On Equality in Distribution of Ratios $\boldsymbol{X\!{/}(X}{+}\boldsymbol{Y)}$ and $\boldsymbol{Y\!{/}(X}{+}\boldsymbol{Y)}$

Motivated by a classical result in the independent identically distributed (i.i.d.) case for a pair random variables X, Y, we look for a simple sufficient condition, allowing for possible dependence between \(X \mbox{and} Y\), under which the ratios of the components X,Y to their sum are equal in distribution. Our finding is easily extended to random vectors of higher (\(n \geq 2\)) dimensions to show that exchangeability of a finite sequence \(X_1, \cdots, X_n\) is sufficient to guarantee the desired result. Any Archimedian copula can be used as a generator of such random vectors. Our main result is applicable in many Bayesian contexts, where the observations are conditionally i.i.d. given an environmental variable with a prior.