Standard Power Distribution / U-Power distribution

< List of probability distributions

The standard power distribution is a continuous probability distribution defined on the interval [0, 1].

Standard power distribution properties

PDF of the standard power distribution.
PDF of the standard power distribution.

The probability density function (PDF) has a single shape parameter, β:

f(x) = βxβ-1; 0 < x < 1.

Standard power distribution uses

The distribution is a rare find in literature, perhaps due to its limited relevance in real-world applications. Use cases include:

  • Random number generation [1]: the standard power distribution can be used to generate random numbers that are uniformly distributed between 0 and 1. This is useful for a variety of applications, such as Monte Carlo simulation and statistical sampling.
  • Statistical analysis: you can use the standard power distribution to model a set of data and estimate distribution parameters, such as the variance and mean.
  • Modeling natural phenomena: For instance, it can model income distribution or earthquake size distribution.
  • Engineering design: Engineers use the standard power distribution to create resilient systems withstanding uncertainties. For instance, they can use it to design a bridge capable of withstanding a wide range of wind speeds.

Despite it’s rarity compared to other distributions, it is a simple and easy-to-use distribution that is well-understood by statisticians and engineers.

U-Power Distribution

The U-power distribution is a versatile tool for modeling various phenomena, such as income, earthquake size, mountain heights, river lengths, car speeds, house prices, and student grades. This distribution can also model complex system behaviors subject to uncertainty. For instance, a bridge’s behavior in the face of windy conditions may be modeled using the U-power distribution. Engineers design a sturdy bridge by analyzing wind gust distributions to avoid collapse in strong winds.

Graph of the U Power distribution PDF.
Graph of the U Power distribution PDF.

The standard U-power distribution is a continuous probability distribution with a U-shape graph and a single shape parameter k ∈ ℕ. This distribution, defined on the interval [-1, 1], is based on a family of power functions that have the distinctive U-shape graph [2].

The PDF of the standard U-power distribution is:

u power distribution pdf g(x) = (2k+1)/2 x^2k

The PDF generally increases monotonically, with a global maximum at the upper boundary of the domain (x = 1 for the standard distribution). Overall shape, such as height, spread, and location of maximum point, is determined by the shape parameter. We call the distribution U-quadratic when k = 1, whereas it reduces to the uniform distribution (which is not u-shaped) when k = 0.

Other notable properties include:

  • Concave up (i.e., shaped like ∪ instead of ∩),
  • Decreasing for x < 0 and increasing for x > 0.
  • Minimum at x = 0,
  • Modes at x = ±1,
  • Symmetry about x = 0.

U power vs. standard power distribution

The main difference between the U-power distribution and the standard power distribution stems from their respective intervals: [-1, 1] for the former and [0, 1] for the latter. Notably, the U-power distribution accommodates negative values, whereas the standard power distribution does not.

Their means are also different, with the U-power distribution possessing a mean of 0, and the standard power distribution, a mean of ½. As a result, the U-power distribution is more evenly distributed around the center, while the standard power distribution is more skewed towards the right.

The variance of the U-power distribution is 2/(α-1)2, while the standard power distribution has a variance of 1/(α-1)2. Hence, the U-power distribution spreads out more than the standard power distribution does.

In general, the U-power distribution provides greater versatility than the standard power distribution since it can model a wider range of phenomena and is more robust to outliers. Nevertheless, the standard power distribution is simpler to understand and use.


[1] numpy.random.power.

[2] Siegrist, K. 5.26: The U Power distribution. Retrieved December 31, 2021 from:

Scroll to Top