Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them?
I'm wondering because I find it difficult to come up with a lot of examples of nontrivial pairwise independent random variables. (by 'nontrivial', i mean not mutually independent)
one example (three r.v.):
X = face of dice 1
Y = face of dice 2
Z = X + Y mod 6
another example (three events) from some book:
Throw two three coins. A = the number of heads is even, B = the first two flips are the same, C = the second two flips are heads.
another example:
$A_{ij}$ = dice i and dice j having the same face
($A_{ij}$, $i\neq j$) form a set of pairwise independent events, but the triple ($A_{ij}$, $A_{jk}$, $A_{ki}$) is not mutually independent.
