> I think if I were in charge of presenting this material to students I'd do it by introducing the concept of memorylessness and by showing how good memorylessness is, how many wonderful things you can do with it. And then one day I'd be like, "well, it sure would be nice if we had any distributions like that!" and then whirl around with my piece of chalk to deliver the exciting news that we do. Exactly one, in fact.

Incidentally, this also goes for the determinant of a matrix. It's got a lot of neat and desirable properties, and it turns out to be the only thing that does. When it was finally taught to me this way, those weird algorithms we use to compute this seemingly-arbitrary number finally made sense. (And, in fact, this is the easiest way to prove that all those algorithms have to be computing the same seemingly-arbitrary number. Because the algorithms preserve the properties that define The Determinant, and The Determinant is the unique thing that preserves all of those properties, so must those algorithms all be computing The Determinant, no matter how different they might look.)

So I can vouch that this style of explanation really does work, at least for people like me.