LetT= (T1,…,Tk;k≥ 2) be a minimal sufficient statistic for ak-parameter natural exponential model. Consider a partition ofTinto (T1,T2), whereT1= (T1,…,Tr) andT2= (Tr+1,…,Tk; 1 <r≤k). It is shown that cumulants of the conditional distribution ofT1, givenT2=t2, can be computed through the marginal distribution ofT2and the norming constant that makes the model a probability model. The results are illustrated by a few examples.