If x, x1,…, xn are independent stochastic variables and if the conditional distribution of x given x1 +…+ xn is known, what can be said about the marginal distributions of x, x1,…, xn? In this paper we will show that if the conditional distribution of x given a subset of x1, x2,…, xn-1, x+ x1 +…+ xn has a certain structural form then x, x1,…, xn are distributed as members of the linear exponential family of distributions and further x1,…, xn are identically distributed.