The random variables U and V are jointly independent of W if their joint probability distribution is unaffected by the value of W. For categorical variables, with pjkl denoting the probability of the combination (j, k, l), the condition for U and V to be jointly independent of W, is that
for all j, k, and l, where![joint independence](Images/oree/doc/10.1093/acref/9780199679188.001.0001/acref-9780199679188-math-0308-full.gif)