# 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 $X$? Sep 27, 2017 at 16:44
• @JuhoKokkala i've added an example
– baxx
Sep 27, 2017 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, 2017 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. Sep 28, 2017 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. Sep 28, 2017 at 8:15

It could be a random variable. Random variable is a variable that attains value according to some random process. You can for example throw a die - every time you throw it, you conduct a random experiment and you can get different values for every throw. Now the values - 1,2,3,4,5,6 - could be values, that a random variable can attain. As you can see, the values that the RV can attain form a set. However, every time the experiment is conducted, RV only attains one specific value (realisation of random variable). So Random variable is not a set. It is a mapping. Technical definitions and more examples have already been discussed on Cross validated and I will omit them here. I will provide one example:

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.

https://en.wikipedia.org/wiki/Random_variable

• I don't think this is very clear - a set in the sense that there are no repeated elements? And find more on Wikipedia - could you provide a link? I'm obviously aware of wikipedia
– baxx
Sep 27, 2017 at 17:19
• OK I will provide some examples. But are you confused about the repetition of elements, or about something else? Sep 27, 2017 at 17:26
• I'm confused why you and others refer to this as a set when it most certainly isn't a set in the mathematical set. And given that it isn't a set - what is it
– baxx
Sep 27, 2017 at 17:28
• Ok I wil, reformulate. How familiar are you with measure theory? Sep 27, 2017 at 17:30
• It's not a set - it's a numerical (one-dimensional) function, assigning one number to each outcome of the random experiment in question. Even if the experiment itself is multi-dimensional, the random variable will result in one number. For example - tossing 2 coins - the results are HH,HT,TH,TT. A random variable X may count the number of tails, and so: X(HH)=0, X(HT)=1, X(TH)=1, X(TT)=2. The function X is the random variable (and not the set of values it assumes). Sep 28, 2017 at 20:18