Independence of a random variable jargon Just a bit confused about whenever "Independent" random variable is used in probability/stats videos/books.
When we say an "Independent" random variable, are we referring to a random variable standalone or in relation to another random variable?
E.g.:
Let's say there is a Random variable X that maps the outcome of a dice roll to the real number i.e
Sample space: {1,2,3,4,5,6}
Random variable X:{1,2,3,4,5,6} --> (1,2,3,4,5,6) on real number line.
Can we say that the Random variable X is an independent random variable? as the outcomes of the experiment don't depend on each other. i.e getting a 6 is not dependent on getting a 4 or 3. (might be confusing this with iid)
Or when we talk about the independence of a random variable it is in relation to another random variable within the same sample space.
Let's say there is another random variable Y within the sample space where
Random variable Y :{2,4,6} --> (0), {1,3,5} --> (1) on a real number line
Now random variables X and Y are dependent random variables as P(X and Y) != P(X).P(Y). So, random variable X is not an independent random variable?
Apologies, if this is naive. Just not able to close this out even after spending a lot of time.
 A: Independence always refers to more than one random variable. A single random variable should not be called independent. What you later write about $X$ and $Y$ is correct, however what you earlier wrtite about $X$ alone is not appropriate.
"Can we say that the Random variable X is an independent random variable? as the outcomes of the experiment don't depend on each other. i.e getting a 6 is not dependent on getting a 4 or 3." If you're talking about a single experiment, these can't be independent, as if you're getting a 6, you know that it isn't a 4 or a 3. Note also that this would be a statement not about $X$ on its own, but rather about the two random variables "indicator of 6" and "indicator of 3 or 4"- statements about dependence or independence are always about the relation between two or more RVs. (There is also (in-)dependence of sets such as $\{6\}$ and $\{3,4\}$ given a certain distribution, but this also always refers to two or more sets, and is in fact equivalent to independence of their indicator variables.)
If you mean that the result of $X$ in one experiment does not depend on the result in another experiment, you are actually making a statement about two different experiments modelled better by random variables $X_1$ and $X_2$, which have the same distribution (namely the distribution of your $X$ you had in mind originally) and are independent, hence "i.i.d." (as you suspected).
