Distributions Template Parameter Method

Method

Distributions

Math Description

uniform_method::standard uniform_method::accurate

uniform(s,d) uniform(i)

Standard method. uniform_method::standard_accurate supported for uniform(s, d) only.

gaussian_method::box_muller

gaussian

Generates normally distributed random number x thru the pair of uniformly distributed numbers \(u_1\) and \(u_2\) according to the formula: \(x = \sqrt{-2lnu_1}\sin(2 \pi u_2)\)

gaussian_method::box_muller2

gaussian lognormal

Generates normally distributed random numbers \(x_1\) and \(x_2\) thru the pair of uniformly distributed numbers \(u_1\) and \(u_2\) according to the formulas: \(x_1 = \sqrt{-2lnu_1}\sin{2\pi u_2}\) \(x_2 = \sqrt{-2lnu_1}\cos{2\pi u_2}\)

gaussian_method::icdf

gaussian

Inverse cumulative distribution function (ICDF) method.

exponential_method::icdf exponential_method::icdf_accurate

exponential

Inverse cumulative distribution function (ICDF) method.

weibull_method::icdf weibull_method::icdf_accurate

weibull

Inverse cumulative distribution function (ICDF) method.

cauchy_method::icdf

cauchy

Inverse cumulative distribution function (ICDF) method.

rayleigh_method::icdf rayleigh_method::icdf_accurate

rayleigh

Inverse cumulative distribution function (ICDF) method.

bernoulli_method::icdf

bernoulli

Inverse cumulative distribution function (ICDF) method.

geometric_method::icdf

geometric

Inverse cumulative distribution function (ICDF) method.

gumbel_method::icdf

gumbel

Inverse cumulative distribution function (ICDF) method.

lognormal_method::icdf lognormal_method::icdf_accurate

lognormal

Inverse cumulative distribution function (ICDF) method.

lognormal_method::box_muller2 lognormal_method::box_muller2_accurate

lognormal

Generated normally distributed random numbers \(x_1\) and \(x_2\) by box_muller2 method are converted to lognormal distribution.

gamma_method::marsaglia gamma_method::marsaglia_accurate

gamma

For \(\alpha > 1\), a gamma distributed random number is generated as a cube of properly scaled normal random number; for \(0.6 \leq \alpha < 1\), a gamma distributed random number is generated using rejection from Weibull distribution; for \(\alpha < 0.6\), a gamma distributed random number is obtained using transformation of exponential power distribution; for \(\alpha = 1\), gamma distribution is reduced to exponential distribution.

beta_method::cja beta_method::cja_accurate

beta

Cheng-Jonhnk-Atkinson method. For \(min(p, q) > 1\), Cheng method is used; for \(min(p, q) < 1\), Johnk method is used, if \(q + K*p2 + C \leq 0 (K = 0.852..., C=-0.956...)\) otherwise, Atkinson switching algorithm is used; for \(max(p, q) < 1\), method of Johnk is used; for \(min(p, q) < 1, max(p, q)> 1\), Atkinson switching algorithm is used (CJA stands for Cheng, Johnk, Atkinson); for \(p = 1\) or \(q = 1\), inverse cumulative distribution function method is used; for \(p = 1\) and \(q = 1\), beta distribution is reduced to uniform distribution.

chi_square_method::gamma_based

chi_square

(most common): If \(\nu \ge 17\) or \(\nu\) is odd and \(5 \leq \nu \leq 15\), a chi-square distribution is reduced to a Gamma distribution with these parameters: Shape \(\alpha = \nu / 2\)Offset \(a = 0\) Scale factor \(\beta = 2\) The random numbers of the Gamma distribution are generated.

binomial_method::btpe

binomial

Acceptance/rejection method for \(ntrial * min(p, 1 - p) \ge 30\) with decomposition into four regions: two parallelograms, triangle, left exponential tail, right exponential tail.

poisson_method::ptpe

poisson

Acceptance/rejection method for \(\lambda \ge 27\) with decomposition into four regions: two parallelograms, triangle, left exponential tail, right exponential tail.

poisson_method::gaussian_icdf_based

poisson

for \(\lambda \ge 1\), method based on Poisson inverse CDF approximation by Gaussian inverse CDF; for \(\lambda < 1\), table lookup method is used.

poisson_v_method::gaussian_icdf_based

poisson_v

for \(\lambda \ge 1\), method based on Poisson inverse CDF approximation by Gaussian inverse CDF; for \(\lambda < 1\), table lookup method is used.

hypergeometric_method::h2pe

hypergeometric

Acceptance/rejection method for large mode of distribution with decomposition into three regions: rectangular, left exponential tail, right exponential tail.

negative_binomial_method::nbar

negative_binomial

Acceptance/rejection method for: \(\frac{(a - 1)(1 - p)}{p} \ge 100\) with decomposition into five regions: rectangular, 2 trapezoid, left exponential tail, right exponential tail.

multinomial_method::poisson_icdf_based

multinomial

Multinomial distribution with parameters \(m, k\), and a probability vector \(p\). Random numbers of the multinomial distribution are generated by Poisson Approximation method.

gaussian_mv_method::box_muller

gaussian_mv

BoxMuller method for gaussian_mv method.

gaussian_mv_method::box_muller2

gaussian_mv

BoxMuller2 method for gaussian_mv method.

gaussian_mv_method::icdf

gaussian_mv

Inverse cumulative distribution function (ICDF) method.

Parent topic: Distributions

Distributions uniform (continuous)