5 ms·
The probability of some point to be chosen is 0 for any probability measure that has a density.
by wolfgke 12y ago
The probability of some point to be chosen is 0 for any probability measure that has a density.
- teraflop 12y agoTechnically true, but not useful. The same is true for, say, choosing a real number uniformly at random on the interval [0,1), but nobody thinks twice about treating that as a well-defined sampling problem.
- wolfgke 12y ago> but nobody thinks twice about treating that as a well-defined sampling problem. This is what I wanted to show the OP: That it's not a well-defined sampling problem.
- colanderman 12y agoWell, obviously. Even a slightly generous reading of the GP's question would indicate that pretending he asked about an algorithm "such that, for subregions a and b of equal volume, P[a]/P[b] approaches 1 as the subregions' volume approaches 0" would give him the answer he was intuitively searching for.
- Xcelerate 12y agoThank you for clarifying that haha. I forgot that on HN one has to articulate all ideas as formally and rigorously as possible in order to prevent the pathological cases from becoming the centerpiece of a conversation.
- Retric 12y agoThis is one of those cases where 0 != 0+. If every point had zero probably then the sum of all of them adds up to zero.
- bandrami 12y agoWhat the sum of an uncountably infinite bunch of zeros is, is the subject of the Calculus. There are systems that treat dx as something strictly different from zero (but not any finite value), though.
- wolfgke 12y agoThat's why sigma additivity (property of a probability measure) only allows union/sum only of a countable number of elements of the sigma algebra: > http://en.wikipedia.org/wiki/Sigma_additivity http://en.wikipedia.org/wiki/Sigma_additivity
- bandrami 12y agowell, it's a specific zero: dx.