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By Pierre Pfeuty

Publication by means of Pfeuty, Pierre, Toulouse, Gerard

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Finally, the expectation of the ratio of the sum X1 + . . + XN divided by Xmax reads E X1 + . . + XN =N max{X1 , . . , XN } +∞ = 1 + N (N − 1) 1 0 dz zG(z) 0 1 dy yP (y)[P< (y)]N −2 0 Using the fact that 1 dz zP (yz) . 8 Maximum of Random Variables and Extreme Value Theory 23 we finally obtain E X 1 + . . + XN =N −N max{X1 , . . , XN } +∞ T (y) d[P< (y)]N −1 . 56) is valid for any pdf of positive random variables. In Sect. 1, we use this expression to discuss the case of pdf with power law tails.

Rather, frequencies in past experiments provide valuable information for updating the probabilities assigned to future trials. Despite this connection, probabilities and frequencies are strictly separate concepts. The simplest operational definition of Bayesian probabilities is in terms of consistent betting behavior, which is decision theory in a nutshell. Consider a bookie who offers a bet on the occurrence of outcome E in some situation. The bettor pays in an amount px – the stake – up front. The bookie pays out an amount x – the payoff – if E occurs and nothing otherwise.

17) −∞ The median x1/2 is the halfway point in a graded array of values, in other words half of all values are below x1/2 and half are above: P≤ (x1/2 ) = 1 . 18) For instance, the IQ median is 100. Finally, the most probable value or mode is the value xmp that maximizes P (x): dP (x) |x=xmp = 0 . 19) If several values satisfy this equation, the most probable is the one with the largest P (x). Fig. 3. Distribution skewed to the right with a thick tail, where xmp , x1/2 and x are represented 14 1.

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