U-quadratic distribution

Probability distribution
name =U-Quadratic
type =density
pdf_

cdf_

parameters =$a:~a\; in\; (-infty,infty)$
$b:~b\; in\; (a,\; infty)$
"or" $alpha:~alphain\; (0,infty)$

support =$xin\; [a\; ,\; b]\; !$
pdf =
cdf =
mean =$\{a+b\; over\; 2\}$
median =$\{a+b\; over\; 2\}$
mode =$a\; ext\{\; and\; \}b$
variance =$\{3\; over\; 20\}\; (b-a)^2$
skewness =$0$
kurtosis =$\{3\; over\; 112\}\; (b-a)^4$
entropy =TBD
mgf = See text

char = 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".

:

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:

:

(gravitational balance center, offset), and

: $alpha\; =\; \{12\; over\; left\; (\; b-a\; ight\; )^3\}$

(vertical scale).

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

$M\_X(t)\; =\; \{-3left(e^\{at\}(4+(a^2+2a(-2+b)+b^2)t)-\; e^\{bt\}\; (4\; +\; (-4b\; +\; (a+b)^2)t)\; ight)\; over\; (a-b)^3\; t^2\; \}$

Characteristic function

$phi\_X(t)\; =\; \{3ileft(e^\{iat\}(-4i+(a^2+2a(-2+b)+b^2)t)+\; e^\{ibt\}\; (4i\; -\; (-4b\; +\; (a+b)^2)t)\; ight)\; over\; (a-b)^3\; t^2\; \}$

Interactive demonstrations

The SOCR tools allow interactive manipulations and computations of the U-quadratic distributions, among other continuous and discrete distributions. Go to [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions] and select the U-quadraticDistribution from the drop-down list of distributions in this Java applet.

External links

* SOCR [http://wiki.stat.ucla.edu/socr/index.php/UQuadraticDistribuionAbout U-Quadratic Distribution Page]