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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 and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a family, then Y =d   (a + bX) is another (where =d means "is equal in distribution to" — that is, "has the same distribution as"), and every distribution in the family is of that form. Moreover, if X is a zero-mean, unit-variance member of the family, then every member Y of the family can be written as Y =d   (μY + σYX), where μY and σY are the mean and standard deviation of Y. 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 a is any real number and b > 0, then G(x) = F(a + bx) is also the cumulative distribution function of a member of Ω. In decision theory, if all alternative distributions available to a decision-maker are in the same location-scale family, then a two-moment decision model applies, and decision-making can be framed in terms of the means and the variances of the distributions.[1][2][3] Examples In general, any random variable X is a member of a location-scale family with an infinite number of members; each of these members Y is distributed the same as a + bX for real numbers a and b > 0. However, the form of the distribution function for a + bX, and hence for Y, may differ from the form of the distribution function for X. Well-known families in which the functional form of the distribution is retained throughout the family include the following: Normal distribution Elliptical distribution Cauchy distribution Uniform distribution (continuous) Uniform distribution (discrete) Logistic distribution Laplace distribution References ^ Meyer, Jack. "Two-moment decision models and expected utility maximization," American Economic Review 77, June 1987, 421-430. ^ Mayshar, J., "A note on Feldstein's criticism of mean-variance analysis," Review of Economic Studies 45, 1978, 197-199. ^ Sinn, H.-W., Economic Decisions under Uncertainty, second English edition, 1983, North-Holland. Further references http://www.ds.unifi.it/VL/VL_EN/special/special1.html v • d • e Probability distributions   Discrete univariate with finite support Benford · Bernoulli · binomial · categorical · hypergeometric · Poisson binomial · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot   Discrete univariate with infinite support Boltzmann · Conway–Maxwell–Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta   Continuous univariate supported on a bounded interval, e.g. [0,1] Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle   Continuous univariate supported on a semi-infinite interval, usually [0,∞) Beta prime · Bose–Einstein · Burr · chi-square · chi · 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 (scaled 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   Continuous univariate supported on the whole real line (−∞, ∞) Cauchy · extreme value · exponential power · Fisher's z  · generalized normal  · generalized hyperbolic  · Gumbel · hyperbolic secant · Landau · Laplace · logistic · noncentral t · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt   Multivariate (joint) Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student  · normal-scaled inverse gamma  · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart   Directional, degenerate, and singular Directional:Dirac comb  · Circular Uniform  · bivariate von Mises  · Kent  · univariate von Mises · von Mises–Fisher  · Wrapped normal  · Wrapped Cauchy  · Wrapped Lévy Degenerate: discrete degenerate · Dirac delta function Singular: Cantor   Families exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie