Abstract
Let L1,L2,...,Ln+1 be the lengths of subintervals created by division of the interval [0,t] by n randomly and independently selected points of this interval. B. de Finetti (1964) proved that F(t;a1,⋯,an+1)=P({L1>a1,⋯,Ln+1>an+1})=t^(−n)(t−a1−...−an+1)^n, where ai≥0,i=1,...,n+1, and a1+...+an+1<t. Let G(t;a1,...,an+1)=P({L1<a1,...,Ln+1<an+1}). We prove that P({a1<L1<b1,...,an+1<Ln+1<bn+1})=F(t;a1,⋯,an+1)G(t−a1−...−an+1;b1−a1,...,bn+1−an+1), where 0≤ai<bi,i=1,...,n+1, and a1+...+an+1<t.
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