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Binomial mixtures and finite exchangeability. (English) Zbl 0759.60010

Let \(\{h(i| p)\}^ n_{i=0}\) be the binomial distribution with parameter \(p\). If \(p\) moves from 0 to 1, the point \((h(0| p),\dots,h(n| p))\) traces a curve \(B_ n\) in the simplex \(T_ n=\{(x_ 0,\dots,x_ n)\): \(\sum^ n_{i=0}x_ i=1\) and \(x_ i\geq 0\) for all \(i\}\). Let \(\text{co} B_ n\) stand for the convex hull of \(B_ n\). The author shows that \[ v(\text{co} B_ n)/v(T_ n)=(n!)^ n\prod^ n_{k=1}1/(2k-1)!, \] where \(v(A)\) stands for the \(n\)- dimensional volume of \(A\). This result is connected with the problem of extending \(n\)-exchangeable sequences of \(\{0,1\}\)-valued random variables to infinite exchangeable sequences.

MSC:

60E05 Probability distributions: general theory
60G09 Exchangeability for stochastic processes
60D05 Geometric probability and stochastic geometry
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