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# NoncentralChiSquareDistribution

 NoncentralChiSquareDistribution represents a noncentral distribution with degrees of freedom and noncentrality parameter .
• The probability density for value is proportional to for and zero otherwise.
Probability density function:
Cumulative distribution function:
Mean and variance:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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 Scope   (7)
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:
Closed form for symbolic order:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (1)
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 :
Find the mean:
Express the moment generating function in terms of the mean:
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 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: