I'd like to create a probability density function where the probability of observing a random variable is proportional to the distance from a circle.
Let the pair represent any location in polar coordinates and
let the pair be the centre of the circle of radius .
The distance between any point and the centre of the circle can be found using the law of cosines:
Which means the distance to the closest point on the circle is
Now the distribution is Gaussian shaped so I set the pdf as:
The probability of a random variable falling in a polar area is found by integration:
But neither Maxima nor Mathematica were able to solve that integral.
I started a Taylor expansion about a fixed point but it got ugly fast.
Any suggestions for another method of solving or approximating this integral are appreciated!