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:

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. And the standard deviation of this N lognormal random variable is found also.

In the classical Drake equation one adds the constraint N ? 1 because Humans exist. But in our statistical Drake equation this N ? 1 fact is just a mathematical consequence of the logs. In fact, ln(N) = Gaussian, and the Gaussian density can never be equal to zero, except for the limiting case where its variance tends to infinity. So, it must be N ? 1.

The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable is ARBITRARY. The CLT in the socalled 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.

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 average distance becomes a new random variable. We derived the relevant probability density function, apparently a previously unknown probability distribution.

DATA ENRICHMENT PRINCIPLE. Please notice that ANY positive number of random variables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known by the scientists. This capability to make room for more future factors in the statistical Drake equation we call the “Data Enrichment Principle”, and it is the key to more profound future results in the field of Astrobiology.
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. For instance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 300 billions with a standard deviation of (say) 100 billions. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows that the mean value of the lognormal random variable N is actually of the same order as the classical N given by the ordinary Drake equation, as one might expect from a good statistical generalization.