# MaxwellDistribution

represents a Maxwell distribution with scale parameter σ.

# Background & Context

• represents a continuous statistical distribution supported over the interval and parametrized by a positive real number σ (called a "scale parameter") that determines the overall behavior of its probability density function (PDF). In general, the PDF of a Maxwell distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (height, spread, and horizontal location of its maximum) is determined by the values of σ. In addition, the PDF of the Maxwell distribution has tails that are "thin" in the sense that its PDF decreases exponentially rather than algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Maxwell distribution is also sometimes referred to as the Maxwell-Boltzmann distribution and as the Maxwell speed distribution.
• The Maxwell distribution was first described in the 1860s by Scottish physicist James Clark Maxwell. It became an indispensable model in statistical mechanics following later investigations by Austrian physicist Ludwig Boltzmann. The Maxwell distribution describes the speeds of particles in ideal gases under the assumption that the particles have reached thermodynamic equilibrium and have minimal interaction with one another. As such, the distribution is considered the foundation of the kinetic theory of gases and is a tool in the related field of Maxwell-Boltzmann statistics that attempts to describe the distribution of non-interacting particles on a more general level. The distribution has also been used to describe phenomena in various fields including chemistry, reliability and risk analysis, signal processing, and Bayesian analysis.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Maxwell distribution. Distributed[x,MaxwellDistribution[σ]], written more concisely as xMaxwellDistribution[σ], can be used to assert that a random variable x is distributed according to a Maxwell distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for Maxwell distributions may be given using PDF[MaxwellDistribution[σ],x] and CDF[MaxwellDistribution[σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a Maxwell distribution, EstimatedDistribution to estimate a Maxwell parametric distribution from given data, and FindDistributionParameters to fit data to a Maxwell distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Maxwell distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Maxwell distribution.
• TransformedDistribution can be used to represent a transformed Maxwell distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Maxwell distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Maxwell distributions.
• MaxwellDistribution is related to a number of other distributions. It is a special case of ChiDistribution (the PDF of is precisely the same as that of ), ChiSquareDistribution (CDF[MaxwellDistribution[1],Sqrt[x]] is identically CDF[ChiSquareDistribution[3],x]), and GammaDistribution (the PDF of and GammaDistribution[3/2,Sqrt[2] σ,2,0] are the same). The exponential decay behavior of MaxwellDistribution makes it qualitatively similar to NormalDistribution, RayleighDistribution, BetaDistribution, and ExponentialDistribution. By way of its relationship to ChiDistribution and ChiSquareDistribution, MaxwellDistribution is also related to NakagamiDistribution, NoncentralChiSquareDistribution, and HalfNormalDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Median:

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