Probability Distributions & Quantiles

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Abstractions about cumulative distributions, quantile transformations, uncertainty bounds, quantization error, and named continuous or discrete probability families.

12 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Cumulative distribution function — For a real-valued random variable X, the function F(x)=P(X≤x), a nondecreasing right-continuous map whose limits are zero and one and which uniquely determines the distribution.
• Davis distribution — A three-parameter continuous income distribution on x>μ with a Planck-like exponential denominator and Pareto upper tail, introduced by Harold T. Davis in 1941.
• Exponentiated Weibull distribution — A positive continuous distribution formed by raising the Weibull cumulative distribution function to an additional positive shape parameter.
• Mean square quantization error — Evaluate a quantizer by averaging the squared difference between each input and its reconstruction value under a declared input distribution, making large reconstruction deviations contribute quadratically.
• Modified Kumaraswamy distribution — A positive continuous two-parameter probability distribution obtained by a Kumaraswamy-type transformation, with an explicit density, distribution function and quantile representation.
• Normal-exponential-gamma distribution — A heavy-tailed continuous location-scale-shape distribution obtained through a normal variance mixture whose variance follows an exponential-gamma hierarchy.
• Probability box — A pair of noncrossing lower and upper cumulative-distribution bounds representing a set of admissible probability distributions for an uncertain quantity.
• Probability integral transform — The result that applying a continuous random variable's own cumulative distribution function produces a standard uniform random variable.
• Q-Weibull distribution — A positive distribution formed by replacing the Weibull exponential with a q-exponential, recovering Weibull at q=1 and allowing compact-support or heavy-tail behavior according to q.
• Quantile function — A generalized inverse of a cumulative distribution function that maps a probability level to the smallest value whose cumulative probability reaches that level.
• Variance-gamma distribution — A continuous heavy-tailed probability family obtained by evaluating Brownian motion with drift at an independent gamma-distributed random time, equivalently a normal variance-mean gamma mixture.
• Zeta distribution — A discrete power-law distribution on positive integers with probability proportional to k^−s and normalized by the Riemann zeta function for s>1.