  In this paper, we apply the combinatorial proof technique of Description,
Involution, Exceptions (DIE) to prove various known identities for the joint
cumulant. Consider a set of random variables $S = \{X_1,..., X_n\} $. Motivated
by the definition of the joint cumulant, we define $ \sC(S) $ as the set of
cyclically arranged partitions of $S$, allowing us to express the joint
cumulant of $ S $ as a weighted, alternating sum over $\sC(S)$. We continue to
define other combinatorial objects that allow us to rewrite expressions
originally in terms of the joint cumulant as weighted sums over the set of
these combinatorial objects. Then by constructing weight-preserving,
sign-reversing involutions on these objects, we evaluate the original
expressions to prove the identities, demonstrating the utility of DIE.
