What is a Parabolic Distribution?
A parabolic distribution is any distribution that has the shape of a parabola. Many different probability distributions from a wide variety of families can be parabolic in shape.
Several quadratic functions and their parabolic graphs.
Examples of Parabolic Distribution
In general, if a probability distribution can be modeled with a quadratic function (creating a U or upside down U), then that distribution is parabolic. For example, the following distribution [1], which is a quadratic function, is parabolic:
f(xi) = 6(xi – A)(B – xi) / (B – A)2.
The beta distribution can take on many different shapes, from u-shaped curves to bell curves. However, when α = β = 2, the distribution becomes parabolic.
The Beta Distribution pdf, showing several different shapes with shape parameters α and β. The pink curve α = β = 2 is parabolic.
The following image shows two examples of a parabolic distribution: a = 5 (red) and a = 10 (blue). The x-axis is a random variable (T), and the y-axis shows the probability density of T [2].
If you have a set of data that makes a parabola and you need to find the best fit equation, use quadratic regression.
Parabolic distributions of order 2 are members of the location-scale family of distributions.
References
Graph created with Desmos.com.
[1] Meloun, M. & Militky, J. Errors in instrumental measurements.
[2] Image: Motohisa Osaka via Researchgate. CC by 4.0.
Stephanie Glen. "Parabolic Distribution" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/parabolic-distribution/
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