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gjm comments on 3AM stats quiz - Less Wrong
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The ellipses of constant probability density are (x/4)^2 + y^2 = constant. The one tangent to x=4 is tangent at (4,0) so it's (x/4)^2 + y^2 = (4/4)^2 = 1. When x=0 this means y^2 = 1, so it passes through y=+-1.
I confess that it doesn't seem particularly amusing.
You're right, and I'm wrong. It was 3AM. It was amusing only because I had convinced myself from some graphs that the answer wasn't y=+-1, which I'd been fairly certain it ought to be.