2 second example typo

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.

1

most general way to generate pairwise independent random variables?

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 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.