# Can the population be smaller than the number of possible outcomes?

Say a random variable $X$ returns one of the values $1, 2, ..., 6$.

Conceptually, does it make sense to speak of a population, where each individual element is just $1$?

In other words, does the population strictly have to cover all possible values for $X$?

• You might like to review descriptions of what a random variable is.
– whuber
Jan 6, 2015 at 19:51
• Using your image, the tickets read 1 to 6 and I pick one of them to get an X. Jan 6, 2015 at 20:57
• OK, maybe we're ready to clarify this question. What do you mean by "population" in this context? To what kind of thing does "individual element" refer? What does "just $1$" mean?
– whuber
Jan 6, 2015 at 21:00
• If you're thinking of X as returning one of six possible values, then it is an inappropriate model for a population of only 1's...that would require a different random variable, one that only returns the value 1. Now, its possible that if you took N measurements of X, you could get all 1's...but that would be a sample of the population of possible numbers. The population itself would be a collection of numbers 1 - 6. Of course, if your population is infinite, you could have an infinte number of 1's and say, 10 of the other numbers...its almost the same as having all 1
– user31668
Jan 7, 2015 at 4:18

The range of a real-valued random variable $X$ (which is a real-valued function) is in principle defined over all $\mathbb R$ (if the random variable is discrete, its range is not $\mathbb R$, but the same approach applies).

But what links the "population" under study and $X$ is the probability distribution, which is a function that describes some characteristics of the range of $X$.

When we therefore define that the probability (density or mass) function, takes the value $0$ for some interval / or set of the range of $X$, this signifies that the population, an aspect of which $X$ models and maps to numeric values, does not take these values for which the density/mass function is zero.

(side-note: strictly speaking "has probability zero of taking these values" -which even more strictly speaking does not make their appearance impossible, only improbable - but I challenge any measure-theorist to come up with an intuitive and practically useful exposition of the difference. It is indispensable for the internal consistency and theoretical validity of the mathematical system, but that's all. But see also @whuber's comment regarding a different and more useful aspect of the distinction "improbable/impossible". End of side-note).

But for every interval/value for which we define a strictly positive density/mass function, it follows that these are values in the range of $X$ that correspond (through the mapping rule that the function $X$ is), to states that members of the population may found themselves into (or so we assume).

Given this general approach, I suspect that when you say that $X \in \{1,..,6\}$, you imply that to each of these six values, strictly positive probability mass is allocated. So the population viewed as a whole, does take these six values.

But maybe you may be asking something a little different : "Does this mean that each member of the population separately can possibly take all of these six values"?

Hmm, what this would imply: Assume that there are only six human-eye colors in the world, and no people have different eye-color per eye. We map each eye-color to one of the six numerical values, and we want to find the proportion of each eye-color in the population. How many eye-colors characterize a single human? Obviously, just one, i.e. conceptually, each human has one eye-color. But to the degree that we do not possess any other relevant information (this is the unconditional probability), is it possible to have any one of them?

The answer touches on how one defines and uses the concept of "statistical population" - a difficult concept. If we answer "no", then in what sense humans belong to the same population as regards their eye color? If 100 humans here can have only the colors A,B,C, and 100 humans there can have only the colors A,B,D, is it valid to "group" them into "one population"?

In other words: should the concept of statistical population includes, as a defining property, the homogeneity (in the sense of possible values/states) of its members as regards the characteristic we want to study?

And what may be the relation of the above to the concept of "identically / non-identically distributed" random variables? When we say "non-identically distributed random variables", are we referring to a different distribution (e.g. same distribution family-different parameters) or possibly also to different ranges with non-zero probability?

• The distinction between having a probability (or probability density) of zero and being impossible comes to the fore in statistics (rather than measure theory). Statistics models the world with families of probability distributions. An impossible value is impossible for all members of a family, whereas a value $x$ where a particular probability (or density) $f$ is zero is one that would provide an arbitrarily large amount of evidence against $f$ (in favor of any other density $g$ with $g(x)\ne 0$).
– whuber
Jan 6, 2015 at 20:01
• @whuber It appears you are invoking the imperfection of human knowledge -"specifying a distribution on a phenomenon does not mean we really know what is possible and what is not with absolute certainty". Granted, and I agree of course. But assuming correct specification, as we so often do, then I was just referring to measure theory and its preoccupation with the distinction about whether the various statements, results etc hold "almost surely". Jan 6, 2015 at 20:34
• I am not making a philosophical statement. But these remarks about sets of measure zero are tangential points. Of greater concern is the potential confusion created at the outset by conflating $X$--referred to as a random variable in the question--with its distribution. That makes it difficult either to understand or to agree with any of the ensuing material, because it is all predicated on that initial solecism.
– whuber
Jan 6, 2015 at 20:47
• @whuber Early on, my answer contains the phrase: "...the probability distribution, which is a function that describes some characteristics of the range of $X$." I do not see how this conflates $X$ with its distribution. Jan 6, 2015 at 20:49
• You are right; I apologize. I was prepared to quote you ("$X$ is the probability distribution") but upon looking more closely I realized how erroneous that would be, taken out of its context! I can only plead that somehow I had overlooked that context and focused only on what remained, which does not do you justice. In my defense I will just say that by the time I reached that section I was struggling to understand phrases like "that the suitable in each case number set includes", which take considerable parsing and still don't quite make sense. Perhaps simplifying the language would help.
– whuber
Jan 6, 2015 at 20:54