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For a 2D Gaussian distribution with
$$ \mu = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{pmatrix}, $$ its probability density function is $$ f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{(x-\mu_x)^2}{\sigma_x^2} + \frac{(y-\mu_y)^2}{\sigma_y^2} - \frac{2\rho(x-\mu_x)(y-\mu_y)}{\sigma_x \sigma_y} \right] \right), $$

I was wondering if there is also a similarly clean formula for 3D Gaussian distribution density? What is it?

Thanks and regards!


EDIT:

What I ask is after taking the inverse of the covariance matrix, if the density has a clean form just as in 2D case?

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1 Answer 1

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There is a standard, general formula for the density of the joint normal (or multivariate normal) distrubution of dimension $n$, provided that the ($n \times n$) covariance matrix $\Sigma$ is non-singular (see, e.g., this or this). In particular, you can apply for $n=3$. When the covariance matrix is singular, the distribution is expressed in terms of the characteristic function.

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  • $\begingroup$ Thanks! What I ask is after taking the inverse of the covariance matrix, if the density has a clean form just as in 2D case? $\endgroup$
    – Tim
    Nov 21, 2010 at 17:08
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    $\begingroup$ Try QuickMath at quickmath.com (a free, excellent application). Using it, calculate the inverse of $\Sigma$ and its determinant. I think it may lead to a "reasonable" expression, taking into account that $\Sigma$ is symmetric. $\endgroup$
    – Shai Covo
    Nov 21, 2010 at 17:33
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    $\begingroup$ @Tim: Ah, I guess we miscommunicated because for most people working in linear algebra, a formula involving matrix inverses does count as clean. What you are asking for is to find the entries of the inverse explicitly, but beyond $2\times 2$ matrices things start to get ugly. $\endgroup$
    – user856
    Nov 21, 2010 at 17:36
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    $\begingroup$ @Tim: If you're taking the computational viewpoint, you're probably better off having an explicit expression for the Cholesky decomposition (your covariance matrices are supposed to be positive (semi)definite) instead of an explicit expression for the inverse. Inverses do not behave very nicely in inexact arithmetic. $\endgroup$ Nov 22, 2010 at 2:27
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    $\begingroup$ @Tim: When something like $A^{-1}b$ appears in a formula, in computational terms you should think of it as "solve $Ax = b$ for $x$" instead. As J.M. said, using a Cholesky decomposition to compute this is much better: having decomposed $A$ as $LL^T$, you just solve $Ly = b$ and then $L^Tx = y$. Solving equations with triangular matrices like $L$ and $L^T$ is very easy, and numerically well-behaved. I would also suggest taking a look at a good textbook on numerical linear algebra. $\endgroup$
    – user856
    Nov 22, 2010 at 6:57

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