en>Kolbasz: Units

2014-01-03T17:14:08Z

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{{Probability distribution |
name =U-Quadratic|
type =density|
pdf_image =[[Image:Distributions UQuadratic PDF.jpg|325px|Plot of the U-Quadratic Density Function]]|
cdf_image =|
parameters =<math>a:~a \in (-\infty,\infty)</math><br><math>b:~b \in (a, \infty)</math><br>''or''<br/><math>\alpha:~\alpha\in (0,\infty)</math><br><math>\beta:~\beta \in (-\infty,\infty),</math>|
support =<math>x\in [a , b]\!</math>|
pdf =<math>\alpha \left ( x - \beta \right )^2 </math>|
cdf =<math>{\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right )</math>|
mean =<math>{a+b \over 2}</math>|
median =<math>{a+b \over 2}</math>|
mode =<math>a\text{ and }b</math>|
variance =<math> {3 \over 20} (b-a)^2 </math>|
skewness =<math>0</math>|
kurtosis =<math> {3 \over 112} (b-a)^4 </math>|
entropy =TBD|
mgf = [[#Moment generating function|See text]]|

char = [[#Characteristic function|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''.

: <math>f(x|a,b,\alpha, \beta)=\alpha \left ( x - \beta \right )^2, \quad\text{for } x \in [a , b].</math>

==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:

: <math>\beta = {b+a \over 2}</math>

(gravitational balance center, offset), and

: <math>\alpha = {12 \over \left ( b-a \right )^3}</math>

(vertical scale).

==Related distributions==
One can introduce a vertically inverted (<math>\cap</math>)-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==
<math>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 }</math>

==Characteristic function==
<math>\phi_X(t) = {3i\left(e^{iate^{ibt}} (4i - (-4b + (a+b)^2)t)\right) \over (a-b)^3 t^2 }</math>

{{ProbDistributions|continuous-bounded}}

{{DEFAULTSORT:U-Quadratic Distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

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en>Kolbasz: Units