For F(x), give the same chances to points A and B:
And the best guess for F(y,x) would be N(y,(x-y)/2,x), because the only information about the dispersion for F(y,x) is the distance between x and y, and the dispersion (x-y)/2 is the best fit. Note, that the function F(y,x) is not proper distribution for variable x, because the probability is given for fixed dispersion and it depends on x, i.e. to be normalized the dispersion has to be fixed. The overall distribution then is:
The graph for function f(x) is presented above. The minimum values are at points: -0.47 and +0.47. This means that the safest place is about a quarter distance from either end.