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I am trying to solve this problem on and off for the past couple of months but to no success. This was supposed to be a very small part of my PhD thesis in navigation but I guess I underestimated the problem. It sounded trivial at the beginning, but now I am not so sure.

Lets say we have two ships, each with its own nominal position (mean). Due to errors in positioning systems we can only be certain that the ships are within 1 mile of the mean with 95% probability. What is the probability that the ships are within 5 miles from each other? Also, same question if the ship's probable position is an ellipse, not a circle.

I asked some people and they told me that there are no analytic solutions. If that is really the case, please explain how to solve it numerically.

As you can already tell, I come from engineering background, therefore my math is more than a bit rusty.

I apologize in advance if the question is too vague or too trivial for this forum. I will be more than happy to explain in more detail if needed.


*EDIT: OK, I found this on statexchange, but it is only for univariate case, and besides I don't know how to implement it in my case where I need to find the probability that the distance between two ships is less than 5 miles.

I imagine this problem as a plane with two hills that intersect and the solution is the volume under the circle with diameter of 5 miles that is located somewhere between the two peaks of hills (means).

Am I on the right track?


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