Cauchy distribution

A continuous probability distribution with density proportional to the reciprocal of one plus squared distance from its location. Its heavy tails make its mean and variance undefined.

Probability density function: A function whose value gives probability density, with probabilities obtained by integrating over intervals. The Cauchy distribution is specified by a density with a distinctive inverse-quadratic shape.
Location parameter: A parameter that shifts a probability distribution along the number line without changing its shape. It identifies the Cauchy distribution’s center of symmetry.
Larmor line shape: A Lorentzian spectral profile associated with resonance broadened by exponential decay of a coherent signal. The Cauchy density describes the idealized line shape of many resonance signals.
Normal distribution: A symmetric, bell-shaped probability distribution with finite mean and variance and exponentially decreasing tails. Unlike the normal distribution, the Cauchy has much heavier tails and no finite moments.
Cauchy–Lorentz distribution: A location-scale family with density proportional to the reciprocal of one plus squared standardized distance. This name emphasizes the same distribution’s location and scale parameters.
Scale parameter: A positive parameter controlling the spread of a location-scale distribution. It sets the Cauchy distribution’s half-width at half its maximum density.
Robust statistics: Statistical methods designed to remain reliable when data contain outliers or violate model assumptions. Cauchy-like tails motivate estimators less dominated by extreme observations than the sample mean.
Student's t-distribution: A symmetric distribution used in inference for means, with tail weight controlled by degrees of freedom. With one degree of freedom, Student’s t-distribution is exactly the standard Cauchy distribution.
Characteristic function: The Fourier transform of a probability distribution, encoding its probability law. The Cauchy characteristic function is an exponential in absolute frequency, unlike a Gaussian’s quadratic exponent.
Median: A value dividing a probability distribution into two halves of equal probability. The Cauchy distribution has a well-defined median even though its mean does not exist.

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