TBan-I: Difference between revisions
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Given a realization, <math> n(x) </math> is defined as the number of random variables above the minimum <math>E_{\min} </math> such that their value is smaller than <math>E_{\min} +x</math>. This quantity is a random variable, and we are interested in its average value: | Given a realization, <math> n(x) </math> is defined as the number of random variables above the minimum <math>E_{\min} </math> such that their value is smaller than <math>E_{\min} +x</math>. This quantity is a random variable, and we are interested in its average value: | ||
<center><math> \overline{n(x)} = \sum_k k \, \text{Prob}[n(x) = k] </math></center> | <center><math> \overline{n(x)} = \sum_k k \, \text{Prob}[n(x) = k] </math></center> | ||
We use the following identity to sum over <math>k</math>: | |||
<center><math> \sum_{k=0}^{M-1} k \binom{M-1}{k} (A-B)^k B^{M-1-k} = (A-B)\frac{d}{dA} \sum_{k=0}^{M-1} \binom{M-1}{k} (A-B)^k B^{M-1-k} = (M-1)(A-B)A^{M-2} </math></center> | |||
Revision as of 19:55, 30 August 2025
Detour: Extreme Value Statistics
Definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) } :
Given a realization, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) } is defined as the number of random variables above the minimum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min} } such that their value is smaller than . This quantity is a random variable, and we are interested in its average value:
We use the following identity to sum over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} :
