U-quadratic distribution

U-quadratic
Probability density function
Parameters	${\displaystyle a:~a\in (-\infty ,\infty )}$ ${\displaystyle b:~b\in (a,\infty )}$ or ${\displaystyle \alpha :~\alpha \in (0,\infty )}$ ${\displaystyle \beta :~\beta \in (-\infty ,\infty ),}$
Support	${\displaystyle x\in [a,b]\!}$
PDF	${\displaystyle \alpha \left(x-\beta \right)^{2}}$
CDF	${\displaystyle {\alpha \over 3}\left((x-\beta )^{3}+(\beta -a)^{3}\right)}$
Mean	${\displaystyle {a+b \over 2}}$
Median	${\displaystyle {a+b \over 2}}$
Mode	${\displaystyle a{\text{ and }}b}$
Variance	${\displaystyle {3 \over 20}(b-a)^{2}}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle {3 \over 112}(b-a)^{4}}$
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 convex quadratic function with lower limit a and upper limit b.

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

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:

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

(gravitational balance center, offset), and

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

(vertical scale).

One can introduce a vertically inverted (${\displaystyle \cap }$)-quadratic distribution in analogous fashion. That inverted distribution is also closely related to the Epanechnikov distribution.

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 and gamma distribution.

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

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