The Generalized Laws of Total Variance and Total Covariance

Charles W. Champ; Andrew V. Sills

doi:10.1007/978-3-031-69706-7_10

The Generalized Laws of Total Variance and Total Covariance

Charles W. Champ, Andrew V. Sills

Mathematical Sciences

Research output: Contribution to book or proceeding › Conference article › peer-review

Abstract

The law of total variance states that the unconditional variance of a random variable Y is the sum of (a) the variance of the conditional expectation of Y given X and (b) the expectation of the conditional variance of Y given the random variable X. We show that the total variance of Y can be partitioned by using the relationship between Y and one or more random variables X₁,…,X_k, where k≥1. An application in multivariate analysis is given. Further, we generalize the total law of total covariance and show that the generalized law of total variance is a special case. Some examples are referenced.

Original language	English
Title of host publication	Applied Mathematical Analysis and Computations I - 1st SGMC
Editors	Divine Wanduku, Shijun Zheng, Zhan Chen, Andrew Sills, Haomin Zhou, Ephraim Agyingi
Publisher	Springer
Pages	243-254
Number of pages	12
ISBN (Print)	9783031697050
DOIs	https://doi.org/10.1007/978-3-031-69706-7_10
State	Published - 2024
Event	1st Southern Georgia Mathematics Conference, SGMC 2021 - Virtual, Online Duration: Apr 2 2021 → Apr 3 2021

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	471
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	1st Southern Georgia Mathematics Conference, SGMC 2021
City	Virtual, Online
Period	04/2/21 → 04/3/21

Scopus Subject Areas

General Mathematics

Keywords

Conditional expectation
Conditional variance
Covariance decomposition
Eve’s law
Law of total covariance
Law of total variance
Variance decomposition

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10.1007/978-3-031-69706-7_10

Cite this

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title = "The Generalized Laws of Total Variance and Total Covariance",

abstract = "The law of total variance states that the unconditional variance of a random variable Y is the sum of (a) the variance of the conditional expectation of Y given X and (b) the expectation of the conditional variance of Y given the random variable X. We show that the total variance of Y can be partitioned by using the relationship between Y and one or more random variables X1,…,Xk, where k≥1. An application in multivariate analysis is given. Further, we generalize the total law of total covariance and show that the generalized law of total variance is a special case. Some examples are referenced.",

keywords = "Conditional expectation, Conditional variance, Covariance decomposition, Eve{\textquoteright}s law, Law of total covariance, Law of total variance, Variance decomposition",

author = "Champ, {Charles W.} and Sills, {Andrew V.}",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.; 1st Southern Georgia Mathematics Conference, SGMC 2021 ; Conference date: 02-04-2021 Through 03-04-2021",

year = "2024",

doi = "10.1007/978-3-031-69706-7_10",

language = "English",

isbn = "9783031697050",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "243--254",

editor = "Divine Wanduku and Shijun Zheng and Zhan Chen and Andrew Sills and Haomin Zhou and Ephraim Agyingi",

booktitle = "Applied Mathematical Analysis and Computations I - 1st SGMC",

address = "Germany",

}

Champ, CW & Sills, AV 2024, The Generalized Laws of Total Variance and Total Covariance. in D Wanduku, S Zheng, Z Chen, A Sills, H Zhou & E Agyingi (eds), Applied Mathematical Analysis and Computations I - 1st SGMC. Springer Proceedings in Mathematics and Statistics, vol. 471, Springer, pp. 243-254, 1st Southern Georgia Mathematics Conference, SGMC 2021, Virtual, Online, 04/2/21. https://doi.org/10.1007/978-3-031-69706-7_10

The Generalized Laws of Total Variance and Total Covariance. / Champ, Charles W.; Sills, Andrew V.
Applied Mathematical Analysis and Computations I - 1st SGMC. ed. / Divine Wanduku; Shijun Zheng; Zhan Chen; Andrew Sills; Haomin Zhou; Ephraim Agyingi. Springer, 2024. p. 243-254 (Springer Proceedings in Mathematics and Statistics; Vol. 471).

Research output: Contribution to book or proceeding › Conference article › peer-review

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The Generalized Laws of Total Variance and Total Covariance

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Scopus Subject Areas

Keywords

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