## The Arithmetic Mean as Approximately the Most Probable Value a Posteriori Under the Gaussian Probability Law |

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American Mathematical Society angle approach zero arith arithmetic mean artiﬁcial assumption black ball Calcul des Probabilités condition constant a priori constant in small DEFECTIVE HYPOTHESES deﬁne a straight deﬁnitely differential form direct probability error-risk fallacies ﬁrm ﬁrst derivative equal ﬁrst sight function when z<M Gaussian law Gaussian Probability Law Hence hypotheses concerning hypothesis ical II(z II(z)=0 outside bl inches long increase indeﬁnitely increasing function inﬁnitesimal inﬂuence interval bl isolated points largest value Let the probability lie in bl MEAN AS APPROXIMATELY measurements with arithmetic median meter-rod metic mean natural number of measurements physicist Poincare postulate principle priori prob priori probability probability a posteriori probability a priori probable value real number satisﬁed Setting the ﬁrst standpoint of probability symbol for certainty take its maximum tangent theory of probability true value lies University of Texas unknown true value uous function Value A Posteriori white ball zero with increasing