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## Contents

### Description

The U quadratic distribution is defined by the following density function

$f(x)=\alpha \left ( x - \beta \right )^2, \forall x \in [a , b], a < b$,

where the relation between the two pairs of parameters (domain-support, a and b) and (range/offset α and β) are given by the following two equations

(gravitational balance center) $\beta = {b+a \over 2}$, and
(vertical scale) $\alpha = {3 \over 2 \left ( b-a \right )^3}$.

### Properties

• Support Parameters: $a < b \in (-\infty,\infty)$
• Range/Offset Parameters: $\alpha \in (0,\infty)$ and $\beta \in (-\infty,\infty)$
• PDF: $f(x)=\alpha \left ( x - \beta \right )^2, \forall x \in [a , b]$
• CDF $F(x)={\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right ), \forall x \in [a , b]$
• Mean: ${a+b \over 2}$
• Median: ${a+b \over 2}$
• Modes: a and b
• Variance: ${\alpha \over 5} \left ( (b-\beta)^5 - (a-\beta)^5 \right )$
• Skewness: TBD
• Kurtosis: TBD
• MGF: TBD