nLab
Gaussian probability distribution

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Definition

A probability distribution on a Cartesian space n\mathbb{R}^n is called Gaussian or a normal distribution if it is of the form

μ A:xdetA(2π) n/2exp(12x,Ax)dx 1dx n. \mu_A \;\colon\; \vec x \mapsto \frac{\sqrt{det A}}{(2\pi)^{n/2}} \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right) \,d x^1 \cdots d x^n \,.

where AA is some n×nn \times n matrix such that ,A\langle -, A-\rangle is a positive definite bilinear form. Here detA\det A denotes the determinant and ,\langle -,-\rangle is the canonical bilinear form on n\mathbb{R}^n.

Since detA\sqrt{\det A} is the coordinate of the volume element vol Avol_A associated with AA, we may equivalently write this as

μ A:x1(2π) n/2exp(12x,Ax)vol A. \mu_A \;\colon\; \vec x \mapsto \frac{1}{(2\pi)^{n/2}} \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right) \,vol_A \,.

The mean of this distribution is 0\vec{0}; for a distribution with mean c\vec{c}, replace x,Ax\langle{\vec{x},A \vec{x}}\rangle with xc,AxAc\langle{\vec{x} - \vec{c},A \vec{x} - A \vec{c}}\rangle.

The matrix AA is the inverse of the covariance matrix?. In particular, for n=1n = 1, we may write x 2/σ 2x^2/\sigma^2 (or (xc) 2/σ 2(x-c)^2/\sigma^2 for mean cc) in place of x,Ax\langle{\vec{x},A \vec{x}}\rangle, where σ\sigma is the standard deviation; similarly, detA\sqrt{\det A} becomes 1/σ1/\sigma. This gives the form

μ σ:x1σ2πexp((xc) 22σ 2)dx, \mu_\sigma \;\colon\; x \mapsto \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-c)^2}{2\sigma^2}\right) \,d x \,,

which may be more familiar to some readers.

References

See also

Last revised on January 17, 2019 at 12:28:34. See the history of this page for a list of all contributions to it.