Standard Power Distribution & U Power Distribution

Probability Distributions >

Standard Power Distribution

The standard power distribution has the probability density function f(x) = ΒxΒ-1; 0 < x < 1.
The distribution has one shape parameter, Β.

standard power distribution
Graph of the standard power distribution with two values for shape parameter, Β.

The distribution isn’t found very often in the literature, possibly because it isn’t a realistic model for real-world applications. One application is in random number generation, where random samples are drawn from the function [1].

U-Power Distribution

U-Power Distribution
Graph of the U-power distribution with k = 1 (red) and k = 2 (blue).

The standard U-power distribution is a U-shaped, continuous probability distribution with one shape parameter k &isin ℕ. The distribution, which is defined on the interval [-1, 1] is based on a family of power functions which take on the distinctive u-shape [2].

The U-power distribution has the probability density function:
U-power distribution probability density function
In general, the PDF is monotone increasing with a global maximum occurring at the upper boundary of the domain (x = 1 for the standard distribution). The shape parameter controls the overall shape, including height, spread, and location of its maximum point.
When k = 1 (graphed in red in the above image), the distribution is called the U-quadratic distribution. When k = 0, the distribution reduces to the uniform distribution, which is not u-shaped.

The distribution function has a few notable properties:

  • Symmetry about x = 0,
  • Concave up (i.e,. a right-side-up-U),
  • Modes at x = ±1,
  • Decreasing for x < 0 and increasing for x>0.
  • Minimum at x = 0.

Related article: Exponential Power Distribution.


Graph created with Desmos.
[1] numpy.random.power.
[2] Siegrist, K. 5.26: The U Power distribution. Retrieved December 31, 2021 from:

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