Synonyms
Abstract
Gaussian distributions are one of the most important distributions in statistics. It is a continuous probability distribution that approximately describes some mass of objects that concentrate about their mean. The probability density function is bell shaped, peaking at the mean. Its popularity also arises partly from the central limit theorem, which says the average of a large number of independent and identically distributed random variables is approximately Gaussian distributed. Moreover, under some reasonable conditions, posterior distributions become approximately Gaussian in the large data limit. Therefore, the Gaussian distribution has been used as a simple model for many theoretical and practical problems in statistics, natural science, and social science.
Definition
The simplest form of Gaussian distribution is the one-dimensional standard Gaussian distribution, which can be described by the probability density function (pdf):
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
For a complete treatment of Gaussian distributions from a statistical perspective, see Casella and Berger (2002), and Mardia et al. (1979) provides details for the multivariate case. Bernardo and Smith (2000) shows how Gaussian distributions can be used in the Bayesian theory. Bishop (2006) introduces Gaussian distributions in Chap. 2 and shows how it is extensively used in machine learning. Finally, some historical notes on Gaussian distributions can be found at Miller et al., especially under the entries “NORMAL” and “GAUSS.”
Recommended Reading
For a complete treatment of Gaussian distributions from a statistical perspective, see Casella and Berger (2002), and Mardia et al. (1979) provides details for the multivariate case. Bernardo and Smith (2000) shows how Gaussian distributions can be used in the Bayesian theory. Bishop (2006) introduces Gaussian distributions in Chap. 2 and shows how it is extensively used in machine learning. Finally, some historical notes on Gaussian distributions can be found at Miller et al., especially under the entries “NORMAL” and “GAUSS.”
Bernardo JM, Smith AFM (2000) Bayesian theory. Wiley, Chichester/New York
Bishop C (2006) Pattern recognition and machine learning. Springer, New York
Casella G, Berger R (2002) Statistical inference, 2nd edn. Duxbury, Pacific Grove
Mardia KV, Kent JT, Bibby JM (1979) Multivariate analysis. Academic Press, London/New York
Miller J, Aldrich J et al. Earliest known uses of some of the words of mathematics. http://jeff560.tripod.com/mathword.html
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this entry
Cite this entry
Zhang, X. (2016). Gaussian Distribution. In: Sammut, C., Webb, G. (eds) Encyclopedia of Machine Learning and Data Mining. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7502-7_107-1
Download citation
DOI: https://doi.org/10.1007/978-1-4899-7502-7_107-1
Received:
Accepted:
Published:
Publisher Name: Springer, Boston, MA
Online ISBN: 978-1-4899-7502-7
eBook Packages: Springer Reference Computer SciencesReference Module Computer Science and Engineering