Uniform Random Variables

📐 Definition

Uniform random variables assign equal probability to any value within a closed interval, often used as the base source for transformed distributions.

For $X\sim U(a,b)$ , the density is

f(x) = \begin{cases} \frac{1}{b-a}, & a \le x \le b,\\ 0, & \text{otherwise} \end{cases}

Domain and Codomain

Inputs are random seeds or generator states; outputs lie within the interval ([a,b]) with constant density.

⚙️ Key Properties

\mathbb{E}[X] = \tfrac{a+b}{2}, \qquad \operatorname{Var}(X) = \tfrac{(b-a)^2}{12}

Independent uniforms are standard primitives for Monte Carlo sampling and for inverse-transform samplers.

$U(0,1)$ is the canonical base distribution. If $a=b$ , the distribution collapses to a deterministic constant. Affine maps $c + d\,U(0,1)$ generate any $U(a,b)$ ; applying $\log$ or trigonometric maps seeds Box–Muller and inverse-transform samplers.

🎯 Special Cases and Limits

$U(0,1)$ is the canonical base distribution.
$a=b$ collapses to a point mass.

Box–Muller Gaussian transforms and Laplace inverse-CDF sampling start from uniforms; Bernoulli or categorical variables arise from thresholding uniforms; low-discrepancy sequences provide quasi-uniform alternatives.

Usage in Oakfield

Oakfield uses uniform RNG streams as the base for several higher-level stochastic features:

Integrator noise hooks use integrator_rng_uniform() as the base RNG in integrators/integrator.c, with integrator_noise_uniform() / Box–Muller Gaussian / inverse-CDF Laplace built on top.
Distribution selection is exposed by name ("uniform", "gaussian", "laplace") and mapped onto the integrator noise function pointers.
Seeded stochastic operators like stimulus_white_noise, stimulus_fbm, and stimulus_random_fourier carry a seed parameter to produce reproducible realizations.

Historical Foundations

📜 Random Sampling

Uniform sampling is the standard starting point for pseudorandom generation and for transforming randomness into other distributions.

🌍 Modern Perspective

Uniform(0,1) variates are the common interface exposed by RNG libraries and underpin a wide range of stochastic simulation pipelines.

📚 References

Devroye, Non-Uniform Random Variate Generation
Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms