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On Fri, 19 Apr 2002 16:39:46 -0400, Noyce, Bill wrote:

>I've also been thinking about George's assertion that the
>cocked hat contains the tru position just 25% of the time,
>and, much to my surprise, I've convinced myself it's true.
>
a counterexample that covers practically all plausible cases:

Taking George's simplifying assumption of a rectangular probability 
distribution, let's be sure we know what that means. Any
probability distribution must sum to 1 over all possible cases. Or, the 
integral from minus infinity to plus infinity of [P(x) dx] =1. Hence,
the probability of the obsevation falling in any intervalof x within the 
non=zero range is exactly the fraction of that range the interval
takes up.

With three lines of position, all independently measured by the same method, 
we must assign the observations the same probability
distribution.

One possibility is that they are just wide enough to make all three 
observations possible. That results in the traditronal center point of
the triangle being the only non-zero probability.

Another possibility is to make the rectangles just cover all the other 
observations. If the cocked hat happens to be equilateral, the
non-zero area expands to fill the triangle, and is zero everywhere outside. As 
the triangle changes away from equilateral, a band of
non-zero appears outside the two long sides, but the inside of the triangle 
remains as strong. Eventually the two ouside triangles will
equal and pass the area inside.

To treat the result that way is to ignore the geometric reason the band got 
wider--that two of the LOPs are nearly parallel, and the
distance of their crossing from the third LOP does not detract from the third 
LOP. A reasonable band to apply in this case would be
the width that allowed the nearly-parallel LOPs to reach the two 
closely-spaced corners on the third (crossing) LOP.





Rodney Myrvaagnes                                  J36 Gjo/a


"Curse thee, thou quadrant. No longer will I guide my earthly way by thee."  Capt. Ahab

   

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