, 95.5% within two standard deviations, and 99.7% within three standard deviations (the "68 - 95.5 - 99.7 rule"). The inflection points of the curve occur at one standard deviation away from the mean.

These statements are also true for non-standard normal distributions.

Cumulative distribution function

The cumulative distribution function of the normal distribution is the probability that a given standard normal variable has a value less than z. Given the probability density function above, the cumulative distribution function has formula:

The following graph represents the cumulative distribution function for values of z from -4 to +4:

For instance, the probability that a standard normal variable has a value less than 0.12 is equal to 0.54776. The cumulative distribution function of the normal distribution does not have an analytic form, and has to be calculated using numerical techniques. It is so commonly used that it is often called "the" error function. A special feature of the normal distribution is that the cumulative distribution function is not needed to simulate normal random variables (see below).

Generating functions

Moment generating function

Characteristic function

The characteristic function of a gaussian random variable X ~ N(μ,σ2) is defined as the expected value of eitX and can be written as

as can be seen by completing the square in the exponent.

Properties

  1. If X ~ N(μ, σ2) and a and b are real numbers, then aX + b ~ N(aμ + b, (aσ)2).
  2. If X1 ~ N(μ1, σ12) and X2 ~ N(μ2, σ22), and X1 and X2 are independent, then X1 + X2 ~ N(μ1 + μ2, σ12 + σ22).
  3. If X1, ..., Xn are independent standard normal variables, then X12 + ... + Xn2 follows a chi-squared distribution with n degrees of freedom.

Standardizing Gaussian random variables

As a consequence of the first listed property, it is possible to relate all gaussian random variables to the standard normal.

If X is a Gaussian random variable with mean μ and variance σ2, then

is a standard normal random variable: Z~N(0,1). Conversely, if Z is a standard normal random variable,

is a Gaussian random variable with mean μ and variance σ2.

The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one. Therefore, if one knows the mean and the standard deviation of a normal distribution, one can use this table to answer all questions about the distribution.

Generating Gaussian random variables

For computer simulations, it is often necessary to generate values that follow a Gaussian distribution. This is best done with the Box-Muller transforms. These methods require two uniformly distributed values as input which can easily be generated by the computer's pseudorandom number generator.

The Box-Muller transform is a beautiful consequence of the third listed property, and the fact that the chi-square distribution with two degrees of freedom is an exponential random variable (which can be easily simulated exactly).

Occurrence

Approximately normal distributions occur in many situations, as a result of the central limit theorem. Simply stated, this theorem says that adding up a large number of small independent variables results in an approximately normal distribution. Therefore, whenever there is reason to suspect the presence of a large number of small effects acting additively, it is reasonable to assume that observations will be normal. This assumption is then subject to empirical test using well-established statistical methods.

It is important to realize, however, that small effects often act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal.

Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting distributions are ideed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the theory of errors (see below).

To summarize, here's a list of situations where approximate normality is expected. For a fuller discussion, see below.

Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality.

Instances of the central limit theorem

Test scores

The IQ score of an individual for example can be seen as the result of many small additive influences: many genes and many environmental factors all play a role.

Criticisms: test scores are discrete variable associated with the number of correct/incorrect answers, and as such they are related to the binomial. Moreover (see this USENET post), raw IQ test scores are customarily 'massaged' to force the distribution of IQ scores to be normal. Finally, there is no widely accepted model of intelligence, and the link to IQ scores let alone a relationship between influences on intelligence and additive variations of IQ, is subject to debate.

Physical characteristics of biological specimens

The overwhelming biological evidence is that bulk growth processes of living tissue proceed by multiplicative, not additive, increments, and that therefore measures of body size should at most follow a lognormal rather than normal distribution. Despite common claims of normality, the sizes of plants and animals is approximately lognormal. The evidence and an explanation based on models of growth was first published in the classic book

Huxley, Julian: Problems of Relative Growth (1932)

Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further make the joint distribution of sizes deviate from lognormality.

The assumption that linear size of biological specimens is normal leads to a non-normal distribution of weight (since weight/volume is roughly the 3rd power of length, and gaussian distributions are only preserved by linear transformations), and conversely assuming that weight is normal leads to non-normal lengths. This is a problem, because there is no a priori reason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed.

Measurement errors

Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not been taken into account. Note that this is the central assumption of the mathematical theory of errors.

Financial variables

Because of the exponential nature of interest and inflation, financial indicators such as interest rates, stock values, or commodity prices make good examples of multiplicative behaviour. As such, they should not be expected to be normal, but lognormal.

Mandelbrot, the popularizer of fractals, has claimed that even the assumption of lognormality is flawed.

Lifetime

Other examples of variables that are not normally distributed include the lifetimes of humans or technical devices. Examples of distributions used in this connection are the exponential distribution (memoryless) and the Weibull distribution. In general, there is no reason that waiting times should be normal, since they are not directly related to any kind of additive influence.

Photon counts

Light intensity from a single source varies with time, and is usually assumed to be normally distributed. However, quantum mechanics interprets measurements of light intensity as photon counting. Ordinary light sources which produce light by thermal emission, should follow a Poisson distribution or Bose-Einstein distribution on very short time scales. On longer time scales (longer than the coherence time), the addition of independent variables generates a gaussian distribution. Laser light, which is definitely a quantum phenomenon, has an exactly gaussian light intensity.

Further reading

External links and references

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Probability

(From Wikipedia, the free Encyclopedia)

Probability derives from the Latin probare (to prove, or to test). The word probable means roughly "likely to occur" in the case of possible future occurrences, or "likely to be true" in the case of inferences from evidence. See also probability theory.

What mathematicians call probability is the mathematical theory we use to describe and quantify uncertainty. In a larger context, (see probability interpretations) the word probability is used with other concerns in mind. Uncertainty can be due to our ignorance, deliberate mixing or shuffling, or due to the essential randomness of Nature. In any case, we measure the uncertainty of events on a scale from zero (impossible events) to one (certain events or no uncertainty).

Probability axioms form the basis for mathematical probability theory. Calculation of probabilities can often be determined using combinatorics or by applying the axioms directly. Probability applications include even more than statistics, which is usually based on the idea of probability distributions and the central limit theorem.

The idea is most often broken into two concepts:

  1. aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon like tossing dice or spinning a wheel; and
  2. epistemic probability, which represents our uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are:
  • Assign a probability to the proposition that a proposed law of physics is true.
Determine how "probable" it is that a suspect committed a crime, based on the evidence presented.

It is an open question whether aleatory probability is reducible to epistemic probability based on our inability to precisely predict every force that might affect the roll of a die, or whether such uncertainties exist in the nature of reality itself, particularly in quantum phenomena governed by Heisenberg's uncertainty principle. Although the same mathematical rules apply regardless of what interpretation you favor, the choice has major implications for the way in which probability is used to model the real world.

Probability in mathematics

While the existence of gambling games of chance shows that there has been a lively interest in quantifying the ideas of probability for millennia, exact mathematical descriptions of use in these types of problems only arose much later.

To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor - certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible for example to flip 10 heads in a row. What then does the number "50%" mean in this context?

One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent - that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let NH be the number of times the coin lands heads, then we can, for any N, consider the ratio NH/N.

As N gets larger and larger, we expect that in our example the ratio NH/N will get closer and closer to 1/2. This allows us to define the probability Pr(H) of flipping heads as the mathematical limit, as N approaches infinity, of this sequence of ratios:

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,

In other words, by saying that "the probability of heads is 1/2", we mean that, if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitraily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.

The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz and Guildenstern are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event - after all, it is possible (although unlikely) that a fair coin would give this result - or whether his assumption that the coin is fair is at fault.

One contribution of Bayesian probability was to provide a philosophical stance which allows us to derive probabilities (according to the above definition) from a set of observations.

Formalization

In general, probabilities of interest regard not just discrete outcomes like "heads/tails", but also more continuous outcomes as well.

In probability theory, an event is a "measurable" subset of a "sample space". "Events" are the things to which probabilities are assigned. A probability is a number in the closed interval from 0 to 1. Probabilities must be assigned to events in such a way that for pairwise disjoint (i.e., no two intersect each other) events A1, A2, A3, ..., the probability of their union is the sum of their probabilities, or, in mathematical notation,

In the special case of a "discrete probability distribution" the sample space is a set of outcomes to each of which a positive number has been assigned as its probability. The one-members sets are "elementary events". One of the simplest of discrete sample spaces is a finite set to each of whose members the same probability 1/n is assigned. An example of a sample space that is not discrete is the closed interval [0, 1] to which the length of any subinterval (a, b) is assigned as the probability of that subinterval. The probability assigned to any one-member subset is 0.

Representation and interpretation of probability values

The value 0 is generally understood to represent impossible events, while the number 1 is understood to represent certain events (though there are more advanced interpretations of probability that use more precise definitions). Values between 0 and 1 quantify the probability of the occurrence of some event. In common language, these numbers are often expressed as fractions or percentages, and must be converted to real number form to perform calculations with them.

For example, if two events are equally likely, such as a flipped coin landing heads-up or tails-up, we express the probability of each event as "1 in 2" or "50%" or "1/2", where the numerator of the fraction is the relative likelihood of the target event and the denominator is the total of relative likelihoods for all events. To use the probability in math we must perform the division and express it as "0.5".

Another way probabilities are expressed is "odds", where the two numbers used represent the relative likelihood of the target event and the likelihood of all events other than the target event. Expressed as odds, tossing a coin will give heads odds of "1 to 1"or "1:1". To convert odds to probability, use the sum of the numbers given as the denominator of a fraction: "1:1" odds make a "1/2" probability; "3:2" odds make a "3/5" probability (or 0.6).

Distributions

The histogram of events versus occurrence is called a probability distribution. There are several important, discrete distributions, such as the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution and the Maxwell-Boltzmann distribution.

Remarks on probability calculations

The difficulty of probability calculations lie in determining the number of possible events, counting the occurrences of each event, counting the total number of possible events. Especially difficult is drawing meaningful conclusions from the probabilities calculated. An amusing probability riddle, the Monty Hall problem demonstrates the pitfalls nicely.

To learn more about the basics of probability theory, see the article on probability axioms and the article on Bayes' theorem that explains the use of conditional probabilities in case where the occurrence of two events is related.

Applications of probability theory to everday life

A major impact of probability theory on everyday life is in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable impact on the population as a whole, statistically. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the impact of such choices, which makes probability measures a political matter.

A good example is the impact of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the impact of such groupthink on pricing, on policy, and on peace and conflict.

It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound impact on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.

See also

External links

Quotations

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Crosswords: Gaussian

Specialty definitions using "Gaussian": difference of gaussians ♦ Gaussian beam, Gaussian constant, Gaussian filter, Gaussian gravitation constant, Gaussian noise, Gaussian process, Gaussian pulse, GDBPSK ♦ random error, random noise, random vibration ♦ Wald distribution. (references)

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Commercial Usage: Gaussian

DomainTitle

Books

  • Asymptotic Methods in the Theory of Gaussian Processes and Fields (Translations of Mathematical Monographs, Vol 148) (reference)

  • Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian (reference)

  • Gaussian 94 Users Reference (reference)

  • Gaussian Self-Affinity and Fractals (reference)

  • Quasioptical Systems: Gaussian Beam Quasioptical Propagation and Applications (IEEE Press/Chapman & Hall Publishers Series on Microwave Technology an (reference)

    (more book examples)

Source: compiled by the editor from various references; see credits.

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Digital Photo Gallery: Gaussian
 

"New Street Station Lights" by Philip Jackson
Commentary: "Lights in Birmingham new street station. Out of focus. fair enough you can gaussian blur stuff but i like to do it properly sometimes. hope its approved..."

Source: photographs selected by the editor, with permission from the photographers.

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Usage Frequency: Gaussian

"Gaussian" is generally used as an adjective (general or positive) -- approximately 95.06% of the time. "Gaussian" is used about 81 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted)
Parts of SpeechPercentUsage per
100 Million Words		Rank in English
Adjective (general or positive)95.06%7737,929
Noun (singular)4.94%4175,879
                    Total100.00%81N/A

Source: compiled by the editor from several corpora; see credits.

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Expression: Gaussian

Expressions using "Gaussian": Gaussian beam ♦ gaussian curve ♦ gaussian distribution ♦ Gaussian filter ♦ Gaussian law ♦ Gaussian noise ♦ Gaussian process ♦ Gaussian pulse ♦ gaussian shape ♦ Gaussian taper ♦ normal Gaussian distribution. Additional references.

Hyphenated Usage

Beginning with "Gaussian": Gaussian-pulse, Gaussian-pulse.

Ending with "Gaussian": non-gaussian.

Source: compiled by the editor from various references; see credits.

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Frequency of Internet Keywords: Gaussian

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.
 
ExpressionFrequency
per Day		ExpressionFrequency
per Day

gaussian

42

curvature gaussian

3

gaussian distribution

25

beam gaussian

3

gaussian elimination

15

gaussian blur

3

gaussian curve

8

gaussian curve fitting

3

gaussian keying minimum shift

8

aperture averaging beam gaussian

2

gaussian function

6

gaussian pulse

2

gaussian generate pick

6

distribution gaussian mean

2

gaussian mixture model

5

algebra elimination gaussian linear matrix strang

2

beam gaussian laser

4

gaussian matlab

2

gaussian filter

4

gaussian noise white

2

98 gaussian

4

fourier gaussian transform

2

gaussian quadrature

4

algorithm elimination gaussian

2

gaussian mixture model speaker

4

equation gaussian

2

basis gaussian set

4

gaussian laplacian

2

beam beam beam diagnostics distribution electron gaussian gaussian imaging

4

definite gaussian integral

2

gaussian noise

3
Source: compiled by the editor from various references; see credits.

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Modern Translation: Gaussian

Language Translations for "Gaussian"; alternative meanings/domain in parentheses.

Chinese 

  

高斯 (Gauss). (various references)

   

Danish

  

gaussisk værdi (gaussian value), Gauss iterationsmetode (Gaussian iterative solution, Gauss-Seidel iterative solution), Gauss-beam (Gaussian beam), Gauss-fading (Gaussian fading, Gaussian filter), gaussfiltreret minimalforskydningstastning (Gaussian filtered minimum shift keying, GMSK), Gauss-fordeling (Gaussian process), Gauss-fordelt hvid stoej (Gaussian white noise), Gauss-formet broend (gaussian well), Gauss-interferens (Gaussian interference), Gaussisk krumningsradius (Gaussian radius, mean radius of curvature), Gaussisk lov (Gaussian law, Gauss'law, normal distribution law), Gauss algoritme (Gaussian algorithm), gaussisk stråle (Gaussian beam), Gaussiske koordinater (Gaussian coordinates), Gauss-kurve (Gaussian curve), Gauss-kurveform (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), Gauss'lov (Gaussian law, Gauss'law, normal distribution law), Gauss-Seidels iterationsmetode (Gaussian iterative solution, Gauss-Seidel iterative solution), Gauss-spektrum (Gaussian spectrum), Gauss-støjeffekt (Gaussian noise power), Gauss-stoej (Gaussian noise), gaussisk puls (Gaussian pulse), transistorens Gaussiske impulsledning (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), normalfordeling (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, normality, second law of Laplace), normalfordelingen (Gaussian law, Gauss'law, normal distribution law), normalfordelingslov (Gaussian law, Gauss'law, normal distribution law). (various references)

   

Dutch

  

Gaussput (gaussian well), Gaussbundel (Gaussian beam), Gauss-distributie (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), Gausse impulsbelasting van een transistor (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), Gauss-fading (Gaussian fading, Gaussian filter), Gaussisch gefilterde minimum shift keying (Gaussian filtered minimum shift keying, GMSK), Gaussisch ruisvermogen (Gaussian noise power), Gaussisch verdeelde witte ruis (Gaussian white noise), Gaussische interferentie (Gaussian interference), Gaussische ruis (Gaussian noise), Gaussische vorm (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), Gauss-Kromme (Gaussian curve), Gausskuil (gaussian well), Gausspuls (Gaussian pulse), Gaussstraal (Gaussian beam), Gauss-verdeling (Gaussian law, Gauss'law, normal distribution law), Gauss-waarde (gaussian value), geruis van Gauss (Gaussian noise), GMSK (Gaussian filtered minimum shift keying, GMSK), Gauss-spectrum (Gaussian spectrum), gauss-proces (Gaussian process), kansverdeling van Gauss-Laplace (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), bundel van Gauss (Gaussian beam), verdeling van Gauss-Laplace (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), relaxatiemethode volgens Gauss Seidel (Gaussian iterative solution, Gauss-Seidel iterative solution), relaxatiemethode volgens Gauss (Gaussian iterative solution, Gauss-Seidel iterative solution), normale verdeling (Gaussian distribution, Gaussian law, Gauss-Laplace distribution, Gauss'law, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal distribution law, normal Gaussian distribution, second law of Laplace), normale distributie (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), kromme van Gauss (Gaussian curve), algoritmes van Gauss (Gaussian algorithm), wet van Gauss (Gaussian law, Gauss'law, normal distribution law), normaalverdeling (Gaussian law, Gauss'law, normal distribution law). (various references)

   

Finnish

  

Gauss-Seidelin menetelmään perustuva ratkaisun iterointi (Gaussian iterative solution, Gauss-Seidel iterative solution), Gaussin algoritmi (Gaussian algorithm), Gaussin jakauma (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), Gaussin käyrä (Gaussian curve), Gaussin koordinaatti (Gaussian coordinates), Gaussin laki (Gaussian law, Gauss'law, normal distribution law), Gaussin menetelmään perustuva ratkaisun iterointi (Gaussian iterative solution, Gauss-Seidel iterative solution), gaussikohina (Gaussian noise), gaussinen säde (Gaussian beam), Gauss-suodatin (Gaussian filter), Gaussin spektri (Gaussian spectrum), valkoinen gaussikohina (Gaussian white noise), kellokäyrä (bell-shaped curve, Gaussian curve), keskikaarevuussäde (Gaussian radius, mean radius of curvature), normaalijakauma (Gaussian distribution, Gaussian process, Gaussian taper, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, normal probability density distribution, second law of Laplace), normaalijakautunut arvo (gaussian value), satunnaiskohina (Gaussian noise). (various references)

   

French

  

loi de distribution normale (Gaussian law, Gauss'law), algorithme de Gauss (Gaussian algorithm), brouillage du type gaussien (Gaussian interference), bruit blanc à répartition gaussienne (Gaussian white noise), bruit gaussien (Gaussian noise), coordonnées de Gauss (Gaussian coordinates), courbe de fréquence de la loi normale (Gaussian distribution, Gauss-Laplace distribution, normal Gaussian distribution), courbe gaussienne (Gaussian curve), distribution de Gauss (Gauss Distribution, Gaussian distribution, Gaussian taper, Gauss-Laplace distribution, normal Gaussian distribution), distribution de Laplace-Gauss (Gaussian distribution, Gauss-Laplace distribution, normal Gaussian distribution), distribution gaussienne (Gaussian distribution, Gaussian taper, Gauss-Laplace distribution, normal Gaussian distribution), distribution normale (Gaussian distribution, Gauss-Laplace distribution, normal Gaussian distribution), données en puissance du transistor en impulsion de Gauss (Gaussian-pulse power, transistor Gaussian-pulse power rating), faisceau gaussien (Gaussian beam), évanouissement gaussien (Gaussian fading), MDMG (Gaussian filtered Minimum Shift Keying), spectre gaussien (Gaussian spectrum), rayon de courbure de Gauss (Gaussian radius), puits de potentiel de forme gaussienne (gaussian well), puits de Gauss (gaussian well), puissance sur bruit gaussien (Gaussian noise power), filtre gaussien (Gaussian filter), modulation à déplacement minimal à filtre gaussien (Gaussian filtered Minimum Shift Keying), impulsion gaussienne (Gaussian pulse), méthode de relaxation de Gauss Seidel (Gaussian iterative solution, Gauss-Seidel iterative solution), méthode de relaxation de Gauss (Gaussian iterative solution, Gauss-Seidel iterative solution), loi normale (Gaussian distribution, Gauss-Laplace distribution, normal Gaussian distribution), loi de Laplace-Gauss (Gaussian distribution, Gauss-Laplace distribution, normal Gaussian distribution), loi de Gauss (Gaussian law, Gauss'law), valeur gaussienne (gaussian value), processus gaussien (Gaussian process). (various references)

   

German

  

Gaussscher Schwund (Gaussian fading, Gaussian filter), Gaußsche Verteilung (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), Gaußscher Algorithmus (Gaussian algorithm), Gauß'scher Impuls (Gaussian pulse), Gaußscher Krümmungshalbmesser (Gaussian radius, mean radius of curvature), Gaußscher Strahl (Gaussian beam), Gauß'scher Strahl (Gaussian beam), Gauß-Verteilung (Gaussian process), Gauß-Wert (gaussian value), gaussisches Rauschen (Gaussian noise), Gausssche Impulsleistung eines Transistors (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), Gausssche Interferenz (Gaussian interference), Gausssche Kurve (Gaussian curve), Gauss'sche Normalverteilung (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), Gaußsche Koordinaten (Gaussian coordinates), Gauss-Verteilung (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), gausssche Potentialmulde (gaussian well), Gesetz von Gauss (Gaussian law, Gauss'law, normal distribution law), Gausssche Verteilung (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), Gaussverteilung (Gaussian distribution, normal distribution), Gausssches weißes Rauschen (Gaussian white noise), Gausssches Spektrum (Gaussian spectrum), Gausssches Rauschen (Gaussian noise power), Gauss'sches Fehlergesetz (Gaussian law, Gauss'law, normal distribution law), Glockenkurve (bell-shaped curve), Relaxationsverfahren nach Gauss-Seidel (Gaussian iterative solution, Gauss-Seidel iterative solution), normale Verteilung (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), normales Verteilungsgesetz (Gaussian law, Gauss'law, normal distribution law), Normalverteilung (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), Normalverteilung-Gesetz (Gaussian law, Gauss'law, normal distribution law), Relaxationsverfahren nach Gauss (Gaussian iterative solution, Gauss-Seidel iterative solution). (various references)

   

Greek 

  

Gaussian παλμός (Gaussian pulse), Gaussian δέσμη (Gaussian beam), νόμος του GAUSS (Gaussian law, Gauss'law, normal distribution law), διαλείψεις Γκάους (Gaussian fading, Gaussian filter), ονομαστική ισχύ λειτουργίας τρανζίστορ σε παλμό του Gauss (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), επαναληπτική μέθοδος του GAUSS (Gaussian iterative solution, Gauss-Seidel iterative solution), επαναληπτική μέθοδος των GAUSS-SEIDEL (Gaussian iterative solution, Gauss-Seidel iterative solution), φάσμα κατά Gauss (Gaussian spectrum), φάσμα κατά Γκάους (Gaussian spectrum), μαντάλωμα ελάχιστης μετατόπισης φιλτραρισμένο κατά Γκάους (Gaussian filtered minimum shift keying, GMSK), κανονική κατανομή (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), νόμος της κανονικής κατανομής (Gaussian law, Gauss'law, normal distribution law), νόμος κανονικής κατανομής (Gaussian law, Gauss'law, normal distribution law), ισχύς θορύβου Γκάους (Gaussian noise power), γκαουσιανή διαδικασία (Gaussian process), γκαουσιανή τιμή (gaussian value), κατανομή GAUSS (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), κατανομή κατά Gauss (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), κατανομή κατά Γκάους (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), παρεμβολή Γκάους (Gaussian interference). (various references)

   

Hungarian

  

normális eloszlási görbe (gaussian curve). (various references)

   

Italian

  

GMSK (Gaussian filtered minimum shift keying, GMSK), limite di funzionamento di potenza a impulso gaussiano del transistore (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), curva di gauss (Gaussian curve), curva gaussiana (Gaussian curve), distribuzione gaussiana (difference of gaussians, Gaussian distribution, Gaussian law, Gauss-Laplace distribution, Gauss'law, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal distribution law, normal Gaussian distribution, second law of Laplace), distribuzione normale (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), evanescenza gaussiana (Gaussian fading, Gaussian filter), fascio gaussiano (Gaussian beam), impulso gaussiano (Gaussian pulse), interferenza gaussiana (Gaussian interference), buca gaussiana (gaussian well), legge di Gauss (Gaussian law, Gauss'law, normal distribution law), variazione ad andamento gaussiano (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), metodo di rilassamento di Gauss (Gaussian iterative solution, Gauss-Seidel iterative solution), metodo di rilassamento di Gauss-Seidel (Gaussian iterative solution, Gauss-Seidel iterative solution), modulazione per diversità minima a filtro gaussiano (Gaussian filtered minimum shift keying, GMSK), potenza del rumore gaussiano (Gaussian noise power), pozzo di Gauss (gaussian well), processo gaussiano (Gaussian process), rumore bianco gaussiano (Gaussian white noise), rumore gaussiano bianco (Gaussian white noise), spettro gaussiano (Gaussian spectrum), legge di distribuzione normale (Gaussian law, Gauss'law, normal distribution law). (various references)

   

Japanese Kanji 

  

ガーター編み (African daisy, Gacrux, garbage, garbology, garden, garden party, garden smoker, garlic, garnet, garter stitch, gaucho, gaucho hat, gaucho look, gaucho pants, gauss, Geiger counter, gerbera, girder bridge, girdle, girl, girl friend, girl hunt, Girl Scouts, gown, guard, guard bunker, guard cable, guardian, guardrail, guidance, guide number, guidebook, guideline, guidepost, guile, guy, security guard, spirit, tour guide). (various references)

   

Japanese Katakana 

  

ガウシアン . (various references)

   

Korean 

  

가우스. (various references)

   

Pig Latin

  

aussiangay.(various references)

   

Portuguese

  

lei de distribuição normal (Gaussian law, Gauss'law, normal distribution law), coordenadas de Gauss (Gaussian coordinates), curva de Gauss (Gaussian curve), desvanecimento gaussiano (Gaussian fading, Gaussian filter), distribuição de Gauss (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), distribuição gaussiana (difference of gaussians, Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), distribuição normal (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), distribuição normal de Laplace-Gauss (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), espectro gaussiano (Gaussian spectrum), feixe gaussiano (Gaussian beam), impulso gaussinao (Gaussian pulse), algoritmo de Gauss (Gaussian algorithm), lei da distribuição normal (Gaussian law, Gauss'law, normal distribution law), valor gaussiano (gaussian value), lei de Gauss (Gaussian law, Gauss'law, normal distribution law), metodo de relaxacao de Gauss (Gaussian iterative solution, Gauss-Seidel iterative solution), metodo de relaxacao de Gauss Seidel (Gaussian iterative solution, Gauss-Seidel iterative solution), modulação de deslocamento mínimo e filtro gaussiano (Gaussian filtered minimum shift keying, GMSK), potência de ruído gaussiano (Gaussian noise power), potencia nominal dum transistor para impulsos gaussianos (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), processo gausseano (Gaussian process), raio de curvatura de Gauss (Gaussian radius, mean radius of curvature), ruído branco gaussiano (Gaussian white noise), ruído gaussiano (Gaussian noise), interferência gaussiana (Gaussian interference). (various references)

   

Spanish

  

ley de la distribución normal (Gaussian law, Gauss'law, normal distribution law), clasificación de potencia del transistor ante impulsos gaussianos (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), coordenadas de Gauss (Gaussian coordinates), curva de Gauss (Gaussian curve), desvanecimiento gaussiano (Gaussian fading, Gaussian filter), distribución de Gauss (difference of gaussians, Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), distribución en campana de Gauss (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), distribución gaussiana (Gaussian distribution, normal distribution), distribución normal (difference of gaussians, Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), espectro gaussiano (Gaussian spectrum), haz de Gauss (Gaussian beam), impulso de Gauss (Gaussian pulse), algoritmo de Gauss (Gaussian algorithm), ley de Gauss (Gaussian law, Gauss'law, normal distribution law), variación gradual de tipo gaussiano (Gaussian distribution, Gaussian taper, normal distribution, normal probability density distribution), método de relajación de Gauss (Gaussian iterative solution, Gauss-Seidel iterative solution), método de relajación de Gauss Seidel (Gaussian iterative solution, Gauss-Seidel iterative solution), MDMG (Gaussian filtered Minimum Shift Keying, GMSK), modulación de desplazamiento mínimo con filtro gaussiano (Gaussian filtered Minimum Shift Keying, GMSK), potencia ante impulsos gaussianos (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), potencia de ruido gaussiano (Gaussian noise power), proceso gaussiano (Gaussian process), radio de Gauss (Gaussian radius, mean radius of curvature), radio medio de curvatura (Gaussian radius, mean radius of curvature), ruido blanco-gaussiano (Gaussian white noise), ruido de Gauss (Gaussian noise), valor gausiano (gaussian value), interferencia gaussiana (Gaussian interference). (various references)

   

Swedish

  

gausstråle (Gaussian beam), gausspuls (Gaussian pulse), Gauss-process (Gaussian process), gaussiskt värde (gaussian value), Gaussiska koordinater (Gaussian coordinates), Gaussformig effektpuls från transistor (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating), Gauss-formad brunn (gaussian well), Gauss metod för lösning av normalekvationer (Gaussian algorithm), Gauss krökningsradie (Gaussian radius, mean radius of curvature), normalfördelning (Gaussian distribution, Gauss-Laplace distribution, Laplacean distribution, Laplace-Gauss distribution, normal distribution, normal Gaussian distribution, second law of Laplace), analog effektpuls (analog-pulse power, analogue-pulse power, Gaussian-pulse power, transistor Gaussian-pulse power rating). (various references)

Source: compiled by the editor from various translation references.

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Anagrams: Gaussian

Scrabble® Enable2K-Verified Anagrams

Words within the letters "a-a-g-i-n-s-s-u"

-1 letter: iguanas.

-2 letters: assign, iguana, saigas, sangas, saunas.

-3 letters: again, angas, assai, gains, gauss, guans, nisus, sagas, saiga, sains, sanga, sasin, sauna, signs, sings, sinus, snags, snugs, suing, unais, using.

-4 letters: agas, agin, ains, anas, anga, anis, ansa, anus, gain, gaun, gins, gnus, guan, guns, nags, saga, sags, sain, sang, sans, sign, sing, sins, snag.

 Words containing the letters "a-a-g-i-n-s-s-u"
 

+1 letter: assuaging.

 

+2 letters: assaulting.

 

+3 letters: anisogamous, salmagundis.

 

+4 letters: glutaminases, sanguinarias, vanguardisms, vanguardists.

 

+5 letters: exsanguinates, gastrulations, guardianships, sagaciousness.

Source: compiled by the editor from various references; see credits.

SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

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INDEX

1. Definition
2. Crosswords
3. Usage: Commercial
4. Images: Digital Art 		5. Usage Frequency
6. Expressions
7. Expressions: Internet
8. Translations: Modern 		9. Anagrams
10. Bibliography


  

Copyright © Philip M. Parker, INSEAD. Terms of Use.