# Pairwise independence

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.

## Example

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

[itex](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\}[itex]

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.

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