# The Nature of Probability Mass/Density Functions

Consider a certain random variable and all its possible probability mass functions (or probability density functions).

What structure does this space have? For example, it can be endowed with a structure of a linear space just like any function of type $X \to \mathbb R\;$, but will that really make sense?

What is essential for pmf's or pdf's, what can be done with them, what are their nature, corresponding category or something?

To make things clear, I'm rather a physicist than a mathematician.

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The set of all the probability density functions is certainly not a linear space, whatever linear space means: if $f$ and $g$ are probability density functions, neither $2f$ nor $0f$ nor $f+g$ nor $3f-4g$ are. – Did Mar 5 '12 at 20:02
Some random variables have neither probability density functions nor probability mass functions: they are known as singular distributions. – Henry Mar 5 '12 at 22:31

One observation is that it's a convex set, as convex combinations of density functions give density functions: for densities $f,g$, $af+bg$ is a density whenever $a+b=1$ with $a,b\geq 0$.