< List of probability distributions

The **sine distribution**, also known as *Gilbert’s sine distribution*, is a probability distribution that is continuous and based on a part of the sine curve. It is useful for fitting data sets that have several peaks, assuming that the data did not come from a mixture distribution but rather a true multimodal distribution [1]. It can also be used to model phenomena that exhibit periodic behavior, such as the tides or the seasons.

This unique distribution’s advantages include that order statistics can be formulated using elementary functions, unlike many other parent distributions. Sine distributions offer advantages like simplicity, flexibility across many phenomena, and resilience to outliers. However, working with this distribution requires familiarity with circular (trigonometric) functions [2]. In addition, they are not always the most accurate way to model phenomena, they can be sensitive to the choice of parameters and difficult to fit to data — partly due to the fact that as they are not well known, not all software packages include the distribution as an option.

## Sine distribution properties

A random variable Z follows a sine distribution if its probability density function (PDF) is

The **standard sine distribution** is [3]

The standard sine distribution has the following properties:

- Symmetry about z = ½
- Increasing, then decreasing with mode at z=½.
- Concave downward (shaped like this: ∩)

All **moments **exist, but they have a complicated general formula which involves special functions such as the beta, gamma, or hypergeometric functions. *Special functions* are important mathematical functions with established names due to their relevance in mathematical analysis, functional analysis, geometry, physics, and other fields. They are mainly considered as a function of a complex variable, possessing analytic properties such as known singularities, cuts, differential and integral representations, and Taylor or asymptotic series. Often, there are relationships between different special functions, where complex ones can be expressed using simpler functions. To evaluate, a function can sometimes be expanded into a Taylor series.

## Sine distribution history

The sine distribution was first introduced in 1892 by geologist G.K. Gilbert who used it for analyzing moon craters. Richard von Mises, in the early 20th century, developed it further for the purpose of modeling the distribution of errors in measurements [4]. John Tukey, definer of the phrase exploratory data analysis (EDA), expanded it once more in the late 20th century to model data that was collected over time [5]. Nowadays, the sine distribution has become a widely used tool to model many phenomena, including tides, seasons and the distribution of errors in measurements.

Although the sine distribution is uncommon in modern literature, it is occasionally referenced. NASA, for instance, has published that a truncated version of the distribution is a suitable match for how temperature is affected by radiation exchange among elements on the interior surface of a tube [6].

## References

[1] Golchrist, W. (2000). Statistical Modelling with Quantile Functions. CRC Press.

[2] Burrows, P. (2012). Extreme Statistics from the Sine Distribution. Retrieved May 26, 2021 from: https://www.tandfonline.com/doi/abs/10.1080/00031305.1986.10475396?journalCode=utas20

[3] Siegrist, K. 5.27: The Sine Distribution. Retrieved May 26, 2023 from: https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.27%3A_The_Sine_Distribution

[4] Mises, R.v. Grundlagen der Wahrscheinlichkeitsrechnung. *Math Z* **5**, 52–99 (1919). https://doi.org/10.1007/BF01203155

[5] Tukey, J. W. (1977). Exploratory data analysis. Reading,

PA: Addison-Wesley[6] NASA Technical Note (1962).