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The Statistical Drake Equation

机译：统计Drake方程

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摘要

We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: (1) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. (2) The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. (3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed "Maccone distribution" by Paul Davies. (4) DATA ENRICHMENT PRINCIPLE. It should be noticed that ANY positive number of random variables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation, we call the "Data Enrichment Principle," and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. Finally, a practical example is given of how our statistical Drake equation works numerically. We work out in detail the case, where each of the seven random variables is uniformly distributed around its own mean value and has a given standard deviation.
机译：我们提供了Drake方程的统计概括。从七个正数的简单乘积，德雷克方程现在变成了七个正数随机变量的乘积。我们称其为“统计德雷克方程”。然后得出该变换的数学结果。我们的结果证明基于统计学的中心极限定理（CLT）。用宽松的术语来说，CLT指出，任意数量的独立随机变量的总和接近高斯（即正态）随机变量，每个变量可以任意分配。这称为CLT的Lyapunov形式，或CLT的Lindeberg形式，这取决于在各种概率分布的第三阶矩上假设的数学约束。总之，我们表明：（1）新的随机变量N产生对数星系中的通信文明，并遵循LOGNORMAL分布。因此，该对数正态分布的平均值为Drake方程中的常态N。还找到了标准偏差，模式和该对数正态N的所有时刻。 （2）现在，普通Drake方程中的七个因子变为七个正随机变量。每个随机变量的概率分布可以是任意的。所谓的李雅普诺夫（Lyapunov）或林德伯格（Lindeberg）形式的CLT（都没有假定因子分布相同）允许这样做。换句话说，通过允许每个因子的任意概率分布，CLT将“转换”为我们的统计Drake方程。当然，这在物理上既现实又实用。 （3）然后应用我们的统计Drake方程。星系中任何两个相邻且通讯的文明之间的（平均）距离可以显示为与N的立方根成反比。然后，在我们的方法中，该距离成为一个新的随机变量。我们推导了相关的概率密度函数，该函数以前似乎是未知的，并被保罗·戴维斯（Paul Davies）称为“ Maccone分布”。 （4）数据丰富原则。应该注意的是，统计Drake方程中任何正数的随机变量都与CLT兼容。因此，只要科学家能够了解关于每个因素的更完善的科学知识，我们的概括就可以在将来添加更多因素。这种为统计Drake方程中的更多未来因素腾出空间的能力，我们称之为“数据丰富原理”，我们将其视为在天体生物学和SETI领域获得更深刻的未来成果的关键。最后，给出一个实际例子，说明我们的统计Drake方程如何进行数值运算。我们详细研究了以下情况：七个随机变量中的每个变量均围绕其平均值均匀分布，并具有给定的标准偏差。

著录项

  • 来源
    《Acta astronautica》 |2010年第12期|p.1366-1383|共18页
  • 作者

    Claudio Maccone;

  • 作者单位

    Technical Director of the International Academy of Astronautics (IAA) and Co-Chair, SETI Permanent Study Croup of the IM Via Martorelli 43, 10155 Torino (Turin), Italy;

  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

    drake equation; statistics; SETI;

    机译：德雷克方程统计;塞蒂;

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