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Gaussian probability distribution
Redirected from "Gaussian probability measure".
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Context
Measure and probability theory
Contents
Definition
A probability distribution on a Cartesian space ℝ n \mathbb{R}^n is called Gaussian or a normal distribution if it is of the form
μ A : x → ↦ det A ( 2 π ) n / 2 exp ( − 1 2 ⟨ x → , A x → ⟩ ) d x 1 ⋯ d x 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 A A is some n × n n \times n matrix such that ⟨ − , A − ⟩ \langle -, A-\rangle is a positive definite bilinear form . Here det A \det A denotes the determinant and ⟨ − , − ⟩ \langle -,-\rangle is the canonical bilinear form on ℝ n \mathbb{R}^n .
Since det A \sqrt{\det A} is the coordinate of the volume element vol A vol_A associated with A A , we may equivalently write this as
μ A : x → ↦ 1 ( 2 π ) n / 2 exp ( − 1 2 ⟨ x → , A x → ⟩ ) 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 → , A x → ⟩ \langle{\vec{x},A \vec{x}}\rangle with ⟨ x → − c → , A x → − A c → ⟩ \langle{\vec{x} - \vec{c},A \vec{x} - A \vec{c}}\rangle .
The matrix A A is the inverse of the covariance matrix? . In particular, for n = 1 n = 1 , we may write x 2 / σ 2 x^2/\sigma^2 (or ( x − c ) 2 / σ 2 (x-c)^2/\sigma^2 for mean c c ) in place of ⟨ x → , A x → ⟩ \langle{\vec{x},A \vec{x}}\rangle , where σ \sigma is the standard deviation ; similarly, det A \sqrt{\det A} becomes 1 / σ 1/\sigma . This gives the form
μ σ : x ↦ 1 σ 2 π exp ( − ( x − c ) 2 2 σ 2 ) d x ,
\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 December 23, 2020 at 08:56:35.
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