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About_pages_for_SOCR_Distributions - U-Quadratic Distribution

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

More information about U-quadratic, and other continuous distribution functions, is available at Wikipedia.

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

Interactive U Quadratic Distribution

You can see the interactive U Quadratic distribution by going to 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.

SOCR Home page: http://www.socr.ucla.edu

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