Random variable as a function I'm reading Schaum's outline of probability, random var. and random processes.
In the second chapter they make it clear that a random variable $X$ is not a variable in the traditional sense but rather a function $X(\zeta)$ where $\zeta$ is a sample point in sample space $S$. So $S$ is the domain of r.v. $X$ and collection of all numbers $X(\zeta)$ is the range of r.v. $X$ 
I'm trying to relate this to distribution of features used on audio. Take MFCCs for example. Let's assume we have two different audio signatures each having a range of MFCC values, so we could have a MFCC distribution for each of the two classes.
If we define the numerical value of r.v. as the values taken by MFCCs, it's clean that range of r.v. $X$ is from $-\infty$ to $+\infty$ (in N dimensions, N = dimensions of MFCC), but what is the sample space here? And what is the $\zeta$? i.e. If $X(\zeta)$ = MFCC_n, what is this $\zeta$ we are referring to? 
 A: Think, a little abstractly, of an MFCC as being a complicated way to generate a vector of numbers from a "sound"--the details do not matter.  The set of "sounds" is the sample space $S$.  Each sound $\zeta\in S$ in principle has an MFCC, $X(\zeta)\in\mathbb{R}^n$.

(When analyzing sounds, one might adopt a (prior) probability distribution to describe assumptions about what sounds might occur.  One might also describe the likely set of sounds associated with a given MFCC by a probability distribution on $S$.  Therefore, at least implicitly, there must be an assumed probability measure $\Pr$ on some field of events $\mathfrak{S}$ and $X$ has to be a measurable function with respect to $\mathfrak{S}$.    That technicality will come to the fore in deeper analyses of sounds as stochastic processes, but can be ignored here.)
In general probability applications, $S$ is a direct mathematical representation of the things that are being studied: orientations of a coin on a table, histories of stock closing prices, sounds, people, countries, or whatever.  The role of a random variable is to capture numerical information about these objects: whether the coin is heads up or not (indicated with a $1$ or $0$ code), how much the stock appreciated, the coefficient of one MFCC component, a person's age, a gross national product, or whatever.
In any situation there often is some important (vector-valued) random variable $Y$ that establishes a one-to-one correspondence between $S$ and a subset of $\mathbb{R}^n$ or at least creates a correspondence between a partition of $S$ and such a subset.  For instance, in studying the outcomes of a coin flip, $S$ might consist of all possible final locations and orientations of the coin on a table.  If all we are really concerned about is whether the coin is heads up, we would focus on the random variable $Y$ defined by $Y(\zeta)=1$ when $\zeta$ is a heads up position and $Y(\zeta)=0$ otherwise.  In such cases $S$ is often replaced by $Y(S) = \{Y(\zeta)\,|\,\zeta\in S\} \subset\mathbb{R}^n$ ($n=1$ for the coin) with no loss of essential information.  This replacement often occurs naturally and without any notice, which can be very confusing.
In the MFCC setting, where MFCCs are used to compress sounds, it likely is important to maintain the distinction between the original sound and its MFCC.  Thus, the set of sounds would probably not be replaced by the set of their MFCCs.  It might, though, be replaced by (say) a set of bounded functions on the interval from 20 Hz to 20 KHz, or perhaps by a set of linear combinations of sinusoidal waves: that is, for further mathematical analysis some reasonably thorough and rich mathematical representation of the original sounds will silently replace the original set of sounds.
