Benini distribution

Top

Home > Library > Miscellaneous > Wikipedia

Benini
Probability density function No image available
Cumulative distribution function No image available
Parameters	$\alpha>0$ shape (real) $\beta>0$ shape (real) $\sigma>0$ scale (real)
Support	$x>\sigma$
PDF	$e^{-\alpha\log{\frac{x}{\sigma}}-\beta\log\left[{\frac{x}{\sigma}}\right]^2} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)$
CDF	$1-e^{-\alpha\log{\frac{x}{\sigma}}-\beta\log[{\frac{x}{\sigma}}]^2}$
Mean	$\sigma+\tfrac{\sigma}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta}}\right)$ where $H_n(x)$ is the "probabilists' Hermite polynomials"
Median	$\sigma \left(e^{\frac{-\alpha+\sqrt{\alpha^2+\beta\log{16}}}{2\beta}}\right)$
Variance	$\left(\sigma^2+\tfrac{2\sigma^2}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-2+\alpha}{\sqrt{2\beta}}\right)\right)-\mu^2$

This article provides insufficient context for those unfamiliar with the subject. Please help improve the article with a good introductory style. (March 2011)

The topic of this article may not meet Wikipedia's general notability guideline. Please help to establish notability by adding reliable, secondary sources about the topic. If notability cannot be established, the article is likely to be merged, redirected, or deleted. (May 2012)

In probability and statistics, the Benini distribution is a continuous probability distribution.

Related distributions

If $X \sim \mathrm{Benini}(\alpha,0,\sigma)\,$ , then X has a Pareto distribution with $x_\mathrm{m}=\sigma$
If $X \sim \mathrm{Benini}(0,\tfrac{1}{2\sigma^2},1),$ then $X \sim e^U$ where $U \sim \mathrm{Rayleigh}(\sigma)$

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2012)

References

Kleiber, Christian; Kotz, Samuel (2003). "Chapter 7.1: Benini Distribution". Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 978-0-471-15064-0.

External links

Benini Distribution at Wolfram Mathematica (definition and plots of pdf)

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte 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]

Arcsine ARGUS Balding-Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Bose–Einstein
Burr
chi-squared
chi
Coxian
Dagum
Davis
Erlang
exponential
F
Fermi–Dirac
folded normal
Fréchet
Gamma
generalized inverse Gaussian
half-logistic
half-normal
Hotelling's T-squared
hyper-exponential
hypoexponential
inverse chi-squared (scaled-inverse-chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell speed
Mittag–Leffler
Nakagami
noncentral chi-squared
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 exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse multivariate gamma inverse-Wishart matrix normal matrix t multivariate gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

This probability-related article is a stub. You can help Wikipedia by expanding it.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Help us answer these

What is a distribution feedr in a distribution substation?

Is f distribution a discrete distribution?

Is the binomial distribution a continuous distribution?

Is the f distribution a continuous distribution?

Post a question - any question - to the WikiAnswers community:

Copyrights:

Cite

Wikipedia on Answers.com This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Benini distribution. Read