AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

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UNCLASSIFIED//FOR OFFICIAL USE ONLY

Random variable | ET_Distance between any two neighboring ET Civilizations in Galaxy assuming they are UNIFORMLY distributed throughout the whole Galaxy volume.
Probability distribution | Unnamed (Paul Davies suggested “Maccone distribution”)
Probability density function | $$f_{ET\_Distance}(r) = \frac{3}{r} \cdot \frac{1}{\sqrt{2\pi}\sigma} \cdot e^{-\frac{\left(\ln\left[\frac{6 R_{Galaxy}^2 h_{Galaxy}}{r^3}\right] - \mu\right)^2}{2\sigma^2}}$$
(Defining the positive numeric constant C) | $$C = \sqrt[3]{6 R_{Galaxy}^2 h_{Galaxy}} \approx 28845 \text{ light years}$$
Mean value | $$\langle ET\_Distance \rangle = C e^{-\frac{\mu}{3}} e^{-\frac{\sigma^2}{18}}$$
Variance | $$\sigma_{ET\_Distance}^2 = C^2 e^{-\frac{2\mu}{3}} e^{-\frac{\sigma^2}{9}} \left( e^{-\frac{\sigma^2}{9}} - 1 \right)$$
Standard deviation | $$\sigma_{ET\_Distance} = C e^{-\frac{\mu}{3}} e^{-\frac{\sigma^2}{18}} \sqrt{e^{-\frac{\sigma^2}{9}} - 1}$$
All the moments, i.e. k-th moment | $$\langle ET\_Distance^k \rangle = C^k e^{-k\frac{\mu}{3}} e^{-\frac{k^2\sigma^2}{18}}$$
Mode (= abscissa of the probability density function peak) | $$r_{\text{mode}} \equiv r_{\text{peak}} = C e^{-\frac{\mu}{3}} e^{-\frac{\sigma^2}{9}}$$
Value of the Mode Peak | Peak Value of $f_{ET\_Distance}(r) = C^k e^{-k\frac{\mu}{3}} e^{-\frac{k^2\sigma^2}{18}}$
$$\equiv f_{ET\_Distance}(r_{\text{mode}}) = \frac{3}{C\sqrt{2\pi}\sigma} \cdot e^{-\frac{\mu}{3}} \cdot e^{-\frac{\sigma^2}{18}}$$
Median (= fifty-fifty probability value for ET_Distance) | $$\text{median} = m = C e^{-\frac{\mu}{3}}$$
Skewness | $$\frac{K_3}{(K_4)^{\frac{3}{2}}} = \frac{e^{-\mu} \left( e^{-\frac{\sigma^2}{2}} - 3 e^{-\frac{5\sigma^2}{18}} + 2 e^{-\frac{\sigma^2}{6}} \right)}{C^3 \left( e^{-\frac{8\sigma^2}{9}} - 4 e^{-\frac{5\sigma^2}{9}} - 3 e^{-\frac{4\sigma^2}{9}} + 12 e^{-\frac{\sigma^2}{3}} - 6 e^{-\frac{2\sigma^2}{9}} \right)^{\frac{3}{2}}}$$
Kurtosis | $$\frac{K_4}{(K_2)^2} = e^{-\frac{4\sigma^2}{9}} + 2 e^{-\frac{\sigma^2}{3}} + 3 e^{-\frac{2\sigma^2}{9}} - 6$$
Expression of $\mu$ in terms of the lower ($a_i$) and upper ($b_i$) limits of the Drake uniform input random variables $D_i$ | $$\mu = \sum_{i=1}^{7} \langle Y_i \rangle = \sum_{i=1}^{7} \frac{b_i[\ln(b_i) - 1] - a_i[\ln(a_i) - 1]}{b_i - a_i}$$
Expression of $\sigma^2$ in terms of the lower ($a_i$) and upper ($b_i$) limits of the Drake uniform input random variables $D_i$ | $$\sigma^2 = \sum_{i=1}^{7} \sigma_{Y_i}^2 = \sum_{i=1}^{7} \frac{a_i b_i [\ln(b_i) - \ln(a_i)]^2}{(b_i - a_i)^2}$$

Table 3. Summary of the properties of the probability distribution that applies to the random variable ET_Distance yielding the (average) distance between any two neighboring communicating civilizations in the Galaxy.

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UNCLASSIFIED//FOR OFFICIAL USE ONLY