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Compare a known stationary distribution with matrix iteration and occupation frequencies from a deterministic-seed Markov-chain simulation.

Integrate y'=-y from zero to one at five resolutions and compare each final value with the analytical exponential solution.

Implement bounded-memory reservoir sampling and record sample integrity plus marginal inclusion frequencies across a fixed-seed simulation.

Compare sequential sum and math.fsum on a repeated cancellation pattern, with a Fraction reference for the exact represented inputs.

Generate 256 paired synthetic measurements through a declared linear transformation of independent standard-normal draws, with known population covariance.

Generate paired complete, independently missing, and value-dependent missing observations from a declared standard-normal process.

Generate 128 regression observations with known coefficients, a declared Gaussian noise process, and a fixed train/test split, together with a small reference fit.

Compare a raw second-moment formula, Welford accumulation, and statistics.pvariance on five large observations with an exact population variance of two.