Success runs in a sequence of exchangeable binary trials

The random variables ξ1,ξ2,…,ξn are said to be exchangeable (or symmetric) if for each n, P{ξ1⩽x1,…,ξn⩽xn}=P{ξπ(1)⩽x1,…,ξπ(n)⩽xn} for any permutation π=(π(1),…,π(n)) of {1,2,…,n} and any xi∈R, i=1,…,n, i.e. the joint distribution of ξ1,ξ2,…,ξn is invariant under permutation of its arguments. In this study, run statistics are considered in the situation for which the elements of an exchangeable sequence ξ1,ξ2,…,ξn are binary with possible values “1” (success) or “0” (failure). The exact distributions of various run statistics are derived using the fact that the conditional distribution of any run statistic given the number of successes is identical to the corresponding distribution in the independent and identically distributed case.