# HoytDistribution

HoytDistribution[q,ω]

represents a Hoyt distribution with shape parameter q and spread parameter ω.

# Details • HoytDistribution is also known as Nakagami- distribution.
• The probability density for value is proportional to for , and is zero for .
• HoytDistribution allows q to be any number between 0 and 1, and ω to be any positive real number.
• HoytDistribution allows ω to be a quantity of any unit dimension, and q to be a dimensionless quantity. »
• HoytDistribution can be used in such functions as Mean, CDF, and RandomVariate.

# Background & Context

• HoytDistribution[q,ω] represents a continuous statistical distribution defined on the interval . The distribution is parametrized by positive real numbers q ( ) and ω, called a "shape parameter" and a "spread parameter," respectively. The probability density function (PDF) of a Hoyt distribution is unimodal, with the parameters q and ω determining the overall height, steepness, and concavity of the PDF. In addition, the tails of the PDF of a Hoyt distribution are "thin," in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Hoyt distribution is sometimes referred to as the Nakagami-q distribution or NakagamiHoyt distribution, though it should not be confused with NakagamiDistribution.
• The Hoyt distribution was independently studied by R. S. Hoyt in the late 1940s and M. Nakagami in the early 1960s. While Hoyt's original work was more theoretical and focused on probability functions related to complex variates, Nakagami devised the distribution to assist in modeling the intensity of radio waves amid extreme fading. Since the work of Nakagami, applications of the Hoyt distribution have included the modeling of wireless cellular and radio signal propagation in the presence of fading.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Hoyt distribution. Distributed[x,HoytDistribution[q,ω]], written more concisely as xHoytDistribution[q,ω], can be used to assert that a random variable x is distributed according to a Hoyt distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[HoytDistribution[q,ω],x] and CDF[HoytDistribution[q,ω],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 Hoyt distribution, EstimatedDistribution to estimate a Hoyt parametric distribution from given data, and FindDistributionParameters to fit data to a Hoyt distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Hoyt distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Hoyt distribution.
• TransformedDistribution can be used to represent a transformed Hoyt 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 Hoyt distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Hoyt distributions.
• HoytDistribution is closely related to a number of other distributions. For example, the work of Hoyt establishes a relationship between HoytDistribution and NormalDistribution by way of complex normal variates. Moreover, HoytDistribution is linked to both GammaDistribution and ChiDistribution, in the sense that if YGammaDistribution[m,ω/m] and ZChiDistribution[2 m], then and satisfy W,XHoytDistribution[m,ω]. HoytDistribution approaches both HalfNormalDistribution and RayleighDistribution, in the sense that the PDF of HoytDistribution[q,ω] is precisely that of both HalfNormalDistribution[ ] as q tends to 0 and RayleighDistribution[ ] as q tends to 1. HoytDistribution is also related to NakagamiDistribution, ExponentialDistribution, ArcSinDistribution, BinormalDistribution, and BeckmannDistribution.

# Examples

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## Basic Examples(3)

Probability density function:

Cumulative distribution function:

Mean and variance:

## Scope(8)

Generate a sample of pseudorandom numbers from a Hoyt distribution:

Compare its 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 first parameter:

Limiting values:

Kurtosis depends only on the first parameter:

Limiting values:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median length:

## Applications(1)

In the theory of fading channels, HoytDistribution is used to model fading amplitude on satellite links in the presence of strong scintillation due to ionosphere. 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 moment-generating function (MGF):

Find the mean:

Express the MGF in terms of the mean:

## Properties & Relations(8)

Hoyt distribution family is closed under scaling by a positive factor:

Relationships to other distributions:

HoytDistribution reduces to HalfNormalDistribution as :

HoytDistribution reduces to RayleighDistribution[Sqrt[ω/2]] as :

NakagamiDistribution is related to Hoyt distribution:

Hoyt distribution can be obtained from ExponentialDistribution and ArcSinDistribution:

Hoyt distribution can be obtained from BinormalDistribution:

Hoyt distribution is a special case of BeckmannDistribution:

## Neat Examples(1)

PDFs for different q values with CDF contours:

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