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## Main Question or Discussion Point

I have a model where the probability is spherically symmetric and follows an exponential law. Now I need the probability density function of this model. The problem is the singularity at the origin. How can I handle this?

P(r) = ∫p(r) dr = exp(-μr)

p(r) = dP(r)/(4πr²dr)

One way I tried to handle this is numerically in Matlab by having the probability at 0 such that the total probability is 1. The problem there is that this depends highly on the mesh you chose, because of the steepness of the pdf close to the origin.

Is there a mathematical way to handle this analytically?

Afterwards I need to combine this pdf with different gaussians in a convolution to get a combined probability map. Obviously I would love to extend this to non-isotropic in carthesian coordinates as a final step. Can this be done with homothety?

P(r) = ∫p(r) dr = exp(-μr)

p(r) = dP(r)/(4πr²dr)

One way I tried to handle this is numerically in Matlab by having the probability at 0 such that the total probability is 1. The problem there is that this depends highly on the mesh you chose, because of the steepness of the pdf close to the origin.

Is there a mathematical way to handle this analytically?

Afterwards I need to combine this pdf with different gaussians in a convolution to get a combined probability map. Obviously I would love to extend this to non-isotropic in carthesian coordinates as a final step. Can this be done with homothety?