< List of probability distributions < Cosine distribution
The cosine distribution is an approximation to the normal distribution. Using a cosine distribution instead of a normal can simplify algebraic manipulations. For example, expressions with products or powers of the normal probability density function (PDF) and its integral are difficult to evaluate but the cosine distribution is much easier to work with in this regard. Numerical evaluations stay the same and can be interpreted with trigonometric tables [1].
The cosine is platykurtic compared to a normal distribution, so the fit isn’t exact. However, it does give reasonable approximations. For example, a cosine function shifted up one period, with the area under the curve normalized to 1, can approximate a normal probability distribution in the range ± 2.5 standard deviations with an accuracy of about ±0.02 [2].
Cosine distributions are used for a wide range of applications including fields as diverse as psychological investigations [1], semiconductor manufacturing, antenna design and atomic sputtering,. In atomic and particle physics, the distribution is required to describe arrival angles of neutral particles or diffusive reemission — a process in which neutral particles that have been absorbed by a surface are reemitted in a random direction.
Cosine Distribution properties
The cosine distribution can be viewed as a special case of the power cosine distribution with ν = 1, which is a common way for sampling a cosine distribution [2]. The power cosine is a generalization of the raised cosine distribution, where the parameter ν controls the shape of the probability distribution; when ν = 1, the power cosine is equal to the raised cosine.
The cosine distribution can also be seen as a special case of the coned cosine distribution with θcone = π / 2 [3]. The coned cosine is also a generalization of the raised cosine, where a parameter θc controls the width of the cone.
The probability density function (PDF) for the cosine distribution is [4]:
Cosine PDF(x) = 1/(2 * π) * (1 + cos(x)), -π ≤ x ≤ π
The cumulative distribution function (CDF) is the integral of the PDF from -π to x [6]:
Cosine CDF(x) = (π + x + sinx ) / 2π, -π ≤ x ≤ π
References
- Raab, D. & Green, E. (1961). A cosine approximation to the normal distribution. Psychometrika, Vol. 26, No. 4. Retrieved December 19, 2021 from: https://link.springer.com/content/pdf/10.1007/BF02289774.pdf
- J. Greenwood. The correct and incorrect generation of a cosine dist. of scattered particles for Monte-Carlo modelling of vacuum systems. Vacuum, 67(2):217-222, 2002.
- 5.3.4 Cosine Dist. Retrieved August 10, 2023 from: https://www.iue.tuwien.ac.at/phd/ertl/node100.html
- Warsza, Z. & Korczynski, J. (2010). Shifted up cosine function as unconventional model of probability distribution. Journal of Automation, Mobile Robotics & Intelligent Systems. Volume 4, No.1.
- Cosine graph created with Desmos.
- Scipy.stats.cosine. Retrieved December 19, 2021 from: https://het.as.utexas.edu/HET/Software/Scipy/generated/scipy.stats.cosine.html