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Gaussian Measure


A Gaussian measure on Euclidean space is a probability measure for which every linear coordinate combination has a normal distribution, allowing constant combinations in the degenerate case. The standard Gaussian measure gamma_n on R^n has mean zero, covariance matrix equal to the identity matrix, and probability density function

 dgamma_n(x)=(2pi)^(-n/2)e^(-||x||^2/2)dx,

where ||x|| is the Euclidean norm. Its coordinates are independent standard normal random variables.


See also

Normal Distribution, Probability Measure

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References

Vershynin, R. High-Dimensional Probability: An Introduction with Applications in Data Science, 2nd ed. Cambridge, England: Cambridge University Press, 2026. https://doi.org/10.1017/9781009490672.

Cite this as:

Weisstein, Eric W. "Gaussian Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GaussianMeasure.html

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