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U-quadratic distribution

Continuous probability distribution

Continuous 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 =a:a \in (-\infty,\infty) b:b \in (a, \infty) or \alpha:\alpha\in (0,\infty) \beta:\beta \in (-\infty,\infty),| support =x\in [a , b]!| pdf =\alpha \left ( x - \beta \right )^2 | cdf ={\alpha \over 3} \left ( (x - \beta)^3 + (\beta - a)^3 \right )| mean ={a+b \over 2}| median ={a+b \over 2}| mode =a\text{ and }b| variance = {3 \over 20} (b-a)^2 | skewness =0| kurtosis = -{38 \over 21} | entropy = \log\left(\frac{e^{2 \over 3}(b-a)}{3} \right) | mgf = See text|

char = See text|

In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique convex quadratic function with lower limit a and upper limit b.

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

: \beta = {b+a \over 2}

(gravitational balance center, offset), and

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

(vertical scale).

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 and gamma distribution.

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

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

References

References

  1. Lakibul, Idzhar. (2023-12-30). "On the Four-Parameter T-extended Standard U-quadratic Exponentiated Weibull Distribution". The Mindanawan Journal of Mathematics.
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