Parabolic distribution

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A parabolic distribution is any distribution that has the shape of a parabola. Many different probability distributions from various families can take on this shape. Let’s dive deeper into why and how this happens.

What is a Parabolic Distribution?

In general, if a probability distribution can be modeled with a quadratic function (which creates a U or upside down U graph), then that distribution is parabolic. For example, the following distribution, which is a quadratic function, is parabolic [1]:

f(xi) = 6(xi – A)(B – xi) / (B – A)2.

parabolic distribution
Examples of parabolic distributions written in the form y = ax2 + bx + c: Graphed with

The degree of curvature in the graph varies based on the coefficients of the equation used to generate it. If all three coefficients are zero (e.g., y = 0), then the graph will be linear (i.e., a straight line) and not curved at all. But if one or more coefficients are nonzero (e.g., y = x2), then the graph will be curved and have a parabolic shape to it. The higher each coefficient is, the sharper the curve will become, which increases its “parabolicity.”

The study of parabolic relationships has proven to be valuable because they appear in many scientific and real-world applications such as in the investigation of projectile motion, the design of satellite dishes and the analysis of economic models. By unraveling the principles behind the parabolic relationship between variables, we can better understand and predict the behavior of various systems and phenomena.

How Does This Relate to Probability Distributions?

There are hundreds of different types of probability distributions, each with its own characteristics and uses within statistics and data analysis. Some of these distributions may take on a parabolic shape when plotted on an X-Y axis due to their internal structure; this means that they can be considered “parabolic” distributions as well. This depends largely on how much skewness (asymmetry) is present in their underlying structure; if there isn’t much skewness present then chances are good that they will take on a parabolic shape when plotted out. So even though all probability distributions aren’t necessarily parabolic in nature, some can still be considered “parabolic” if they meet certain criteria.

As an example, the beta distribution can take on many different shapes, from u-shaped curves to bell curves. However, when α = β = 2 (shown in pink below), the distribution becomes parabolic.

The beta distribution is parabolic when α = β = 2 (pink line). [2]

Parabolic distributions of order 2 are members of the location-scale family of distributions. These distributions are so-named because they are parameterized by a location parameter and a scale parameter. The probability density function of a parabolic distribution of order 2 is given by:

f(x;μ,σ) = (1 + (x – μ)22)-1


  • μ is the location parameter
  • σ is the scale parameter.


[1] Meloun, M. & Militky, J. Errors in instrumental measurements.

[2] Image: ​wikipedia user Nusha, CC BY-SA 3.0, via Wikimedia Commons

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