Pairwise independence

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In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not independent.

[edit] Example

Here is perhaps the simplest example. Suppose X, Y, and Z have the following joint probability distribution:

(X,Y,Z)=\left\{\begin{matrix} (0,0,0) & \mbox{with}\ \mbox{probability}\ 1/4, \\ (0,1,1) & \mbox{with}\ \mbox{probability}\ 1/4, \\ (1,0,1) & \mbox{with}\ \mbox{probability}\ 1/4, \\ (1,1,0) & \mbox{with}\ \mbox{probability}\ 1/4. \end{matrix}\right\}

Then

  • X and Y are independent, and
  • X and Z are independent, and
  • Y and Z are independent, but
  • X, Y, and Z are not independent, since any of them is just the mod 2 sum of the other two, and so is completely determined by the other two. That is as far from independence as random variables can get. However, X, Y, and Z are pairwise independent, i.e. in each of the the pairs (X, Y), (X, Z), and (Y, Z), the two random variables are independent.