what is meaning of "independent values of a random variable"? I need some help with basic statistics terminology. Could someone please explain in layman terms the meaning of "independent values" regarding a random variable?
Perhaps a six-sided die (with sides numbered 1 to 6) is good to use as an example. 
Assume a random variable is the time function x defined as the number resulting from the throw of the die, when the die is thrown multiple times.
If we throw the die twice, what are the independent values of the time function x?
Is it the set of values 1,2,3,4,5,6,1,2,3,4,5,6 (so, there are 12 independent values?)?
Or, if the result of the two die throws is 2 and 5, is the set of independent values for x equal to 2,5 (so, there are 2 independent values?)?
Or, something else?
 A: Two events $E_1$ and $E_2$ are called "independent" if knowing the result of one event gives you no information about the other, that is, the probability that $E_2$ happens is unchanged by the knowledge of whether or not $E_1$ happens, and vice versa. Formally this is written as the four conditions:
$P(E_1|E_2) = P(E_1)$
$P(E_1|\neg E_2) = P(E_1)$
$P(E_2|E_1) = P(E_2)$
$P(E_2|\neg E_1) = P(E_2)$
In fact, using $P(E_1|E_2) = \frac{P(E_1 \cap E_2)}{P(E_2)}$ and $P(\neg E) = 1-P(E)$, assuming none of the relevant probabilities in the denominators are zero, one can see that these conditions are each equivalent to $P(E_1)P(E_2) = P(E_1 \cap E_2)$, which is the usual definition of independence of the two events.
Two random variables are then independent if each event in terms of one random variable is independent from each event in terms of the other. For example, if $X$ is the roll of the first die and $Y$ is the roll of the second, we would expect the events $\{X>3\}$ and $\{Y \text{ is even}\}$ to be independent.
A: The set of independent values are just the possible outcomes of two die rolls:
$\{(1,1),(1,2).....(5,6),(6,6)\}$ The "independent" part of this really only means that the value of the first die roll does not affect the value of the second die roll. 
An example of where this is not the case, i.e. not independent throws, would be the an unusual situation like below:


*

*If the first die throw is an even number, the second throw will be an odd number

*Vice versa


In that case, the set of dependent values would be:
$\{(1,2), (1,4), (1,6), (2,1), (2,3), (2,5)....(5,6),(6,5)\}$
