Post date: Mar 22, 2021 6:59:38 PM
Michael Hardy is a guardian angel for mathematics on Wikipedia, but his StackExchange:Mathematics answer to “Distinguishing probability measure, function and distribution” seems quite disappointing, especially because this question is so important and his answer has been viewed 13,000 times
I disagree that “nobody cares” what the space is in the definition for a probability measure. And he seems to underemphasize or miss entirely the difference between a measure and a distribution. I also agree a pdf has a precise definition, but it’s not what he says. I think a probability density is best described as a derivative of the probability distribution, otherwise you’re going to have trouble explaining how you get “probabilities” bigger than unity.
Maybe my caveman understanding is totally wrong. Is it wrong to just say the following?
Each of these functions is defined with respect to some space X. I agree with Michael that X is often the reals R, or R2, etc., which creates ordering among the elements of X.
A probability measure P is a function into [0,1] from the power set 2X of that space X (or, rather a sigma algebra which is just a subset of the power set) such that P(X) = 1 and P(Uei)=åP(ei) where ei are countably many disjoint sets in X.
A probability distribution F is a function from X to [0,1] defined by F(x) = P([-¥, x]), for every x in X, which is defined whenever X is ordered. Thus, a distribution is a restriction function of P.
A probability density function is the derivative of a probability distribution when it exists, and when the topology of X is discrete, a probability mass function is the restriction function of the measure function applied to singletons from X. (If you wanna wrangle measure theory, you can find a way to say that these two kinds of functions are special cases of some more general structure.)
Measures, distributions and probability mass functions (but not density functions) have the same codomain [0,1], the unit interval (a subset of the reals).