(Difference between revisions)
 Revision as of 18:47, 6 November 2007 (view source)Jenny (Talk | contribs)← Older edit Revision as of 19:45, 6 November 2007 (view source)IvoDinov (Talk | contribs) m (→Properties)Newer edit → Line 14: Line 14: ===Properties=== ===Properties=== * Support Parameters: $a < b \in (-\infty,\infty)$ * Support Parameters: $a < b \in (-\infty,\infty)$ - * Range/Offset Parameters: $\alpha \in (0,\infty)$ and $\beta \in (-\infty,\infty)$ + * Scale/Offset Parameters: $\alpha \in (0,\infty)$ and $\beta \in (-\infty,\infty)$ * PDF: $f(x)=\alpha \left ( x - \beta \right )^2, \forall x \in [a , b]$ * PDF: $f(x)=\alpha \left ( x - \beta \right )^2, \forall x \in [a , b]$ * CDF  $F(x)={\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right ), \forall x \in [a , b]$ * CDF  $F(x)={\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right ), \forall x \in [a , b]$

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

### Properties

• Support Parameters: $a < b \in (-\infty,\infty)$
• Scale/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: ${3 \over 20} (b-a)^2$
• Skewness: 0 (distribution is symmetric around the mean)
• Kurtosis: ${3 \over 112} (b-a)^4$ 