(Difference between revisions)
 Revision as of 18:47, 6 November 2007 (view source)Jenny (Talk | contribs)← Older edit Current revision as of 19:09, 28 December 2009 (view source)IvoDinov (Talk | contribs) m (4 intermediate revisions not shown) Line 10: Line 10:
(vertical scale) $\alpha = {12 \over \left ( b-a \right )^3}$.
(vertical scale) $\alpha = {12 \over \left ( b-a \right )^3}$.
- + More information about U-quadratic, and other continuous distribution functions, is available at [http://en.wikipedia.org/wiki/UQuadratic_distribution Wikipedia]. ===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  $F(x)={\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right ), \forall x \in [a , b]$ Line 23: Line 23: * Skewness: 0 (distribution is symmetric around the mean) * Skewness: 0 (distribution is symmetric around the mean) * Kurtosis: ${3 \over 112} (b-a)^4$ * Kurtosis: ${3 \over 112} (b-a)^4$ + * 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: ${3i\left(e^{iat}(-4i+(a^2+2a(-2+b)+b^2)t)+ e^{ibt} (4i - (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }$ ===Interactive U Quadratic Distribution=== ===Interactive U Quadratic Distribution=== - You can see the interactive ''U Quadratic'' distribution by going to [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] and selecting from the drop down list of distributions ''U Quadratic''. Then follow the '''Help''' instructions to dynamically set parameters, compute critical and probability values using the mouse and keyboard. + You can see the interactive ''U Quadratic'' distribution by going to [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html SOCR Distributions] and selecting from the drop down list of distributions ''U Quadratic''. Then follow the '''Help''' instructions to dynamically set parameters, compute critical and probability values using the mouse and keyboard.
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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) $\alpha = {12 \over \left ( b-a \right )^3}$.

### 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: ${3 \over 112} (b-a)^4$
• 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: ${3i\left(e^{iat}(-4i+(a^2+2a(-2+b)+b^2)t)+ e^{ibt} (4i - (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }$ 