samplingrandom

Samplingrandom is a general term describing methods for drawing random samples from probability distributions or data streams in order to reflect a specified model. It underpins statistical estimation, computer simulations, Bayesian inference, and data analysis by providing representative values drawn from targeted distributions.

Sampling can be done for both finite discrete and continuous distributions. In discrete cases, one often seeks samples from a finite set with given probabilities. In continuous cases, samples are drawn from distributions such as uniform, normal, exponential, or custom models. Key goals include unbiasedness, efficiency, and reproducibility, often achieved through careful generator design and proper seeding of pseudo-random number generators.

Common algorithms for simple distributions include the inverse transform method, which uses the cumulative distribution function to map uniform samples to the target distribution; acceptance-rejection sampling, which enforces a bounding condition to generate samples from complex densities; and the alias method for efficient discrete sampling. For data streams, reservoir sampling allows obtaining a representative subset without knowing the stream length in advance.

For more complex targets, especially in high dimensions, techniques such as Markov Chain Monte Carlo (MCMC), including Metropolis-Hastings and Gibbs sampling, are used to generate dependent sample sequences that approximate the target distribution. Importance sampling and sequential Monte Carlo (particle methods) address weighting and dynamic models.

Applications span statistics, physics simulations, finance, machine learning, and graphics. Practical considerations include the quality of the random number generator, initialization, convergence diagnostics for iterative methods, and balancing bias, variance, and computational cost.