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Nitpicking, but you can have a uniform distribution on the unit interval, which is infinite :). What you cannot have is a uniform distribution on an unbounded s
by obastani 10y ago
Nitpicking, but you can have a uniform distribution on the unit interval, which is infinite :). What you cannot have is a uniform distribution on an unbounded set.
Edit: As pointed out, the correct assumption is not unboundedness, but having infinite measure (since then no normalization constant exists). I thought they were equivalent, but a simple counterexample is [0,1] \cup Z.
- kutkloon7 10y agoWell, I was criticising the article in a condescending way, on very similar grounds, so you make a fair point ;)
- kutkloon7 10y agoIn my defense, I was using the context of hash functions, in which the input set of the hash is discrete. You certainly can't hash any real number.
- SomeStupidPoint 10y agoIt's an interesting exercise in measure to figure out what falls apart between a uniform distribution on [0,1] and the lack of one on [0, inf]. (Adding a point at infinity to compactify the set, which makes the two intervals topologically equivalent.)
- klodolph 10y agoTo elaborate, the reason topological equivalence doesn't help here is because probability is defined in terms of measure spaces, not topological spaces.
- klodolph 10y agoMathematician here. You can't have a uniform distribution on a set with infinite measure. Unbounded sets are fine as long as they have finite measure.
- MrQuincle 10y agoA point distribution (single point) might be considered uniformly distributed. :-)
- klodolph 10y agoNot just "might", but is. It is a standard case of the discrete uniform distribution.
- inimino 10y agoWhat is an unbounded set with finite measure?
- obastani 10y agoI suppose you could take [0,1] \cup Z, and put a uniform measure on that set.
- vog 10y agoAs a somewhat extreme case, the rationals (Q) are unbounded but their measure is 0, hence finite.
- klodolph 10y agoSorry, by "finite" I also mean non-zero. You can't have a uniform distribution on a measure zero set.