Location-scale family

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In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter μ and a scale parameter σ ≥ 0; if X is any random variable whose probability distribution belongs to such a family, then Y = μ + σX is another, and every distribution in the family is of that form.

In other words, a class Ω of probability distributions is a location-scale family if whenever F is the cumulative distribution function of a member of Ω and μ is any real number and σ > 0, then G(x) = F(μ + σx) is also the cumulative distribution function of a member of Ω.

Examples

References

http://www.ds.unifi.it/VL/VL_EN/special/special1.html

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Probability distributions

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Discrete univariate with finite support
Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Discrete univariate with infinite support
Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Continuous univariate supported on a bounded interval
Beta · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

[[Template: {{{name}}}|v]] • [[Template talk: {{{name}}}|d]] • [{{fullurl:Template: {{{name}}}|action=edit}} e]

Continuous univariate supported on a semi-infinite interval

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Continuous univariate supported on the whole real line
Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

[[Template: {{{name}}}\|v]] • [[Template talk: {{{name}}}\|d]] • [{{fullurl:Template: {{{name}}}\|action=edit}} e] Multivariate (joint)
Discrete: Ewens · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart