Parameters Probability density function ${\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 ),}$ ${\displaystyle x\in [a,b]\!}$ ${\displaystyle \alpha \left(x-\beta \right)^{2))$ ${\displaystyle {\alpha \over 3}\left((x-\beta )^{3}+(\beta -a)^{3}\right)}$ ${\displaystyle {a+b \over 2))$ ${\displaystyle {a+b \over 2))$ ${\displaystyle a{\text{ and ))b}$ ${\displaystyle {3 \over 20}(b-a)^{2))$ ${\displaystyle 0}$ ${\displaystyle {3 \over 112}(b-a)^{4))$ TBD See text 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.

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

## Parameter relations

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).

## Related distributions

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

## Applications

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.

## Moment generating function

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

## Characteristic function

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

### Beamforming

The quadratic U and inverted quadratic U distribution has an application to beamforming and pattern synthesis.[1][2]

## References

1. ^ Buchanan, Kristopher; Wheeland, Sara (July 2022). "Comparison of the Quadratic U and Inverse Quadratic U Sum-Difference Beampatterns". 2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI). pp. 1828–1829. doi:10.1109/AP-S/USNC-URSI47032.2022.9887273. ISBN 978-1-6654-9658-2. S2CID 252411058.
2. ^ Buchanan, Kristopher; Wheeland, Sara (July 2022). "Investigation of the Sum-Difference Beampatterns Using the Quadratic U Distribution". 2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (AP-S/URSI). pp. 135–136. doi:10.1109/AP-S/USNC-URSI47032.2022.9886771. ISBN 978-1-6654-9658-2. S2CID 252410725.