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].

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:

\beta = {b+a \over 2}

(gravitational balance center, offset), and

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

(vertical scale).

Differential equation

\left\{(-a-b+2 x) f'(x)-4 f(x)=0,f(0)=-\frac{3 (a+b)^2}{(a-b)^3}\right\}
\left\{(x-\beta ) f'(x)-2 f(x)=0,f(0)=\alpha \beta ^2\right\}

Related distributions

One can introduce a vertically inverted (\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, Gamma distribution, etc.

Moment generating function

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

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