1
vote
0answers
19 views

Projecting non-central Gaussian on a sphere?

Suppose I have an $n$-dimensional Gaussian random variable with mean $\mu\in \mathbb{R}^n$ and covariance matrix $\sigma^2I_n$, where $I_n$ is the identity matrix. If I condition on the distribution ...
1
vote
0answers
64 views

Joint PDF for spherical region

A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere. What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly ...
1
vote
2answers
572 views

Probability density function for radius within part of a sphere

I would like to find the probability density function for radius within a given section of a sphere. For example, suppose I specify $\pi / 4 < \theta < \pi / 3$ and $\pi /7 < \phi < \pi ...
2
votes
2answers
798 views

Projection of a 3D spherical distribution function in to a 2D cartesian plane

Consider a 3D spherical Gaussian distribution function that depends on radius only, $$f(r) = \frac{1}{N} e^{-(\frac{r-R_\mu}{\sigma})^2}$$ where $R_\mu$ is the radial offset of the distribution and ...
1
vote
0answers
154 views

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 ...