Distributions & Tests

Random Variates from Statistical Distributions

Generation of pseudo-random numbers from over 20 continuous and discrete distributions using the Mulberry32 PRNG.

Statusvalidated

Version1.0.0

Minitab equivalentCalc > Random Data

Description

This module implements reproducible random number generation using the Mulberry32 PRNG (32-bit, deterministic). Continuous distributions are generated via inversion, Box-Muller, and Marsaglia & Tsang methods. The normal distribution uses the SinhArcSinh transform (Jones & Pewsey 2009), which enables a flexible family of asymmetric distributions via skewness (ε) and tailweight (δ) parameters.

Formulas

Mulberry32

s_{n + 1} = s_{n} + 0x6D2B79F5, U = \frac{t \oplus ( t ≫ 14 )}{2 ^{32}}

Deterministic PRNG: same seed always produces the same sequence of uniformly distributed U(0,1) values.

Box-Muller

Z_{1} = - 2 ln S \cdot \frac{U _{1}}{S}, Z_{2} = - 2 ln S \cdot \frac{U _{2}}{S}

Generation of standard normal pairs (Z₁, Z₂) from uniform U₁, U₂ with S = U₁² + U₂² < 1.

SinhArcSinh

X = μ + σ \cdot sinh (\frac{arcsinh ( Z ) + ε}{δ}), Z \sim N (0, 1)

Jones & Pewsey transform (2009): ε controls skewness, δ controls tailweight. With ε=0, δ=1 this reduces to the standard normal.

Gamma (Marsaglia–Tsang)

X = d \cdot v^{3} \cdot β, d = α - \frac{1}{3}, c = \frac{1}{9 d}

Marsaglia & Tsang method for gamma variates with shape α ≥ 1. For α < 1, uses the identity Gamma(α) = Gamma(α+1) · U^(1/α).

Chi-Quadrat

X \sim χ^{2} (k) = Gamma (\frac{k}{2}, 2)

Chi-square as special case of the Gamma distribution.

F-Verteilung

X = \frac{χ ^{2} ( d _{1} ) / d _{1}}{χ ^{2} ( d _{2} ) / d _{2}}

Ratio of two scaled chi-square variates.

t-Verteilung

X = \frac{Z}{χ ^{2} ( ν ) / ν}, Z \sim N (0, 1)

Ratio of standard normal to scaled chi-square root.

Beta

X = \frac{G _{1}}{G _{1} + G _{2}}, G_{1} \sim Gamma (α, 1), G_{2} \sim Gamma (β, 1)

Beta variate as ratio of two independent gamma variates.

Exponential (Inversion)

X = - \frac{ln ( 1 - U )}{λ}, U \sim U (0, 1)

Inversion method for the exponential distribution.

Weibull (Inversion)

X = β \cdot (- ln (1 - U))^{1/ α}

Inversion method for the Weibull distribution with shape α and scale β.

Lognormal

X = e^{μ + σ Z}, Z \sim N (0, 1)

Exponential transform of a normally distributed variable.

Cauchy (Inversion)

X = x_{0} + γ tan (π (U - 0, 5))

Inversion method. Mean and variance do not exist.

Laplace

X = μ - b \cdot sign (U - 0, 5) \cdot ln (1 - 2∣ U - 0, 5∣)

Inversion method for the Laplace distribution.

Gumbel Max

X = μ - β ln (- ln U)

Inversion method for the Gumbel maximum distribution (Type I extreme value).

Logistic (Inversion)

X = μ + s ln (\frac{U}{1 - U})

Inversion method via the logit function.

Dreieck

X = {a + U (b - a) (c - a) b - (1 - U) (b - a) (b - c) U < \frac{c - a}{b - a} sonst

Inversion method for the triangular distribution with minimum a, maximum b, and mode c.

Binomial

X = i = 1 \sum n 1 (U_{i} < p), U_{i} \sim U (0, 1)

Direct simulation as sum of n Bernoulli trials.

Poisson

X = min {k : i = 1 \prod k U_{i} \leq e^{- λ}} - 1

Knuth algorithm for λ < 30; normal approximation for λ ≥ 30.

Neg. Binomial

X \sim Poi (G), G \sim Gamma (r, \frac{1 - p}{p})

Gamma-Poisson mixture: first generate gamma variate, then Poisson variate with that mean.

Hypergeometrisch

P (X_{i} = 1) = \frac{K _{i}}{N _{i}}, sequentielles Ziehen ohne Zur \overset{u}{¨} cklegen

Sequential simulation: drawing without replacement from finite population.

Assumptions

PRNG is deterministic — same seed produces identical sequences
Mulberry32 has a period of 2³² — sufficient for samples up to ~10⁵
All distribution parameters must be within their valid range

Limitations

Mulberry32 is not a cryptographically secure generator
Binomial distribution with large n is computationally expensive (O(n) per value)
Cauchy distribution: empirical mean and variance are unstable

References

Jones, M. C. & Pewsey, A. (2009). Sinh-arcsinh distributions. Biometrika, 96(4), 761–780.
Marsaglia, G. & Tsang, W. W. (2000). A simple method for generating gamma variables. ACM TOMS, 26(3), 363–372.
Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. 3rd Ed. — Chapter 3.4