[syntaxfree]

The pdf is just the pullback measure.

A random variable is a function X(w) taking (eg) real values. In your probability space you already have an ambient measure space and an ambient probability measure P which takes sets in the measure space to [0,1]. The pdf is then the function defined on sets P(invX(q)). invX is a set valued inverse.

Ok, consider coin flips. Then X takes each element of sample space either to 1 or -1. Set values inverse of 1 is the sets that map to 1. Then we get the ambient probability measure of them.

You don’t really have to cope with measure theory in full to take this tiny step.

[travisjungroth]

I don’t think the set of people who couldn’t understand the quoted paragraph but could understand your comment is very large.

[patrick451]

I'd be shocked if somebody knows what an ambient measure space is but doesn't understand the nth moment of a random variable.

[rhymer]

1/ I think you are referring to pushforward measure (https://en.wikipedia.org/wiki/Pushforward_measure): the random variable "pushes" the probability measure to its codomain. 2/ pdf requires a stronger condition: the pushforward measure needs to be absolutely continuous with respect to the sigma-finite measure (usually the Lebegue measure) on the codomain.