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ThmDex – An index of mathematical definitions, results, and conjectures.
Upper bound to the probability that two finite random euclidean real sums equal each other
Formulation 0
Let P=(Ω,F,P) be a D1159: Probability space such that
(i) X1,Y1,,XN,YN:ΩRD are each a D4383: Random euclidean real number on P
Then P(Nn=1XnNn=1Yn)P(Nn=1{XnYn})
Proofs
Proof 0
Let P=(Ω,F,P) be a D1159: Probability space such that
(i) X1,Y1,,XN,YN:ΩRD are each a D4383: Random euclidean real number on P
Let ωΩ be an outcome such that Nn=1Xn(ω)Nn=1Yn(ω). Then Xn(ω)Yn(ω) for at least one n1,,N, for otherwise the sums Nn=1Xn(ω) and Nn=1Yn(ω) would be equal. Thus, by definition of a union, ωNn=1{XnYn}. Since ωΩ was arbitrary, we have the inclusion {Nn=1XnNn=1Yn}Nn=1{XnYn} Result R2090: Isotonicity of probability measure now implies P(Nn=1XnNn=1Yn)P(Nn=1{XnYn}) which is what was required to be shown.