# SuzukiDistribution

SuzukiDistribution[μ,ν]

represents the Suzuki distribution with shape parameters μ and ν.

# Details # Background & Context

• SuzukiDistribution[μ,ν] represents a continuous statistical distribution supported on the interval and parametrized by a real number μ and by a positive real number ν (both called "shape parameters"), which together determine the overall behavior of its probability density function (PDF). Depending on the values of μ and ν, the PDF of a Suzuki distribution may have any of a number of shapes, including unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with potential singularities approaching the lower boundary of its domain. In addition, the tails of the PDF may be "fat" (i.e. the PDF decreases non-exponentially for large values ) or "thin" (i.e. the PDF decreases exponentially for large ), depending on the values of μ and ν. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
• The Suzuki distribution was first proposed in the late 1970s by Hirofumi Suzuki as a model for urban radio propagation. Since its inception, the distribution has become a staple for modeling various phenomena related to wireless communications, though it has also proven useful in areas such as macroeconomics.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Suzuki distribution. Distributed[x,SuzukiDistribution[μ,ν]], written more concisely as xSuzukiDistribution[μ,ν], can be used to assert that a random variable x is distributed according to a Suzuki 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 Suzuki distributions may be given using PDF[SuzukiDistribution[μ,ν],x] and CDF[SuzukiDistribution[μ,ν],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 Suzuki distribution, EstimatedDistribution to estimate a Suzuki parametric distribution from given data, and FindDistributionParameters to fit data to a Suzuki distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Suzuki distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Suzuki distribution.
• TransformedDistribution can be used to represent a transformed Suzuki 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 Suzuki distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Suzuki distributions.
• SuzukiDistribution is related to a number of other distributions. By definition, SuzukiDistribution is a parameter mixture (ParameterMixtureDistribution) of RayleighDistribution and LogNormalDistribution, in the sense that SuzukiDistribution[μ,ν] is equivalent to provided that σLogNormalDistribution[μ,ν]. By way of these relationships, SuzukiDistribution is also related to NormalDistribution, BinormalDistribution, LaplaceDistribution, KDistribution, StableDistribution, RiceDistribution, MaxwellDistribution, LevyDistribution, ChiDistribution, and ChiSquareDistribution.

# Examples

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## Basic Examples(3)

Probability density function:

Cumulative distribution function:

Mean and variance:

## Scope(7)

Generate a sample of pseudorandom numbers from a Suzuki distribution:

Compare the histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare a density histogram of the sample with the PDF of the estimated distribution:

Skewness depends only on the second parameter:

Skewness is an increasing function with limiting values:

Kurtosis depends only on the second parameter:

Kurtosis is an increasing function with limiting values:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function does not have closed form but can be evaluated numerically:

Quantile function:

## Applications(1)

In the theory of fading channels, SuzukiDistribution is used to model fading amplitude. Find the distribution of instantaneous signal-to-noise ratio where , is the energy per symbol, and is the spectral density of white noise:

Find the mean:

Limiting values:

## Properties & Relations(3)

Suzuki distribution is closed under scaling by a positive factor:

SuzukiDistribution is a parameter mixture of RayleighDistribution and LogNormalDistribution:

Suzuki distribution can be obtained from RayleighDistribution and LogNormalDistribution:

## Neat Examples(1)

PDFs for different μ values with CDF contours:

Introduced in 2010
(8.0)
|
Updated in 2016
(10.4)