

Normal (Gauss) Distribution 
Normal (or Gauss) distributions are availlable through the functions
normal_distribution or gauss_distribution which are equivalent.
The functions return the Math::Symbolic representation of a
gauss distribution.
The gauss distribution has three parameters: The mean mu, the root mean square deviation sigma and the variable x. The functions take two optional arguments: The Math::Symbolic trees (or strings) to be plugged into the formula for 1) mu and 2) sigma. If any argument is undefined or omitted, the corresponding variable will remain unchanged. The variable x always remains in the formula. Please refer to the literature referenced in the SEE ALSO section for details. 
Bivariate Normal Distribution 
Bivariate normal distributions are availlable through the function
bivariate_normal_distribution.
The function returns the Math::Symbolic representation of a
bivariate normal distribution.
The bivariate normal distribution has seven parameters: The mean mu1 of the first variable, the root mean square deviation sigma1 of the first variable, the mean mu2 of the second variable, the root mean square deviation sigma2 of the second variable, the first variable x1, the second variable x2, and the correlation of the first and second variables, sigma12. The function takes five optional arguments: The Math::Symbolic trees (or strings) to be plugged into the formula for 1) mu1, 2) sigma1, 3) mu1, 4) sigma1, and 5) sigma12. If any argument is undefined or omitted, the corresponding variable will remain unchanged. The variables x1 and x2 always remain in the formula. Please refer to the literature referenced in the SEE ALSO section for details. 
Cauchy Distribution 
Cauchy distributions are availlable through the function
cauchy_distribution.
The function returns the Math::Symbolic representation of a
cauchy distribution.
The cauchy distribution has three parameters: The median m, the full width at half maximum lambda of the curve, and the variable x. The function takes two optional arguments: The Math::Symbolic trees (or strings) to be plugged into the formula for 1) m and 2) lambda. If any argument is undefined or omitted, the corresponding variable will remain unchanged. The variable x always remains in the formula. Please refer to the literature referenced in the SEE ALSO section for details. 
Boltzmann Distribution 
Boltzmann distributions are availlable through the function
boltzmann_distribution.
The function returns the Math::Symbolic representation of a
Boltzmann distribution.
The Boltzmann distribution has four parameters: The energy E, the weighting factor gs that describes the number of states at energy E, the temperature T, and the chemical potential mu. The function takes fouroptional arguments: The Math::Symbolic trees (or strings) to be plugged into the formula for 1) E, 2) gs, 3) T, and 4) mu If any argument is undefined or omitted, the corresponding variable will remain unchanged. The formula used is: N = gs * e^((Emu)/(k_B*T)). Please refer to the literature referenced in the SEE ALSO section for details. Boltzmann’s constant k_B is used as 1.3807 * 10^23 J/K. 
Fermi Distribution 
Fermi distributions are availlable through the function
fermi_distribution.
The function returns the Math::Symbolic representation of a
Fermi distribution.
The Fermi distribution has four parameters: The energy E, the weighting factor gs that describes the number of states at energy E, the temperature T, and the chemical potential mu. The function takes fouroptional arguments: The Math::Symbolic trees (or strings) to be plugged into the formula for 1) E, 2) gs, 3) T, and 4) mu If any argument is undefined or omitted, the corresponding variable will remain unchanged. The formula used is: N = gs / ( e^((Emu)/(k_B*T)) + 1). Please refer to the literature referenced in the SEE ALSO section for details. Boltzmann’s constant k_B is used as 1.3807 * 10^23 J/K. 
Have a look at Math::Symbolic, Math::Symbolic::Parser, Math::SymbolicX::ParserExtensionFactory and all associated modules.New versions of this module can be found on http://steffenmueller.net or CPAN.
Details on several distributions implemented in the code can be found on the MathWorld site:
Eric W. Weisstein. Normal Distribution. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/NormalDistribution.html
Eric W. Weisstein. Bivariate Normal Distribution. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/BivariateNormalDistribution.html
Eric W. Weisstein. Cauchy Distribution. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
The Boltzmann, Bose, and Fermi distributions are discussed in detail in N.W. Ashcroft, N.D. Mermin. Solid State Physics. Brooks/Cole, 1976
Steffen Mueller, <symbolicmodule at steffenmueller dot net>
Copyright (C) 2005, 2006 by Steffen MuellerThis library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.6.0 or, at your option, any later version of Perl 5 you may have available.
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