NoncentralChiSquareDistribution

Generate a set of pseudorandom numbers that have a noncentral distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness varies with degrees of freedom and noncentrality :

Kurtosis varies with degrees of freedom and noncentrality :

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Closed form for symbolic order:

Hazard function:

Quantile function:

In the theory of fading channels, scaled NoncentralChiSquareDistribution is the distribution of instantaneous signal-to-noise ratio when signal fading amplitude is modeled by RiceDistribution. 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 for :

Show is a scaled NoncentralChiSquareDistribution:

Find the mean:

Express the moment generating function in terms of the mean:

Find the amount of fading:

Limiting value:

Parameter influence on the CDF for each :

CDF of NoncentralChiSquareDistribution admits closed-form approximations:

Compare the CDF to its approximation:

Relationships to other distributions:

Noncentral distribution simplifies to ChiSquareDistribution:

Sum of squares of variables with NormalDistribution has NoncentralChiSquareDistribution:

Noncentral distribution is related to BeckmannDistribution:

Noncentral distribution can be obtained from RiceDistribution:

Quotient of two variables from noncentral distribution follows NoncentralFRatioDistribution:

NoncentralChiSquareDistribution is a parameter mixture of ChiSquareDistribution and PoissonDistribution:

NoncentralBetaDistribution can be obtained as a transformation of ChiSquareDistribution and NoncentralChiSquareDistribution:

NoncentralChiSquareDistribution is not defined when is not a positive real number:

NoncentralChiSquareDistribution is not defined when is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: