# Why are we using population symbols ($\sigma^2$, $\mu$, …) instead of sample symbols ($s^2$, $\bar{x}$, …) for Random Variables?

In my Stats course (can't ask prof, it's an online course), we got to random variables and the notation has changed from using $s^2$, $\bar{x}$, etc...to using what we were initially taught are the symbols used to describe population as opposed to sample; namely $\sigma$, $\mu$, etc...

Why are we using population symbols when talking about random variables? Do we not use random variables for prediction or inference? And if we do, then aren't those samples?

I thought I understood that population is the entire data set we're interested in and that inference is done using a sample of that population, because if we already had the entire data set we're interested in there's no inference to be done since we have all the information we need.

It seems to me that if we want to talk about the probability of something happening, we're interested in the behavior of elements we don't currently have in our data set; that should imply that what we currently have is not the population, but a sample.

In this case, why not use population symbols and for example $\frac{1}{n-1}$ instead of $\frac{1}{n}$?

EDIT: For example, when talking about the variance of a random variable, the formula given to us is: $$\sigma^2 = Var(X) = E[X^2] - (\mu_x)^2$$

• Could you edit your post to include a quote where your lecturer uses $\sigma$, $\mu$ etc. in a way you don't understand? – S. Kolassa - Reinstate Monica Jul 4 '16 at 18:56
• @StephanKolassa, I edited my post to include an example. Specifically I just don't understand why we're using the greek laters supposedly reserved for describing population tendencies, instead of using sample symbols. I thought any time we are interested in doing prediction/inference we're basically dealing with a sample; so are random variables not used for any of that? – jeremy radcliff Jul 4 '16 at 19:07
• In your example, although $X$ is a random variable, $\operatorname{Var}(X)$ is a number: it is a property of that random variable. Thus, there seems to be no violation of the convention of using lower case Greek letters for properties and other letters for statistics and random variables. And please beware of the "population" metaphor: although it may be linguistically and conceptually convenient, it simply does not apply in many situations where a process or system is being studied or where random variables are used to model other things besides finite samples. – whuber Jul 4 '16 at 19:15
• Just as English often uses the same word to mean different things--and the listener is assumed to develop the correct understanding from context--so it goes in statistics (and even mathematics). In your example, "Var" is also a statistic: a calculation performed on a sample. ("Mean" also behaves similarly.) In your question, though, "$X$" was explicitly stipulated to be a random variable, not a sample. Thus "Var" is understood to mean a property of $X$ qua random variable: the other sense of "Var" as a statistic just doesn't apply. – whuber Jul 4 '16 at 19:23
• @whuber, I think this makes sense. And you're right with the comment you left under Kostia's answer. I think rigor is what I need because I keep finding myself looking for some structural coherence that I can't find given the lack of rigor in the way the concepts are initially explained to me (not prof's fault, it's supposed to be an intro course, so i think the expectation is that you won't go crazy on your own all the time.). I'll keep in mind what you said and keep learning, I'm sure things will fall in place more and more. – jeremy radcliff Jul 4 '16 at 19:31

You can think of a set of values a random variable can take (called sample space) as a population. For example, if $X\sim\mathcal{N}(\mu,\sigma^2)$ is a normal random variable with mean $\mu$ and variance $\sigma^2$, then the underlying population (sample space) is the set of all real numbers $\mathbb{R}$. Distribution $\mathcal{N}(\mu,\sigma^2)$ is essentially a sampling law: it describes how sample are obtained from the population. If $x_1,\ldots,x_n$ is a sample form $\mathcal{N}(\mu,\sigma^2)$, then the sample mean $\bar{x}$ is an approximation of the "theoretical" mean $\mu$.