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Consider the following standard definition of mutual independence of (discrete) random variables:

"A set of random variables is mutually independent iff for any finite subset $X_1, ..., X_n$ and any finite set of numbers $a_1, ..., a_n$, the events $\{X_1 \le a_1\}, ..., \{X_n \le a_n\}$ are independent events (as defined above)." Taken from Wikipedia.

My question is: Shouldn't we also have specified that we have to have $n>1$ ? Since otherwise for $n=1$ it is not defined what it mean for just one random variable to be independent. As far as I can see, we need at least two random variables (even if they are the same).

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    @DilipSarwate Agreed. Until Hagens edit come into place (as of present I can't detect any changes), could you please give me a reference to a book where I can find a proper definition of mutual independence ?2013-01-02

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The definition refers to independence of events. Indeed, independent is only defined for exactly two events. Thus (apparently) we should rather refer to mutually independent events here, a concept that is defined ("above" in Wikipedia) for arbitrary finite numbers of events, including one, via the multiplication rule, in spite of the section heading speaking only of "more than two events". A set (or as I'd prefer: family) consisting only of a single event is of course always mutually independent.

EDIT: I hope you agree to my recent edits at WP.

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    Sorry I meant to "more than two variables" in WP. How can they be mutually independent if there is only one ? Then I can only one set $A$ and geht only on set $\{ X\in A\}$ - how can that be independent ?2012-12-31
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As far as I can see, the point of the question is that the mathematical definition of "mutually independent" includes some situations that don't fit your intuition of what "independent" should mean. Such things happen fairly often in other parts of mathematics too, because we use ordinary English words that might carry some unwanted connotations. An extreme example is that, in ancient times (perhaps even in medieval times --- I'm not sure about that), "number" meant what we now call "natural number greater than or equal to 2". The idea that 1 is a number still clashes with some people's intuition; if I say "a number of people think I'm a genius" and it turns out that only one person (guess who!) thinks so, was I lying? The idea that 0 is a number is, for some people, even more counter-intuitive. Nevertheless, broader notions of "number", including not only 1 and 0 but negative numbers, real numbers, and complex numbers, have been fruitful enough that mathematicians adopt these meanings and discard the old, more intuitive meaning.

In the case at hand, something similar may be at work; the official definition is (at least) easier to work with than a variant that requires at least two random variables. For example, we can say that, if a set of random variables is mutually independent, then so is any subset.

In addition, though, I think it is possible to explain the official definition in a way that makes intuitive sense. Consider what dependence (the opposite of mutual independence) of a family of random variables should mean intuitively. To me, it means that, if we're given the values of some of the random variables in the family, then this information influences the distribution of some other random variable in the family. Now this will never happen when the family consists of only one random variable. Since we can't have (my intuitive notion of) dependence in this case, it makes sense to declare any family of just one random variable to be mutually independent. (For the same reason, I would consider the empty family of random variables to be mutually independent.)

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    @user26698: If $n=1$ the definition you quoted requires independence for the certain families of events. First, for each $a$, there is the family $\mathcal F_a$ consisting of just one event $\{X_1\leq a\}$. Second, there is the empty family. Now go back to the definition of independence for families of events, and check that the empty family is independent (because the intersection of no events is the whole probability space, whose probability, 1, is the product of the empty family) and so is any family consisting of a single event (because a product of one factor equals that factor).2013-02-13