UQuadraticDistribuionAbout

From Socr

Revision as of 19:07, 28 December 2009 by IvoDinov (Talk | contribs)
(diff) ← Older revision | Current revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Contents

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.




Translate this page:

(default)

Deutsch

Español

Français

Italiano

Português

日本語

България

الامارات العربية المتحدة

Suomi

इस भाषा में

Norge

한국어

中文

繁体中文

Русский

Nederlands

Ελληνικά

Hrvatska

Česká republika

Danmark

Polska

România

Sverige

Personal tools