From: Bill Lionheart
Date: 2019 Mar 30, 05:48 -0700
NavList member Robin Stuart has had a paper published in the Journal of Navigation
"Probabilities in a Gaussian Cocked Hat"
(Stuart, R. . Probabilities in a Gaussian Cocked Hat. Journal of Navigation, 1-17. doi:10.1017/S0373463319000110)
We all know that before the lines are chosen the probability of being in the hat (assuming equal probability of being on either side is equal) is 1/4. What Robin addressses is how that changes when you know what the lines are. He assumes zero mean Gaussian errors, and he has a very cunning way to integrate a bivariate Gaussian over a polygon.
(This DOI may not work without a subscription but it gives an link that can be freely shared "by email", so presumably not in a public forum?)
Here is the Abstract:
A round of three celestial sights yields three lines of position along which the observer's true position could lie. Due to measurement errors, the lines of position do not intersect at a point but rather form a triangle called the “cocked hat”. The probability that this encloses the observer's true position is well known to be 25% which is the average over all possible cocked hats that could arise when the sights are made. It does not apply to any specific set of sights and in that case the probabilities depend on the statistical distribution of the measurement errors. With fixed azimuths for the observed celestial bodies and assuming a normal distribution for the errors in their measured altitudes, a closed form analytic expression is derived for the probability of the observer's position falling inside the cocked hat and this is related back to the global average. Probabilities for exterior regions bounded by the lines of position are also obtained. General results are given that apply for any number of lines of position
(Thanks to Robin for sending me an earlier draft and David Burch for pointing out it has appeared!)