U-quadratic distribution

U-Quadratic
Probability density function
Parameters	$a:~a \in (-\infty,\infty)$ $b:~b \in (a, \infty)$ or $\alpha:~\alpha\in (0,\infty)$ $\beta:~\beta \in (-\infty,\infty),$
Support	$x\in [a , b]\!$
pdf	$\alpha \left ( x - \beta \right )^2$
CDF	${\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right )$
Mean	${a+b \over 2}$
Median	${a+b \over 2}$
Mode	$a\text{ and }b$
Variance	${3 \over 20} (b-a)^2$
Skewness	$0$
Ex. kurtosis	${3 \over 112} (b-a)^4$
Entropy	TBD
MGF	See text
CF	See text

In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique quadratic function with lower limit a and upper limit b.

$f(x|a,b,\alpha, \beta)=\alpha \left ( x - \beta \right )^2, \quad\text{for } x \in [a , b].$

1 Parameter relations
2 Related distributions
3 Applications
4 Moment generating function
5 Characteristic function

Parameter relations [edit]

This distribution has effectively only two parameters a, b, as the other two are explicit functions of the support defined by the former two parameters:

$\beta = {b+a \over 2}$

(gravitational balance center, offset), and

$\alpha = {12 \over \left ( b-a \right )^3}$

(vertical scale).

Related distributions [edit]

One can introduce a vertically inverted ( $\cap$ )-quadratic distribution in analogous fashion.

Applications [edit]

This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution - e.g., Beta distribution, Gamma distribution, etc.

Moment generating function [edit]

$M_X(t) = {-3\left(e^{at}(4+(a^2+2a(-2+b)+b^2)t)- e^{bt} (4 + (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }$

Characteristic function [edit]

$\phi_X(t) = {3i\left(e^{iate^{ibt}} (4i - (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }$

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,∞)

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 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham