Hello!

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!