Trapezoidal distribution

Trapezoidal
	Probability density function
	Cumulative distribution function
Parameters	- lower bound; - level start; - level end; - upper bound;
Support
PDF
CDF
Mean
Entropy

In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution the graph of whose probability density function resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateaus.

Each trapezoidal distribution has a lower bound ${\textstyle a}$ and an upper bound ${\textstyle d}$ , where ${\textstyle a<d}$ , beyond which no values or events on the distribution can occur (i.e. beyond which the probability is always zero). In addition, there are two sharp bending points (non-differentiable discontinuities) within the probability distribution, which we will call $b$ and ${\textstyle c}$ , which occur between ${\textstyle a}$ and ${\textstyle d}$ , such that $a\leq b\leq c\leq d$ .

The image to the right shows a perfectly linear trapezoidal distribution. However, not all trapezoidal distributions are so precisely shaped. In the standard case, where the middle part of the trapezoid is completely flat, and the side ramps are perfectly linear, all of the values between $c$ and ${\textstyle d}$ will occur with equal frequency, and therefore all such points will be modes (local frequency maxima) of the distribution. On the other hand, though, if the middle part of the trapezoid is not completely flat, or if one or both of the side ramps are not perfectly linear, then the trapezoidal distribution in question is a generalized trapezoidal distribution,^[1]^[2] and more complicated and context-dependent rules may apply. The side ramps of a trapezoidal distribution are not required to be symmetric in the general case, just as the sides of trapezoids in geometry are not required to be symmetric.

Special cases of the trapezoidal distribution include the uniform distribution (with $a=b$ and ${\textstyle c=d}$ ) and the triangular distribution (with $b=c$ ). Trapezoidal probability distributions seem to not be discussed very often in the literature. The uniform, triangular, Irwin-Hall, Bates, Poisson, normal, bimodal, and multimodal distributions are all more frequently discussed in the literature. This may be because these other (non-trapezoidal) distributions seem to occur more frequently in nature than the trapezoidal distribution does. The normal distribution in particular is especially common in nature, just as one would expect from the central limit theorem.

References

↑ "Generalized Trapezoidal Distributions" (PDF). Semantic Scholar. March 2003.
↑ Dorp, J. René van; Kotz, Samuel (2003-08-01). "Generalized trapezoidal distributions". Metrika. 58 (1): 85–97. doi:10.1007/s001840200230. ISSN 0026-1335.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "Generalized Trapezoidal Distributions" (PDF). Semantic Scholar. March 2003.

[2] Dorp, J. René van; Kotz, Samuel (2003-08-01). "Generalized trapezoidal distributions". Metrika. 58 (1): 85–97. doi:10.1007/s001840200230. ISSN 0026-1335.

Probability distributions
List
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 Borel 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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Trapezoidal
Probability density function
Cumulative distribution function
Parameters	$a\;(a<d)$ - lower bound $b\;(a\leq b<c)$ - level start $c\;(b<c\leq d)$ - level end $d\;(c\leq d)$ - upper bound
Support	$x\in [a,d]$
PDF	${\begin{cases}{\frac {2}{d+c-a-b}}{\frac {x-a}{b-a}}&{\text{for }}a\leq x<b\\{\frac {2}{d+c-a-b}}&{\text{for }}b\leq x<c\\{\frac {2}{d+c-a-b}}{\frac {d-x}{d-c}}&{\text{for }}c\leq x\leq d\end{cases}}$
CDF	${\begin{cases}{\frac {1}{d+c-a-b}}{\frac {1}{b-a}}(x-a)^{2}&{\text{for }}a\leq x<b\\{\frac {1}{d+c-a-b}}(2x-a-b)&{\text{for }}b\leq x<c\\1-{\frac {1}{d+c-a-b}}{\frac {1}{d-c}}(d-x)^{2}&{\text{for }}c\leq x\leq d\end{cases}}$
Mean	${\frac {1}{3(d+c-b-a)}}\left({\frac {d^{3}-c^{3}}{d-c}}-{\frac {b^{3}-a^{3}}{b-a}}\right)$
Entropy	${\frac {d-c+b-a}{2(d+c-b-a)}}+\ln \left({\frac {d+c-b-a}{2}}\right)$

Trapezoidal distribution

See also

References