*μ*and unit variance, but with a twist: the sample space has a gap in it between -1 and 3. Accounting for the gap, the resulting probability density becomes:

In this model *μ* is not quite a location parameter; when it’s far from the gap the density is effectively a normal centered at *μ* but when it’s close to the gap its shape is distorted. It becomes a half-normal at the gap boundary and then something like an extra-shallow exponential (log-quadratic instead of log-linear like an actual exponential) as *μ* moves toward the center of the gap. At *μ* = 1 the probability mass flips from one side of the gap to the other. Here’s a little web app in which you can play around with this statistical model (don’t neglect the play button under the slider on the right hand side).

Now the question; I ask my readers to report their gut reaction in addition to any more considered conclusions in comments.

Suppose *μ* is unknown and the data is a single observation *x*. Consider two scenarios:

*x*= -1 (the left boundary)*x*= 3 (the right boundary)

*μ*≤ 0 vs.

*> 0. Should it make a difference to our inference whether we’re in scenario (i) or scenario (ii)?*

*μ*