# What is $X$, it doesn't seem to be a set in the usual mathematical sense, even though it's often referred to as one [duplicate]

It's often used as a "set of values", however, it's not a set as there's no repetition in a set.

Also operations such as $X^2$ are carried out, which will apply the operation to each element in the "set".

So I'm wondering what this $X$ is actually referred to as, whether references to it as a set are informal, or if there's some different meaning of a set in statistics.

For example $( X - \bar{X})^2$

• Do you have an example of this use of $X$? – Juho Kokkala Sep 27 '17 at 16:44
• @JuhoKokkala i've added an example – baxx Sep 27 '17 at 17:13
• One can think of several different senses in which a random variable might be referred to as a "set of values" or being closely related to such a set. In the formal setting where $X:(\Omega,\mathfrak{S},\mathbb{P})\to\mathbb{R}$ is a measurable function, these sets might be an event of the form $X^{-1}(A)$ where $A$ is an interval of possible values of $X$, for instance. Or "set" might refer to the range (aka "support") of $X$. Or possibly its domain $\Omega$. Or maybe the sets in the sub-sigma algebra of $\mathfrak S$ generated by $X$. Do you have a specific quotation or context in mind? – whuber Sep 27 '17 at 21:24
• I recall reading that "set" was the single word with the highest number of meanings in the English language. I don't know if that's true, but it certainly has a lot; e.g. en.oxforddictionaries.com/definition/set . Your statement that "there's no repetition in a set" assumes some particular - unstated - meaning of the word. – user20637 Sep 28 '17 at 8:05
• Restricting to "in the ... mathematical sense" is still very general, and does not necessarily imply unique values e.g. encyclopediaofmath.org/index.php/Set It might help if you gave your definition of the word, and why you think that definition applies to the usage you quote. – user20637 Sep 28 '17 at 8:15

Let $X$ be the random variable from the die throwing. It can attain values $1,2,3,4,5,6$. If we square it ($Y = X^2$), then $Y$ can attain values $1,4,9,16,25,36$. However, when zou throw a die, you only get one outcome - one number, not the whole set of possible numbers.