26 Numerics library [numerics]

A normal_­distribution random number distribution produces random numbers x distributed according to the probability density function
$$p(x|\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}\cdot \exp (-\frac{(x-\mu {)}^2}{2\sigma {}^2})\text{.}$$

The distribution parameters μ and σ are also known as this distribution's mean and standard deviation.

A lognormal_­distribution random number distribution produces random numbers $x>0$ distributed according to the probability density function
$$p(x|m,s)=\frac{1}{sx\sqrt{2\pi }}\cdot \exp (-\frac{(\ln x-m{)}^2}{2s^2})\text{.}$$

A chi_­squared_­distribution random number distribution produces random numbers $x>0$ distributed according to the probability density function
$$p\left(x|n\right)=\frac{x^{\left(n/2\right)-1}\cdot e^{-x/2}}{\Gamma \left(n/2\right)\cdot 2^{n/2}}\text{.}$$

A cauchy_­distribution random number distribution produces random numbers x distributed according to the probability density function
$$p(x|a,b)={\left(\pi b(1+{\left(\frac{x-a}{b}\right)}^2)\right)}^{-1}\text{.}$$

A fisher_­f_­distribution random number distribution produces random numbers $x\ge 0$ distributed according to the probability density function
$$p(x|m,n)=\frac{\Gamma ((m+n)/2)}{\Gamma \left(m/2\right)\Gamma \left(n/2\right)}\cdot {\left(\frac{m}{n}\right)}^{m/2}\cdot x^{\left(m/2\right)-1}\cdot {(1+\frac{mx}{n})}^{-(m+n)/2}\text{.}$$

A student_­t_­distribution random number distribution produces random numbers x distributed according to the probability density function
$$p\left(x|n\right)=\frac{1}{\sqrt{n\pi }}\cdot \frac{\Gamma ((n+1)/2)}{\Gamma \left(n/2\right)}\cdot {(1+\frac{x^2}{n})}^{-(n+1)/2}\text{.}$$