User:Miraceti/sandbox

$$ N = R \cdot f_\mathrm p \cdot n_\mathrm e \cdot f_\mathrm l \cdot f_\mathrm i \cdot f_\mathrm c \cdot L. $$

$$ N = N_\mathrm s \cdot f_\mathrm p \cdot n_\mathrm e \cdot f_\mathrm l \cdot f_\mathrm i \cdot f_\mathrm c \cdot f_\mathrm L, $$

$$ N = \prod_{k=0}^7 D_k, $$

$$ Y = \sum_{k=0}^7 Y_k. $$

$$ \begin{align} \langle D_k \rangle &= \frac{a_k+b_k}{2},\\ \sigma_{D_k} &= \frac{b_k-a_k}{2\sqrt{3}}.\\ \end{align} $$

$$ f_{Y_k}(y)=\frac{\mathrm e^y}{b_k-a_k}\qquad i=k,\ldots, 7\qquad \ln{a_k} \le y \le \ln{b_k}. $$

$$ \begin{align} \langle Y_k \rangle &= \frac{b_k [\ln{b_k}-1]-a_k[\ln{a_k}-1]}{b_k-a_k},\\ \sigma_{Y_k} &= \sqrt{1-\frac{a_{k}b_{k}[\ln{b_k}-\ln{a_k}]^2}{\left( b_k-a_k \right)^2}}.\\ \end{align} $$

$$ f_N(y)=\frac{1}{\left | y \right |}f_Y\left(\ln{y}\right). $$

$$ \mathit\Phi_{Y_k}\left(\zeta\right) = \frac{b_k^{1+i\zeta}-a_k^{1+i\zeta}}{\left( b_k-a_k \right)\left( 1+i\zeta \right)}. $$

$$ \mathit\Phi_Y\left(\zeta\right) = \prod_{k=1}^7 \frac{b_k^{1+i\zeta}-a_k^{1+i\zeta}}{\left( b_k-a_k \right)\left( 1+i\zeta \right)}. $$

$$ \begin{align} f_N(n) &= \frac{1}{n\sqrt{2\pi\sigma^2}}\, \mathrm e^{-\frac{\left(\ln n-\mu\right)^2}{2\sigma^2}},\\ \langle N \rangle &= \mathrm e^{\mu+\sigma^2/2} ,\\ \tilde N &= \mathrm e^{\mu},\\ \hat{N} &= \mathrm e^{\mu-\sigma^2},\\ \sigma_N^2 &= (\mathrm e^{\sigma^2}\!\!-1) \mathrm e^{2\mu+\sigma^2}.\\ \end{align} $$

$$ \begin{align} S_\mathrm{civ} &= \frac{C}{\sqrt[3]{N}},\\ C &= \sqrt[3]{6{R_G}^2h_G}.\\ \end{align} $$

$$ f_{S_\mathrm{civ}}(r) = \frac{3}{\sqrt{2\pi}\sigma r} \mathrm e^{-\frac{\left(\ln{\frac{c^3}{r^3}}-\mu\right)^2}{2\sigma^2}}. $$

$$ \begin{align} \langle S_\mathrm{civ} \rangle &= C \mathrm e^{-\frac{\mu}{3}+\frac{\sigma^2}{18}},\\ {\tilde{S}}_\mathrm{civ} &= C \mathrm e^{-\frac{\mu}{3}},\\ {\hat{S}}_\mathrm{civ} &= C \mathrm e^{-\frac{\mu}{3}-\frac{\sigma^2}{9}},\\ \sigma_{S_\mathrm{civ}}^2 &= C^2 \mathrm e^{-\frac{2\mu}{3}+\frac{\sigma^2}{9}} \left( \mathrm e^{ \frac{ \sigma^2 }{ 9 }} - 1 \right) .\\ \end{align} $$