U-quadratic distribution

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.

U-quadratic
Probability density function
Parameters

or

Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy TBD
MGF See text
CF See text

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:

(gravitational balance center, offset), and

(vertical scale).

One can introduce a vertically inverted ()-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

how do you get this??

Characteristic function

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