U-quadratic distribution

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.

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.

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

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