Density above the minimum

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 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} +x} . This quantity is a random variable, and we are interested in its average value:

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 \overline{n(x)} = \sum_k k \, \text{Prob}[n(x) = k] }

The key relation for this quantity is:

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 \text{Prob}[n(x) = k] = M \binom{M-1}{k}\int_{-\infty}^\infty dE \; p(E) [P^>(E) - P^>(E+x)]^{k} P^>(E+x)^{M - k - 1} }

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} :

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 \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} }

to arrive at the form:

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 \overline{n(x)} = M (M-1) \int_{-\infty}^\infty dE \; p(E) \left[P^>(E) - P^>(E+x)\right] P^>(E)^{M-2} }

which simplifies further to:

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 \overline{n(x)} = M \int_{-\infty}^\infty dE \; \left[P^>(E) - P^>(E+x)\right] \frac{d P^>(E)^{M-1}}{dE} = M \int_{-\infty}^\infty dE \; \left[P^<(E+x) - P^<(E)\right] \frac{d Q_{M-1}(E)}{dE} }

Using asymptotic forms: So far, no approximations have been made. To proceed, we use 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 Q_{M-1}(E)\approx Q_M(E)} and its asymptotics:

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 \frac{d Q_{M-1}(E)}{dE} \; dE \sim \exp(\frac{E-a_M}{b_M}) \exp(-\exp(\frac{E-a_M}{b_M})) \frac{dE}{b_M} = P(z) dz }

where 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 z=(E-a_M)/b_M } . The contribution to the integral comes then form the region near 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 a_M} where 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 P^<(E) \sim e^{A(a_M) +A'(a_M) (E-a_M)} } . We can then arrive to:

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 \overline{n(x)} = \left(e^{x/b_M}-1\right) \int_{-\infty}^\infty dz \; e^{2z - e^z} = \left(e^{x/b_M}-1\right) }