# Cross-section of a circle with a three-dimensional Gaussian

Suppose I have a three-dimensional Gaussian with mean $\bar{\mu}$, volume $A$ and covariance matrix $\Sigma$ $$G(X)=\frac{A}{\sqrt{(2\pi)^{3}\det(\Sigma)}}e^{-\frac{1}{2}(X-\mu)^{T}\cdot \Sigma^{-1}\cdot (X-\mu)}$$ Now consider this Gaussian in spherical coordinates $G(R,\phi,\theta)$ where $\phi$ the azimuth and $\theta$ the polar angle. If $\phi_{\mu}$ and $\theta_{\mu}$ are azimuth an polar angle of $\bar{\mu}$, I want to consider the function $g(\theta)=G(\|\bar{\mu}\|,\phi_{\mu},\theta)$. The question is: is $g$ a (one-dimensional) Gaussian and if yes, how are its parameters (mean, variance, area) related to parameters of the three-dimensional Gaussian (mean, covariance matrix, volume)?

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Writing \det instead of det not only prevents italicization but also provides proper spacing before and after it in things like $5\det a$ (as opposed to $5 det a$). (I changed it in the posting.) –  Michael Hardy Sep 6 '12 at 18:22