Talk:List of integrals of Gaussian functions

Move to List of integrals of the Gaussian function
It is not commonly called the normal function but the normal distribution. The function is the Gaussian function. The current name would also be confusing to people new to integrals, they might think normal functions means the standard functions. Dmcq (talk) 12:27, 24 February 2011 (UTC)


 * In fact better how about 'List of integrals of Gaussian functions'. Dmcq (talk) 12:30, 24 February 2011 (UTC)


 * I'm ok with that.  //  st pasha  »  19:42, 24 February 2011 (UTC)

Formulas for multivariate Gaussians?
Maybe it is worth to add some formulas for multivariate Gaussians?

$$\rho_{\mu, \Sigma}(x):= \frac{1}{\sqrt{|2\pi\Sigma|}} e^{-\frac 12 (x-\mu)^T\Sigma^{-1} (x-\mu)}$$

Integral of product of Gaussian distributions in $$\mathbb{R}^D$$ with covariance matrix $$\Sigma$$ and $$\Gamma$$, shifted by $$\mu$$ vector (derivation):

$$\int_{\mathbb{R}^D} \rho_{\mu, \Sigma}(x)\cdot\rho_{\mathbf{0},\Gamma}(x)\,dx=\frac{\exp\left(-\frac 12 (\mu^T\Sigma^{-1}(\Sigma^{-1}+\Gamma^{-1})^{-1} \Gamma^{-1}\mu) \right)} {\sqrt{(2\pi)^D |\Sigma||\Gamma||\Sigma^{-1}+\Gamma^{-1}|}}$$

For spherically symmetric $$\Sigma=\sigma^2 \mathbf{I},\ \Gamma=\gamma^2 \mathbf{I}$$ shifted by any length $$l$$ vector it becomes: $$\frac{\exp \left(-\frac 12 \frac {l^2}{\sigma^2+\gamma^2} \right)} {\sqrt{2\pi \left(\sigma^{2}+\gamma^{2}\right)}^D}$$

Jarek Duda (talk) 09:43, 13 November 2018 (UTC)