Geodesic
On the Structural Properties of the Conditional Distributions
Canadian mathematical bulletin, Tome 10 (1967) no. 2, pp. 239-245

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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. 
Mathai, A.M. On the Structural Properties of the Conditional Distributions. Canadian mathematical bulletin, Tome 10 (1967) no. 2, pp. 239-245. doi: 10.4153/CMB-1967-022-2
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