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The wrapped normal distribution (or wrapped up normal distribution) occurs via Brownian motion on a circle. It is the circular analog of a normal distribution, achieved by wrapping the probability density function (PDF) of a real-valued, linear random variable to the circumference of a unit circle an infinite number of times [1].

## Formula of the Wrapped Normal Distribution

The density of the wrapped normal distribution can be defined as the series [2]

Where N is the normal distribution being wrapped and

- μ = mean (taken as modulo 2 π)
- σ = standard deviation.

The density is normalized because integrating it over 2 π is equivalent to integrating the normal distribution over the reals.

Early works define the distribution in different ways. For example, Haight’s 1958 reference work for the National Bureau of Statistics [3] defines the wrapped up normal distribution as:

Stephens [4] made a brief mention of the wrapped up normal distribution in his 1958 paper *Random Walk on a Circle*. He defines it slightly differently, as “of the type”

Stephens also notes that when on a sphere, the wrapped up normal distribution is similar to the Riemann theta (presumably, the Riemann theta function).

## References

[1] Jorge , A. et al (Eds.) (2015). Machine Learning and Knowledge Discovery in Databases. European Conference, ECML PKDD 2015, Porto, Portugal, September 7-11, 2015, Proceedings, Part II · Part 2. Springer International.

[2] Pfaff, F. (2019). Multitarget Tracking Using Orientation Estimation for Optical Belt Sorting. KIT Scientific Publishing

[3] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[4] Stephens, M.A. (1963). Random Walk on a Circle. *Biometrika* Vol. 50, No. 3/4 (Dec.), pp. 385-390 (6 pages)