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The Statistical Fermi Paradox

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In this paper is provided the statistical generalization of the Fermi paradox. The statistics of habitable planets may be based on a set of ten (and possibly more) astrobiological requirements first pointed out by Stephen H. Dole in his book Habitable planets for man (1964). The statistical generalization of the original and by now too simplistic Dole equation is provided by replacing a product of ten positive numbers by the product of ten positive random variables. This is denoted the SEH, an acronym standing for “Statistical Equation for Habitables”. The proof in this paper is based on the Central Limit Theorem (CLT) of Statistics, stating 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 (Lyapunov form of the CLT). It is then shown that: 1. The new random variable NHab, yielding the number of habitables (i.e. habitable planets) in the Galaxy, follows the log- normal distribution. By construction, the mean value of this log-normal distribution is the total number of habitable planets as given by the statistical Dole equation. 2. The ten (or more) astrobiological factors are now 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 the SEH by allowing an arbitrary probability distribution for each factor. This is both astrobiologically realistic and useful for any further investigations. 3. By applying the SEH it is shown that the (average) distance between any two nearby habitable planets in the Galaxy may be shown to be inversely proportional to the cubic root of NHab. This distance is denoted by new random variable D. The relevant probability density function is derived, which was named the "Maccone distribution" by Paul Davies in 2008. 4. A practical example is then given of how the SEH works numerically. Each of the ten random variables is uniformly distributed around its own mean value as given by Dole (1964) and a standard deviation of 10% is assumed. The conclusion is that the average number of habitable planets in the Galaxy should be around 100 million ±200 million, and the average distance in between any two nearby habitable planets should be about 88 light years ±40 light years. 5. The SEH results are matched against the results of the Statistical Drake Equation from reference 4. As expected, the number of currently communicating ET civilizations in the Galaxy turns out to be much smaller than the number of habitable planets (about 10,000 against 100 million, i.e. one ET civilization out of 10,000 habitable planets). The average distance between any two nearby habitable planets is much smaller that the average distance between any two neighbouring ET civilizations: 88 light years vs. 2000 light years, respectively. This means an ET average distance about 20 times higher than the average distance between any pair of adjacent habitable planets. 6. Finally, a statistical model of the Fermi Paradox is derived by applying the above results to the coral expansion model of Galactic colonization. The symbolic manipulator "Macsyma" is used to solve these difficult equations. A new random variable Tcol, representing the time needed to colonize a new planet is introduced, which follows the lognormal distribution, Then the new quotient random variable Tcol/D is studied and its probability density function is derived by Macsyma. Finally a linear transformation of random variables yields the overall time TGalaxy needed to colonize the whole Galaxy. We believe that our mathematical work in deriving this STATISTICAL Fermi Paradox is highly innovative and fruitful for the future.
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222
Claudio Maccone
1. INTRODUCTION TO SETI
SETI is an acronym for “Search for Extra Terrestrial Intelli-
gence”, in which scientific research began only in 1959.
The goal of SETI is to ascertain whether Alien Civiliza-
tions exist in the universe, how far from Earth they exist, and
possibly how much more advanced than mankind they may
be.
The only physical tools available that could help mankind
THE STATISTICAL FERMI PARADOX
JBIS, Vol. 63, pp.222-239, 2010
CLAUDIO MACCONE
Via Martorelli, 43 - Torino (Turin) 10155, Italy.
Email: clmaccon@libero.it and claudio.maccone@iaamail.org
In this paper is provided the statistical generalization of the Fermi paradox. The statistics of habitable planets may be based on
a set of ten (and possibly more) astrobiological requirements first pointed out by Stephen H. Dole in his book Habitable planets
for man (1964). The statistical generalization of the original and by now too simplistic Dole equation is provided by replacing
a product of ten positive numbers by the product of ten positive random variables. This is denoted the SEH, an acronym
standing for “Statistical Equation for Habitables”. The proof in this paper is based on the Central Limit Theorem (CLT) of
Statistics, stating 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 (Lyapunov form of the CLT). It is then shown that:
1. The new random variable NHab, yielding the number of habitables (i.e. habitable planets) in the Galaxy, follows the log-
normal distribution. By construction, the mean value of this log-normal distribution is the total number of habitable planets
as given by the statistical Dole equation.
2. The ten (or more) astrobiological factors are now 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 the SEH by allowing an arbitrary
probability distribution for each factor. This is both astrobiologically realistic and useful for any further investigations.
3. By applying the SEH it is shown that the (average) distance between any two nearby habitable planets in the Galaxy may be
shown to be inversely proportional to the cubic root of NHab. This distance is denoted by new random variable D. The
relevant probability density function is derived, which was named the “Maccone distribution” by Paul Davies in 2008.
4. A practical example is then given of how the SEH works numerically. Each of the ten random variables is uniformly
distributed around its own mean value as given by Dole (1964) and a standard deviation of 10% is assumed. The conclusion
is that the average number of habitable planets in the Galaxy should be around 100 million ±200 million, and the average
distance in between any two nearby habitable planets should be about 88 light years ±40 light years.
5. The SEH results are matched against the results of the Statistical Drake Equation from reference 4. As expected, the
number of currently communicating ET civilizations in the Galaxy turns out to be much smaller than the number of
habitable planets (about 10,000 against 100 million, i.e. one ET civilization out of 10,000 habitable planets). The average
distance between any two nearby habitable planets is much smaller that the average distance between any two neighbouring
ET civilizations: 88 light years vs. 2000 light years, respectively. This means an ET average distance about 20 times higher
than the average distance between any pair of adjacent habitable planets.
6. Finally, a statistical model of the Fermi Paradox is derived by applying the above results to the coral expansion model of
Galactic colonization. The symbolic manipulator “Macsyma” is used to solve these difficult equations. A new random
variable Tcol, representing the time needed to colonize a new planet is introduced, which follows the lognormal distribution,
Then the new quotient random variable Tcol/D is studied and its probability density function is derived by Macsyma. Finally
a linear transformation of random variables yields the overall time TGalaxy needed to colonize the whole Galaxy. We believe
that our mathematical work in deriving this STATISTICAL Fermi Paradox is highly innovative and fruitful for the future.
Keywords: Drake equation, statistics, habitable planets, Fermi Paradox
contact Aliens are the electromagnetic waves that an Alien
Civilization could emit and we could detect. This forces us
to use the largest radiotelescopes on Earth for SETI research
because the larger the collecting area of electromagnetic
radiation the higher is the sensitivity, i.e. the further in space
we can probe. Yet, even by using the largest radiotelescopes
on Earth (e.g. the 305 metre dish at Arecibo) it is not possi-
ble to search for Aliens beyond, say, a few hundred light
years away. This is a very small amount of space around our
223
The Statistical Fermi Paradox
Galaxy, the Milky Way in comparison is about a hundred
thousand light years in diameter. Thus, current SETI can
cover only a very tiny fraction of the Galaxy, and it is not
surprising that in the past 50 years of SETI searches no
Extraterrestrial Civilization have been discovered. Quite sim-
ply, we did not reach far enough.
This demands the construction of much more powerful and
radically new radiotelescopes. Rather than big and heavy metal
dishes, whose mechanical problems greatly hamper SETI re-
search, “software radiotelescopes” are being used, where a
large number of small dishes (ATA = Allen Telescope Array and
ALMA = Atacama Large Millimeter/submillimeter Array) or
even just simple dipoles (LOFAR = Low Frequency Array)
using state-of-the-art electronics and very high-speed comput-
ing can outperform the classical radiotelescopes in many re-
gards. The current aim in this field is the SKA (Square Kilometer
Array), currently being designed and expected to be completed
around 2020.
2. THE KEY QUESTION: HOW FAR ARE THEY?
The key question remains: how far away are they?
Or, more correctly, how far do we expect the nearest Extra-
terrestrial Civilization to be from the Solar System in the
Galaxy?
This question was first addressed in a scientific manner in
1961 by the first experimental SETI radio astronomer: Frank
Donald Drake. He first considered the shape and size of the
Milky Way, which is a spiral galaxy where:
1. The diameter of the Galaxy is (about) 100,000 light
years, (abbreviated ly) i.e., its radius, RGalaxy, is about
50,000 ly.
2. The thickness of the Galactic Disk at half-way from its
centre, hGalaxy, is about 1,600 ly.
3. The volume of the Galaxy may then be approximated as
the volume of the corresponding cylinder, i.e.
2
Galaxy Galaxy Galaxy
VRh
π
=(1)
4. Consider the sphere around the Earth having a radius of
half the distance between Earth and the nearest ET
Civilization, the “ET_Distance”. Its volume is given by
3
4ET_Distance
32
Our _ Sphere
V
π

=

(2)
The assumption used is that all ET Civilizations are equally
spaced from each other in the Galaxy This assumption should
be replaced by more scientifically-grounded assumptions when
more is known about our Galactic Neighbourhood. Further-
more, let N denote the total number of Civilizations now living
in the Galaxy, including mankind. Therefore N 1 since one
Civilization does at least exist!
Having assumed that ET Civilizations are uniformly spaced
in the galaxy:
1
Galaxy Our _ Sphere
VV
N=(3)
That is, upon replacing both (1) and (2) into (3):
3
24 ET_Distance
32
1
Galaxy Galaxy
Rh
N
π
π



=(4)
Where N and ET_Distance are both unknown.
It is assumed that by resorting to the (rather uncertain)
knowledge on the Evolution of the Galaxy through the last 10
billion years or so, an approximate value for N can be com-
puted.
Then, equation (4) is solved for the ET_Distance thus ob-
taining the average distance between any pair of neighbouring
civilizations in the galaxy.
2
3
33
6
ET_Distance( ) Galaxy Galaxy
Rh C
NNN
==
(5)
where the positive constant C is defined by
2
36 28845 light years
Galaxy Galaxy
CRh=≈ (6)
Equations (5) and (6) are the starting point in understanding
the origin of the Drake equation (see Section 3).
Three different numerical cases of the distance law (5) are of
interest:
1. Mankind exists, so N is not smaller than 1. Suppose that
mankind is alone in the Galaxy, i.e., that N = 1. The
distance law (5) gives the distance to the nearest
civilization as the constant C, i.e., 28,845 light years.
This is about equivalent to the distance between Mankind
and the centre of the Galaxy (i.e. the Galactic Bulge).
This result suggests that, if no extraterrestrial civilization
is found in the outskirts of the Galaxy close to Earth,
then the search should concentrate around the Galactic
Centre. And this is indeed what is happening, i.e., many
SETI searches are pointing antennas towards the Galactic
Centre, looking for beacons (e.g. Ref. [1]).
2. Suppose that N = 1000, i.e. assume there are about a
thousand extraterrestrial communicating civilizations in
the whole Galaxy now. Then equation (5) yields an
average distance of 2,885 light years which is beyond
the capability of most radiotelescopes on: hence the
need to build larger radiotelescopes, like ALMA, LOFAR
and the SKA.
3. Suppose finally that N = 1000000, i.e., there are a million
communicating civilizations in the Galaxy. Equation (5)
yields an average distance of 288 light years. This is
within the (upper) range of distances that our current
radiotelescopes may reach for SETI searches, and that
justifies all SETI searches that have been done so far in
the first fifty years of SETI (1960-2010).
In conclusion, interpolating the results from equation (5)
of the three special cases of N, yields the following key
diagram of the average ET distance vs. the assumed number
of communicating civilizations, N, in the Galaxy (Fig. 1).
This is called the DISTANCE LAW, i.e., the average dis-
tance (plot along the vertical axis in light years) versus the
number of communicating civilizations assumed to exist in
the Galaxy.
224
Claudio Maccone
3. COMPUTING N USING THE DRAKE EQUATION
In the previous section, the problem of finding how close the
nearest ET Civilization may be was “solved” by reducing it to
the computation of N, the total number of Extraterrestrial Civi-
lizations now existing in this Galaxy. In this section the Drake
equation is used to estimate the numerical value of N. The
author believes that the best introductory description of the
Drake equation is given by Carl Sagan (Ref. [2]) and is repro-
duced below unabridged.
But is there anyone out there to talk to? With a third or a
half a trillion stars in our Milky Way Galaxy alone, could
ours be the only one accompanied by an inhabited planet?
How much more likely it is that technical civilizations are a
cosmic commonplace, that the Galaxy is pulsing and hum-
ming with advanced societies, and, therefore, that the near-
est such culture is not so very far away – perhaps transmit-
ting from antennas established on a planet of a naked-eye
star just next door. Perhaps when we look up the sky at
night, near one of those faint pinpoints of light is a world on
which someone quite different from us is then glancing idly
at a star we call the Sun and entertaining, for just a moment,
an outrageous speculation.
It is very hard to be sure. There may be several impediments
to the evolution of a technical civilization. Planets may be rarer
than we think. Perhaps the origin of life is not so easy as our
laboratory experiments suggest. Perhaps the evolution of ad-
vanced life forms is improbable. Or it may be that complex life
forms evolve more readily, but intelligence and technical socie-
ties require an unlikely set of coincidences – just as the evolu-
tion of the human species depended on the demise of the
dinosaurs and the ice-age recession of the forests in whose trees
our ancestors screeched and dimly wondered. Or perhaps civi-
lizations arise repeatedly, inexorably, on innumerable planets
in the Milky Way, but are generally unstable; so all but a tiny
fraction are unable to survive their technology and succumb to
greed and ignorance, pollution and nuclear war.
It is possible to explore this great issue further and make a
crude estimate of N, the number of advanced civilizations in the
Galaxy. We define an advanced civilization as one capable of
radio astronomy. This is, of course, a parochial if essential
definition. There may be countless worlds on which the inhab-
itants are accomplished linguists or superb poets but indifferent
radio astronomers. We will not hear from them. N can be
written as the product or multiplication of a number of factors,
each a kind of filter, every one of which must be sizable for
there to be a large number of civilizations:
Ns, the number of stars in the Milky Way Galaxy;
fp, the fraction of stars that have planetary systems;
ne, the number of planets in a given system that are
ecologically suitable for life;
fl, the fraction of otherwise suitable planets on which life
actually arises;
fi, the fraction of inhabited planets on which an intelligent
form of life evolves;
fc, the fraction of planets inhabited by intelligent beings
on which a communicative technical civilization
develops; and
fL, the fraction of planetary lifetime graced by a technical
civilization.
Written out, the equation reads
NNsfpneflfifcfL=⋅(7)
All of the f’s are fractions, having values between 0 and 1;
they will pare down the large value of Ns.
To derive N we must estimate each of these quantities. We
know a fair amount about the early factors in the equation, the
number of stars and planetary systems. We know very little
about the later factors, concerning the evolution of intelligence
or the lifetime of technical societies. In these cases our esti-
mates will be little better than guesses. I invite you, if you
disagree with my estimates below, to make your own choices
and see what implications your alternative suggestions have for
the number of advanced civilizations in the Galaxy. One of the
great virtues of this equation, due to Frank Drake of Cornell, is
that it involves subjects ranging from stellar and planetary
astronomy to organic chemistry, evolutionary biology, history,
politics and abnormal psychology. Much of the Cosmos is in
the span of the Drake equation.
Fig. 1 Distance law, i.e. average
distance between two nearby
civilizations vs assumed number of
civilisation in the Galaxy (that is, N in
the Drake equation).
225
The Statistical Fermi Paradox
We know Ns, the number of stars in the Milky Way Galaxy,
fairly well, by careful counts of stars in a small but representa-
tive region of the sky. It is a few hundred billion; some recent
estimates place it at 4 x 1011. Very few of these stars are of the
massive short-lived variety that squander their reserves of ther-
monuclear fuel. The great majority have lifetimes of billions or
more years in which they are shining stably, providing a suit-
able energy source for the energy and evolution of life on
nearby planets.
There is evidence that planets are a frequent accompaniment
of star formation: in the satellite systems of Jupiter, Saturn and
Uranus, which are like miniature solar systems; in theories of
the origin of the planets; in studies of double stars; in observa-
tions of accretion disks around stars; and is some preliminary
investigations of gravitational perturbations of nearby stars1.
Many, perhaps even most, stars may have planets. We take the
fraction of stars that have planets, fp, as roughly equal to 1/3.
Then the total number of planetary systems in the Galaxy
would be Ns fp ~ 1.3 x 1011 (the symbol ~ means “approxi-
mately equal to”). If each system were to have about ten
planets, as ours does, the total number of worlds in the Galaxy
would be more than a trillion, a vast arena for the cosmic
drama.
In our own solar system there are several bodies that may be
suitable for life of some sort: the Earth certainly, and perhaps
Mars, Titan and Jupiter. Once life originates, it tends to be very
adaptable and tenacious. There must be many different envi-
ronments suitable for life in a given planetary system. But
conservatively we choose ne = 2. Then the number of planets in
the Galaxy suitable for life becomes Ns fp ne ~ 3 x 1011.
Experiments show that under the most common cosmic
conditions the molecular basis of life is readily made, the
building blocks of molecules able to make copies of them-
selves. We are now on less certain grounds; there may, for
example, be impediments in the evolution of the genetic code,
although I think this is unlikely over billions of years of prime-
val chemistry. We choose fl ~ 1/3, implying a total number of
planets in the Milky Way on which life has arisen at least once
as Ns fp ne fl ~ 1 x 1011 , a hundred billion inhabited worlds.
That in itself is a remarkable conclusion. But we are not yet
finished.
The choices of fi and fc are more difficult. On the one hand,
many individually unlikely steps had to occur in biological
evolution and human history for our present intelligence and
technology to develop. On the other hand, there must be quite
different pathways to an advanced civilization of specified
capabilities. Considering the apparent difficulty in the evolu-
tion of large organisms, represented by the Cambrian explo-
sion, let us choose fi x fc = 1/100, meaning that only 1 per cent
of planets on which life arises actually produce a technical
civilization. This estimate represents some middle ground among
the varying scientific options. Some think that the equivalent of
the step from the emergence of trilobites to the domestication
of fire goes like a shot in all planetary systems; others think
that, even given ten or fifteen billion years, the evolution of a
technical civilization is unlikely. This is not a subject on which
we can do much experimentation as long as our investigations
are limited to a single planet. Multiplying these factors to-
gether, we find Ns fp ne fl fi fc ~ 1 x 109, a billion planets on
which technical civilizations have arisen at least once. But that
is very different from saying that there are a billion planets on
which technical civilizations now exist. For this we must also
estimate fL.
What percentage of the lifetime of a planet is marked by a
technical civilization? The Earth has harboured a technical civili-
zation characterized by radio astronomy for only a few decades
out of a lifetime of a few billion years. So far, then, for our planet fL
is less than 1/108, a millionth of a percent. And it is hardly out of
the question that we might destroy ourselves tomorrow. Suppose
this were a typical case, and the destruction so complete that no
other technical civilization – of the human or any other species –
were able to emerge in the five or so billion years remaining before
the Sun dies. Then Ns fp ne fl fi fc fL ~ 10, and, at a given time there
would be only a tiny smattering, a handful, a pitiful few technical
civilizations in the Galaxy, the steady state number maintained as
emerging societies replace those recently self-immolated. The
number N might be even as small as 1. if civilizations tend to
destroy themselves soon after reaching a technological phase,
there might be no one for us to talk with but ourselves. And that we
do but poorly. Civilizations would take billions of years of tortuous
evolution, and then snuff themselves out in an instant of unforgiv-
able neglect.
But consider the alternative, the prospect that at least some
civilizations learn to live with technology; that the contradic-
tions posed by the vagaries of past brain evolution are con-
sciously resolved and do not lead to self destruction; or that,
even if major disturbances occur, they are reveres in the subse-
quent billions of years of biological evolution. Such societies
might live to a prosperous old age, their lifetimes measured
perhaps on geological or stellar evolutionary time scales. If 1
percent of civilizations can survive technological adolescence,
take the proper fork at this critical historical branch point and
achieve maturity, then fL ~ 1/100, N ~ 107, and the number of
extant civilizations in the Galaxy is in the millions. Thus, for all
our concern about the possible unreliability of our estimates of
the early factors in the Drake equation, which involve as-
tronomy, organic chemistry and evolutionary biology, the prin-
cipal uncertainty comes to economics and politics and what, on
Earth, we call human nature. It seems fairly clear that if self-
destruction is not the overwhelmingly preponderant fate of
galactic civilizations, then the sky is softly humming with
messages from the stars.
These estimates are stirring. They suggest that the receipt of
a message from space is, even before we decode it, a pro-
foundly hopeful sign. It means that someone has learned to live
with high technology; that it is possible to survive technologi-
cal adolescence. This alone, quite apart from the contents of the
message, provides a powerful justification for the search for
other civilizations.”
4. THE DRAKE EQUATION IS OVER-SIMPLIFIED
In the fifty years (1961-2010) since Frank Drake proposed his
1. Carl Sagan was writing these lines back in the 1970’s, when no
extrasolar planets had been discovered. The first such discovery
occurred in 1995, when two Swiss astronomers from the Geneva
Observatory, Michel Mayor and Didier Queloz, working at the
“Observatoire de Haute Provence” in France, discovered the first
extrasolar planet orbiting nearby star 51 Peg. This first extrasolar
planet was hence named 51 Peg B. Many more extrasolar planets
have been discovered around nearby stars ever since. As of April
2009, 347 extrasolar planets (exoplanets) are listed in the Extrasolar
Planets Encyclopaedia [1] maintained by Jean Schneider of the Paris
Observatory at Meudon.
226
Claudio Maccone
equation, a number of scientists and writers have tried to find
out which numerical values of its seven independent variables
are more realistic in agreement with our present-day knowl-
edge. Thus there is a considerable amount of literature about
the Drake equation, and the results obtained by the various
authors differ largely from one another. In other words, the
value of N, that various authors obtained by different assump-
tions about the astronomy, the biology and the sociology im-
plied by the Drake equation, may range from a few tens (in the
pessimist’s view) to some million or even billions in the opti-
mist’s opinion. Much uncertainty is thus affecting our knowl-
edge of N. In all cases, however, the final result about N has
always been a sheer number, i.e., a positive integer number
ranging from 1 to millions or billions. This is precisely the
aspect of the Drake equation that this author regards as “too
simplistic” and improved mathematically [4].
5. THE STATISTICAL DRAKE EQUATION
BY MACCONE (2008)
Consider the first independent variable in the Drake equation
(7), i.e., Ns, the number of stars in the Milky Way Galaxy.
Astronomers estimate that there should be about 350 billion
stars in the Galaxy (difficult considering e.g. the dust clouds
blocking view of the Galactic Bulge in the visible light -
although it can be “seen” at radio frequencies like the famous
neutral hydrogen line at 1420 MHz). So, it cannot be said that
Ns = 350 x 109, or, that the number of stars in the Galaxy is
354,233,321,123.Much more scientific, on the contrary, is to
say that the number of stars in the Galaxy is 350 billion plus or
minus, say, 50 billion (or whatever values the astronomers may
regard as more appropriate, since this is just an example to let
the reader understand the difficulty).
Thus, it makes sense to replace each of the seven independ-
ent variables in the Drake equation (7) by a mean value (350
billion, in the above example) plus or minus a certain standard
deviation (1 billion, in the above example).
By doing so, the statistical Drake equation is created. In
other words, the classical and simplistic Drake equation (7) has
been transformed into an advanced statistical tool for the inves-
tigation of a host of facts not known in detail.
1. Each independent variable in (7) is replaced by a random
variable, labelled Di (from Drake);
2. It is assumed that the mean value of each Di is the same
numerical value previously attributed to the
corresponding independent variable in (7);
3. A standard deviation i
D
σ
is added to each side of the
mean value, that is provided by the knowledge gathered
by scientists in each discipline encompassed by each Di.
Having so done, the next question is: How can the probabil-
ity distribution be found for each Di?
Will it be a Gaussian distribution?
This is a difficult question, for nobody knows the probabil-
ity distribution of the number of stars in the Galaxy, not to
mention the probability distribution of the other six variables in
the Drake equation (7).
There is a brilliant way to get around this difficulty.
The Gaussian distribution can be excluded because each
variable in the Drake equation is a positive random variable,
while the Gaussian applies to real positive and negative
random variables only. Then, one might consider the large
class of well-studied and positive probability densities called
“the gamma distributions,” but it is then unclear why one
should adopt the gamma distributions and not any other one.
The solution to this apparent conundrum comes from Shan-
non’s Information Theory and a theorem that he proved in
1948: “The probability distribution having maximum en-
tropy (= uncertainty) over any finite range of real values is
the uniform distribution over that range”.
Assume that each of the seven Di in (7) is a uniform random
variable, whose mean value and standard deviation is known by
the scientists working in the respective field (let it be as-
tronomy, or biology, or sociology). Notice that, for such a
uniform distribution, the knowledge of the mean value i
D
µ
and
of the standard deviation i
D
σ
automatically determines the
range of that random variable in between its lower (called ai)
and upper (called bi) limits: in fact these limits are given by the
equations
3
3
ii
ii
iD D
iD D
a
b
µ
σ
µ
σ
=−
=+
(8)
(the factor 3 in the above equations comes from the defini-
tions of mean value and standard deviation: (see equations
(12), (15) and (17) in Ref. [4] for the relevant proof). So the
uniform distribution of each random variable Di is perfectly
determined by its mean value and standard deviation, and so
are all its other properties.
Since everything is known about each uniformly distributed
Di, what is the probability distribution of N, given that N is the
product (7) of all the Di?
In other words, not only is the analytical expression of the
probability density function of N required, but it should be
related by its mean value
µ
N to all mean values i
D
µ
of the Di,
and its standard deviation
σ
N to all standard deviations i
D
σ
of
the Di.
This is a difficult problem which has occupied the Author’s
mind for over a decade.
To the best of this author’s knowledge, it is impossible to
find an analytic expression for any finite product of uniform
random variables Di. This result is proven in Sections 2 to 3.3
of Ref. [4].
6. SOLVING THE STATISTICAL DRAKE
EQUATION BY VIRTUE OF THE CENTRAL
LIMIT THEOREM (CLT)
The solution to the problem of finding the analytical expression
for the probability density function of N in the statistical Drake
equation was found by this author in September 2007. The key
steps are the following:
1. Take the natural logarithms of both sides of the statistical
Drake equation (7). This changes the product into a sum.
2. The mean values and standard deviations of the logs of
the random variables Di may all be expressed analytically
in terms of the mean values and standard deviations of
the Di.
227
The Statistical Fermi Paradox
3. Recall the Central Limit Theorem (CLT) of statistics,
stating that if there is a sum of independent random
variables, each of which is arbitrarily distributed (hence,
also including uniformly distributed), then, when the
number of terms in the sum increases indefinitely the
sum of the random variables tends to a Gaussian
distribution.
4. Thus, the natural log of N tends to a Gaussian.
5. N tends to the log-normal distribution.
6. The mean value and standard deviations of this log-
normal distribution of N may all be expressed analytically
in terms of the mean values and standard deviations of
the logs of the Di already found previously. This result is
fundamental.
All the relevant equations are summarized in Table 1 (repro-
duced from [4]).
To sum up, not only is it shown that N approaches the known
lognormal distribution for an infinite number of factors in the
statistical Drake equation (7), but it enables further applica-
tions by removing the condition that the number of terms in the
product (7) must be finite.
This possibility of adding any number of factors into the
Drake equation (7) was not envisaged by Frank Drake back in
1961, when “summarizing” the evolution of life in the Galaxy
in seven simple steps. But today, the number of factors in the
Drake equation should be increased: for instance, there is no
mention in the original Drake equation of the possibility that
asteroidal impacts might destroy life on Earth at any time, and
this is because the demise of the dinosaurs at the K/T impact
was not understood until 1980.
In practice, the number of factors should be increased to
produce better estimates of N as scientific knowledge increases.
We term this the “Data Enrichment Principle” and believe it
should be the next important goal in the study of the statistical
Drake equation.
In the next section a numerical example is given showing
how the statistical Drake equation works in practice.
7. AN EXAMPLE ILLUSTRATING THE
STATISTICAL DRAKE EQUATION
To understand how the statistical Drake equation works in
practice, consider Table 2. The first column on the left lists the
seven input sheer numbers, the input mean values are given in
the middle column. Finally the right hand column lists the
seven input standard deviations.
The bottom line is the classical Drake equation (7). We see
that, for this particular set of seven inputs, equation (7) predicts
a total of 3500 communicating extraterrestrial civilizations
existing in the Galaxy right now.
The statistical Drake equation, however, provides a much
more articulated answer than just the above sheer number N =
3500. Coding the statistical Drake equation for a given set of
seven input mean values plus seven input standard deviations
into MathCad, yields for N the lognormal distribution (thin
curve) plotted in Fig. 2. The peak of this thin curve (i.e. the
mode) falls at about
2
mode peak 250nnee
µσ
≡= ≈
(Equation (99) of Ref. [4]), while the median is
nmedian e
µ
1740
Although these are smaller in value than N, it is N that is the
important value, i.e. the mean value. The thin curve (i.e. the
lognormal distribution arising from the Central Limit Theorem)
in Fig. 2 has a long tail to the right. In other words, it does not
immediately approach zero beyond the peak of the mode. Thus,
when the mean value is computed, it equals
2
24589 559Nee .
σ
µ
=≈
~ 4590 communicating civilizations now in the Galaxy. This is
the important number, and it is higher than the 3500 provided
by the classical Drake equation. Thus, in conclusion, the statis-
tical extension of the classical Drake equation produces higher
predictions for finding an extraterrestrial civilization.
Next consider the standard deviation associated with the
mean value 4590. This yields
2
2
21 11195
Nee e
σ
µσ
σ
=−=
and so the expected number for N may actually be higher than
the 4590 provided by the mean value. The “upper limit of the
one-sigma confidence interval” i.e. the sum 4590+11195 =
15,785, yields a higher number still. (Note: the “lower limit of
the one-sigma confidence interval, i.e. -6605, is zero because
the log-normal distribution is defined as positive.) Finally,
although the numerical solution here for the statistical Drake
equation is for a finite number of 7 input factors, this curve “is
well interpolated” by the lognormal distribution (thin curve).
8. FINDING THE PROBABILITY DISTRIBUTION
OF THE ET-DISTANCE BY VIRTUE OF THE
STATISTICAL DRAKE EQUATION
Having solved the statistical Drake equation by finding the log-
normal distribution, the ET-DISTANCE problem can also be
solved using statistics, rather than using the deterministic Dis-
tance Law (5) (Section 2). The new statistical Distance Law
will yield a probability distance, with the relevant mean value
and standard deviation. The Distance becomes a random vari-
able whose probability distribution, mean value and standard
deviation are computed by substituting into (5) a log-normal
distribution for N (ref.[4]).
The probability density for the distance, designated “the
Maccone distribution” by Paul Davies, is:
2
2
3
2
6
2
ET_Distance
31
2
Galaxy Galaxy
Rh
ln r
f(r) e
r
µ
σ
πσ








=⋅ ⋅ (9)
and holds for r 0.
The mean value of the random variable ET_DISTANCE is
expressed as:
2
318
ET_Distance Ce e
µ
σ
=(10)
and the ET_DISTANCE standard deviation as:
228
Claudio Maccone
TABLE 1: Summary of the Properties of the Lognormal Distribution That Applies to the Random Variable N =
Number of ET Communicating Civilizations in the Galaxy.
Random variable N = number of communicating ET civilizations in Galaxy
Probability distribution Log-normal
Probability density function
()
()
()
2
2
2
11 0
2
ln n
N
f
ne(n)
n
µ
σ
πσ
=⋅ ≥
Mean value
2
2
Nee
σ
µ
=
Variance
()
22
22 1
Nee e
µσ σ
σ
=−
Standard deviation
2
2
21
Nee e
σ
µσ
σ
=−
All the moments, i.e. k-th moment
2
2
2
k
kk
Nee
σ
µ
=
Mode (= abscissa of the lognormal peak) 2
mode peak
nnee
µ
σ
≡=
Value of the Mode Peak
2
2
mode
1
2
N
f
(n ) e e
σ
µ
πσ
=⋅
Median (= fifty-fifty probability value for N) median = m = e
µ
Skewness
()
()()()
2
2
2222
63
3
35 3
32
2
4
2
1366
Kee
e
eeee
K
µσ
σ
σσσσ
−−
=+
−+++
Kurtosis
()
222
432
4
2
2
236
Keee
K
σσσ
=+ +
Expression of m in terms of the lower (ai) and upper (bi)
() ()
77
11
11
ii i i
i
ii
ii
blnb alna
Yba
µ
==

−− −

==
∑∑
limits of the Drake uniform input random variables Di
Expression of s2 in terms of the lower (ai) and upper (bi)
() ()
()
2
77
22
2
11
1
i
ii i i
Y
ii ii
ab ln b ln a
ba
σσ
==




==


∑∑
limits of the Drake uniform input random variables Di
22
318 9
ET_Distance 1Ce e e
µσ σ
σ
=−
(11)
All other descriptive statistical quantities, such as moments,
cumulants etc. can be computed from the probability density
(9) and are given in Table 3.
The numerical values that equations (10) and (11) yield for
the input values in Table 2 are, respectively:
2
318 2 670 light years
mean_value
rCee,
µσ
=≈ (12)
and
22
318 9
ET_Distance 1 1 309 light yearsCe e e ,
µσ σ
σ
=−(13)
Figure 3 graphs the ET_Distance probability density (9).
From Fig. 3 it is seen that the probability of finding
ExtraTerrestrials is practically zero up to a distance of about
500 light years from Earth. Then it starts increasing with the
increasing distance from Earth, and reaches its maximum
at
229
The Statistical Fermi Paradox
2
39
mode 1 933 light years
peak
rrCee ,
µσ
−−
≡= ≈ (14)
This is the most likely value of the distance at which we can
expect to find the nearest ExtraTerrestrial civilization.
It is not the mean value of the probability distribution (9) for
fET_Distance(r). In fact, the probability density (9) has an infinite
tail on the right, as clearly shown in Fig. 5, and hence its mean
value must be higher than its peak value. As given by (10) and
(12), its mean value is
2
318 2670 light years
mean_value
rCee
µσ
=≈
TABLE 2: Input Values (i.e. Mean Values and Standard
Deviations) for the Seven Drake Uniform Random Variables
Di. The First Column on the Left Lists the Seven Input Sheer
Numbers That Also Become the Mean Values (Middle Col-
umn). Finally the Last Column Lists the Seven Input Standard
Deviations. The Bottom Line is the Classical Drake Equation
(7).
Ns = 350109
µ
Ns = Ns
σ
Ns = 1109
50
100
fp =
µ
fp = fp 10
100
fp
σ
=
ne = 1
µ
ne = ne 1
3
ne
σ
=
50
100
fl =
µ
fl = fl 10
100
fl
σ
=
20
100
fi =
µ
fi = fi 10
100
fi
σ
=
20
100
fc =
µ
fc = fc 10
100
fc
σ
=
10
10000
10
fL =
µ
fL = fL 10
1000
10
fL
σ
=
N = nsfpneflfifcfL N = 3500
Fig. 2 Comparing the two probability
density functions of the random
variable N found: 1. Numerical
approach from section 3 (thick curve),
and 2. Analytical approach by using
the CLT and the relevant lognormal
approximation (thin curve).
This is the mean distance at which we can expect to find
ExtraTerrestrials.
After having found the above two distances (1933 and 2670
light years, respectively), the range around the mean value of the
distance, within which ExtraTerrestrials are expected to be found
is calculated using the standard deviation from (11) and (13),
22
318 9
ET_Distance 1 1309 light yearsCe e e
µσ σ
σ
=−
More precisely, this is the so-called 1-sigma (distance) level.
Probability theory shows that the nearest ExtraTerrestrial civi-
lization is expected to be located within this range, i.e. within
the two distances of (2670-1309) = 1361 light years and
(2670+1309) = 3979 light years, with probability given by the
integral of fET_Distance(r) taken in between these two lower and
upper limits, that is:
()
3979 light years
ET_Distance
1361 light years 075 75
f
rdr . %≈=
(15)
9. THE “DATA ENRICHMENT PRINCIPLE”
AS THE BEST CLT CONSEQUENCE
UPON THE STATISTICAL DRAKE
EQUATION (ANY NUMBER OF
FACTORS ALLOWED)
Our “DATA ENRICHMENT PRINCIPLE is that “The Higher
the Number of Factors in the Statistical Drake equation, The
Better.”
The CLT lets the random variable Y approach the normal
distribution when the number of terms in the product (7) ap-
proaches infinity. However, the Enrichment principle has more
profound methodological consequences which will be expanded
upon in future publications.
10. HABITABLE PLANETS FOR MANKIND
The search for Habitable Planets for Mankind in the Galaxy is
now considered. How many are there ? And how far from Earth
is the nearest such Habitable Planet?
These topics were investigated for the first time in 1964 by
Stephen H. Dole of the Rand Corporation. Dole [5] used the
230
Claudio Maccone
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 Neighbouring Communicating Civilizations in the Galaxy.
Random variable ET_Distance between any two neighboring ET Civilizations in Galaxy
assuming they are UNIFORMLY distributed throughout the whole
Galaxy volume.
Probability distribution Paul Davies suggested “Maccone distribution”
Probability density function
2
2
3
2
6
2
ET_Distance
31
2
Galaxy Galaxy
Rh
ln r
f(r) e
r
µ
σ
πσ








=⋅ ⋅
Numerical constant C related to the Milky Way size 2
36 28 845 light years
Galaxy Galaxy
CRh ,=≈
Mean value
2
318
ET_Distance Ce e
µ
σ
=
Variance
22
2
22
39 9
ET_Distance 1Ce e e
σσ
µ
σ


=−


Standard deviation
22
318 9
ET_Distance 1Ce e e
µσ σ
σ
=−
All the moments, i.e. k-th moment
2
2
318
ET_Distance kk
kk
Ce e
µ
σ
−⋅
=
Mode (= abscissa of the lognormal peak)
2
39
mode peak
rrCee
µσ
−−
≡=
Value of the Mode Peak Peak Value of
2
318
ET_Distance ET_Distance mode
3
2
f
(r) f (r ) e e
C
µ
σ
πσ
≡=
Median (= fifty-fifty probability value for N)3
median mCe
µ
==
Skewness
()
222
222 22
5
18 6
2
3
33
854 2
22
43999 39
32
43126
ee e e
K
K
Ce e e e e
σσσ
µ
σσσ σσ


−+


=


−−+


Kurtosis
()
22 2
42
4939
2
2
23 6
Keee
K
σσ σ
=++ −
Expression of
µ
in terms of the lower (ai) and
() ()
77
11
11
ii i i
i
ii
ii
blnb alna
Yba
µ
==

−− −

==
∑∑
upper (bi) limits of the Drake uniform input random
variables Di
Expression of σ2 in terms of the lower (ai) and
() ()
()
2
77
22
2
11
1
i
ii i i
Y
ii ii
ab ln b ln a
ba
σσ
==




==


∑∑
upper (bi) limits of the Drake uniform input random
variables Di
231
The Statistical Fermi Paradox
same mathematical structure as the Drake equation (7) in order
to find the number of habitable planets for Mankind in the
Galaxy.
The classical Dole equation is made up by ten factors:
Hab
N NsPpPiPDPM PePBPRPAPL=⋅⋅ ⋅⋅⋅⋅⋅ (16)
Here NHab is the total number of Habitable Planets for
Mankind in the Galaxy, and it is given by the product of the
following input numbers:
1. Ns is the number of stars in the suitable mass range 0.35
to 1.43 solar masses (this is Dole’s assumption about to
the mass of “habitable stars”).
2. Pp is the probability that a given star has planets in orbit
around it.
3. Pi is the probability that the inclination of the planet’s
equator is correct for its orbital distance.
4. PD is the probability that at least one planet orbits within
an ecosphere.
5. PM is the probability that the planet has a suitable mass,
0.4 to 2.35 Earth masses.
6. Pe is the probability that the planet’s orbital eccentricity
is sufficiently low.
7. PB is the probability that the presence of a second star
has not rendered the planet uninhabitable.
8. PR is the probability that the planet’s rate of rotation is
neither too fast nor too slow.
9. PA is the probability that the planet is of the proper age.
10. PL is the probability that, all astronomical conditions
being proper, life has developed on the planet.
11. THE STATISTICAL DOLE EQUATION
The above ten input variables of the classical Dole equation
(16) are now renamed as follows:
1
2
3
4
5
6
7
8
9
10
D
Ns
D
Pp
D
Pi
D
PD
D
PM
D
Pe
D
PB
D
PR
D
PA
D
PL
=
=
=
=
=
=
=
=
=
=
(17)
so that equation (16) may be rewritten as
10
1
Hab i
i
ND
=
=(18)
The same procedure is now applied to (18) as previously
applied to the classical Drake equation.
1. All the input variables on the right-hand side of (18) now
become positive random variables.
2. All these random variables are assumed to be uniformly
distributed with assigned mean values i
D
µ
and standard
deviations i
D
σ
. It can then be shown that assigning
them amounts to assigning the lower and upper limits (ai
and bi, respectively) of each uniform random variable
Di.
3. As a consequence of these assumptions, the total number
of Habitable Planets in the Galaxy, NHab, is also a random
variable with a log-normal distribution.
Equation (18) can now be denoted the statistical Dole equa-
tion.
Applying the same arguments as developed for the Drake
Fig. 3 This is the probability of
finding the nearest ExtraTerrestrial
Civilization at the distance r from
Earth (in light years) if the values
assumed in the Drake Equation are
those shown in Table 2. The relevant
probability density function
fET_Distance(r) is given by equation (9).
Its mode (peak abscissa) equals 1933
light years, but its mean value is
higher since the curve has a long tail
on the right: the mean value equals
in fact 2670 light years. Finally, the
standard deviation equals 1309 light
years:
232
Claudio Maccone
equation leads to the conclusion that the total number of habit-
able planets in the Galaxy follows the log-normal distribution
shown in Table 1.
And by repeating the same arguments developed for the
Drake equation it can be concluded that the distance between
any two nearby habitable planets follows the Maccone distribu-
tion.
12. A NUMERICAL EXAMPLE: A HUNDRED
MILLION HABITABLE PLANETS
EXIST IN THE GALAXY!
Consider the Input variables in Table 4.
These are in principle comparable to the variables in Table 2
for the Statistical Drake equation. The arguments developed by
Dole [5] provide the mean values of each Di.
For the Statistical Dole Equation, values must be assigned to
the ten standard deviations. Values of 1/10 (i.e. 10%) are
assumed adequate for each of the ten standard deviations listed
in Table 4.
Solving via MathCad, the lognormal probability density for
the random variable NHab is shown in Fig. 4. The peak (i.e. the
mode) corresponds to about ten million planets, but the tail is
rather long.
To quantify these remarks, the MathCad code yields the
following numerical values for the two parameters
µ
and
σ
given by the last two columns in both Tables 2 and 4:
1
0
1 76268289631314 10
1 27010132908265 10
Hab
Hab
.
..
µ
σ
=⋅
=⋅
(19)
Then, the mean value of the random variable NHab, is given
by
2
8
21 012 10 100 million
Hab
Hab
Hab
Nee .
σ
µ
==(20)
Thus the statistical treatment of the Dole equation yields
100 million expected Habitable Planets in the Galaxy. This
figure is higher than the 35 million given by the classical Dole
equation, and much higher than the value of the mode (10
million) shown by the lognormal curve in Fig. 4.
The last result, stating that there are about 100 million
Habitable Planets in the Galaxy, is of course good news for the
future “human conquest of the Galaxy”. The standard deviation
around the mean value (20) of the random variable NHab is
given by
2
28
21 2 0 10 200 million
Hab
Hab Hab
Hab
Nee e .
σ
µσ
σ
=−=
(21)
And so, with probability 1-sigma, we might expect the ac-
tual number of Habitable Planets to rise up 100 million plus
200 million = 300 million.
Finally, the median yields a value of
7
median 4 521 10 45 million
Hab
me .
µ
== = ⋅ ≈ (22)
TABLE 4: Input Values (i.e. Mean Values and Standard
Deviations) for the ten Dole Uniform Random Variables Di.
The First Column on the Left Lists the ten Input Sheer Numbers
That Also are the Mean Values (Middle Column). The Last
Column on the Right Lists the ten Input Standard Deviations.
The Bottom Line is the Classical Dole Equation (16). So, the
Number of Habitable Planets in the Galaxy, Given by the
Classical Dole Equation Just as a Sheer Number, is 35 Mil-
lions 171Hundred Thousand and 930.
Ns = 6.448108
µ
Ns = Ns
σ
Ns = 1107
Pp = 1.0
µ
Pp = Pp 10
100
Pp
σ
=
Pi = 0.81
µ
Pi = Pi 10
100
Pi
σ
=
PD = 0.63
µ
PD = PD 10
100
PD
σ
=
PM = 0.19
µ
PM = PM 10
100
PM
σ
=
Pe = 0.94
µ
Pe = Pe 10
100
Pe
σ
=
PB = 0.95
µ
PB = PB 10
100
PB
σ
=
PR = 0.9
µ
PR = PR 10
100
PR
σ
=
PA = 0.7
µ
PA = Pe 10
100
PA
σ
=
PL = 1
µ
PL = PL 10
100
PL
σ
=
NHab = nsPpPiPDPMPePBPRPA PL
NHab = 3.5171930508624 × 107
14. DISTANCE (MACCONE) DISTRIBUTION
OF THE NEAREST HABITABLE PLANET
TO EARTH ACCORDING TO THE
PREVIOUS NUMERICAL INPUT
Assuming that the distribution of Habitable Planets in the
Galaxy is uniform, the distance distribution of the nearest
Habitable Planet to Earth follows the Maccone distribution,
and this is plotted in Fig. 5.
The mean value of the Maccone distribution using the data
given by the Table 4 is:
2
1
318
Hab_Distance 8 8 10 ly 88 ly
Hab Hab
Ce e .
µσ
==
(23)
The relevant standard deviation is
22
1
318 9
Hab_Distance 13910ly 40 ly
Hab Hab Hab
Ce e e .
µσ σ
σ
=−=
(24)
233
The Statistical Fermi Paradox
Thus, with probability 1 sigma, the detection of a Habitable
Planet even at, say, just 88-40 = 48~50 light years from Earth
might be possible.
Figure 5 shows that it is “hopeless” to expect to detect a
Habitable Planet at distances smaller than 25 light years from us,
since the value of the Maccone distribution is practically zero at
such distances. Thus, future Interstellar Spacecraft designers should
keep this lower bound in mind wished they land on Habitable
Planets, rather than just on “any Planet”. Also, the curve reaches its
peak (mode) at about 67 light years from Earth, its median at about
80 light years and, above all, its mean value at 88 light years. The
relevant standard deviation turns out to be about 40 light years,
since the distribution tail is rather “short”.
15. COMPARING THE STATISTICAL DOLE
AND DRAKE EQUATIONS: NUMBER OF
HABITABLE PLANETS VS. NUMBER
OF ET CIVILIZATIONS IN THIS
GALAXY
It is now appropriate to make a comparison between the number
of Habitable Planets and the number of expected ET Civiliza-
tions in the Galaxy, i.e.comparing:
1. the mean value and standard deviation of the total number
of both Habitable Planets and ET Civilizations, and
2. the mean value and standard deviation of their respective
distances from us (of course, under the hypothesis that
both of them are uniformly scattered throughout the
Galaxy).
The result is the following Table 5, clearly showing that how
much “more rare” the ET Civilizations are with respect to the
Habitable Planets. Roughly, one has:
100 20 202
4590
Hab
ET
Nmillion ,
N=≈ (25)
so that the Habitable Planets seem to be 20,000 more frequent
than ET Civilizations, or, only one ET Civilization emerges out
of 20,000 Habitable Planets.
As for the distances, the ratio is the other way round:
Hab_Distance 2670 ly 30 340
ET_Distance 88 ly .=≈ (26)
implying that ETs are, on the average, 30 times further out that
Habitable Planets.
16. SEH, THE “STATISTICAL EQUATION
FOR THE HABITABLES” IS JUST THE
STATISTICAL DOLE EQUATION
Equation (18) will now be denoted by the acronym SEH, for
“Statistical Equation for the Habitables”. This will be clear in
the future papers by the author, where a number possibly higher
than ten will be the new number of independent, uniform
random variables describing the equation inputs.
17. THE CLASSICAL CORAL MODEL
OF GALACTIC COLONIZATION
The following description of the Coral Model of Galactic
Colonization relies on reference [3].
Assume that another civilization has decided to start sending
out spacecraft to colonize other habitable planets. How long
Fig. 4 The lognormal probability
density of the overall number of
Habitable Planets in the Galaxy
(Dole, [5]), implemented by assigning
a 10% standard deviation to all the
ten input random variables listed in
Table 4.
Fig. 5 The Maccone probability
distribution of the distance of the
nearest habitable planet to Earth for
the data of the Table 4 assumed as
inputs to the Statistical Dole Equation
(18).
234
Claudio Maccone
would it take for this civilization to colonize the entire Galaxy?
The answer clearly depends on the civilization’s technologi-
cal capabilities. For example, if it has the technology to build
spacecraft that can travel at speeds close to the speed of light,
then it could add colonies throughout the Galaxy fairly quickly,
since trips between nearby stars would take only a few years.
Perhaps surprisingly, the conclusion is not that much different
if we assume much lower speeds.
Consider a civilization that has nuclear rockets such as
described in Project Orion (1958-63) or Project Daedalus (1973-
78): such rockets might attain speeds of about 10% of the speed
of light (0.1 c). Given that a typical distance between star
system in our region of the Galaxy is about 5 light-years, a
nuclear spacecraft travelling at 10% of the speed of light could
journey from one star system to the next in about 50 years. This
trip would be possible in a human lifetime and might be practi-
cal if the colonizers have found ways to hibernate during the
voyage or if they have somewhat longer life spans than we do
(either naturally or through medical intervention).
After arriving at a new star system, the colonists establish
themselves and begin to increase the population. Once the
population has grown sufficiently, these colonists send their
own pilgrims into space, adding yet more star systems to the
growing civilization. Thus, the process starts at the home star
system and the first few colonies are located within just a few
light-years. These colonies then lead to other colonies at greater
distances, as well as at unexplored locations in between. The
growth tends to expand the empire around the edges of the
existing empire, much like the growth of coral in the sea. For
this reason, this type of colonization model is often called a
coral model of Galactic colonization.
The overall result is a gradually expanding region in which
all habitable planets are colonized. The colonization rate de-
pends on the speed of spacecraft and the time it takes each
colony to start sending their own spacecraft to other stars. For
travel 10% the speed of light and assuming that it takes 150
years before each colony’s population grows enough to send
out more colonists, the calculations in the next section show
that the inhabited region of the Galaxy expands outwards from
the home world at about 1% of the speed of light. Thus, if the
home star is near one edge of the Galactic disk, so that coloniz-
ing the entire Galaxy means inhabiting star systems 100,000
light-years away, the civilization would expand through the
entire Galaxy in about 10 million years. The required time
would be a few million years less if the home star is in a more
central part of the Galaxy.
For an even more conservative estimate, suppose the colo-
nists have rockets that travel at only 1% of the speed of light
and that it takes each new colony 5,000 years until it is ready to
send out additional colonists. Even in this case, the region
occupied by this civilization would grow at a rate of roughly 1/
1000 (0.1 %) the speed of light and the entire Galaxy would be
colonized in 100 million years. This is still a very short time
compared to the time that has been available for civilizations to
arise (4.5 billion years for Humanity), further deepening the
mystery of why we see no evidence that anyone else has done it
by now. This is, of course, the well-known Fermi Paradox.
The overall expansion speed of the empire, vexp, is the ratio
of the average distance among any two nearby stars, D, to the
sum of two times:
1. the time of actual spaceflight from one star to the next
one, tflight, plus
2. the time tcol requested to colonize a planet, i.e. to develop
there a civilization until the time is ripe for one more
spaceflight jump to the next star.
exp
f
light col
D
vk
tt
=+(27)
The factor k is included to take into account the “zigzag”
motion of expansion from one star to the next in three-dimen-
sional space. Bennett and Shostak [3] explain that the purely
numerical factor k would be equal to 1 only if the colonization
was always directed straight outward from the home star. In
reality, the colonists will sometimes go to uncolonized star
systems in other directions, so constant k will accounts for this
zigzag motion in three-dimensional space, we assume that
1
2
k=(28)
The flight time tflight from one star to the next one, and the
corresponding spaceship (average) speed, vss, is
flight
s
s
D
tv
=(29)
Inserting (28) and (29) into (27), yields
TABLE 5: Comparing the Results of the Statistical Dole and Drake Equation Found by
Inputting to Them the Tables 4 and 2, Respectively.
Statistical Dole Equation Statistical Drake Equation
Mean Value of the Habitable Planets in the Galaxy ET Civilizations in the Galaxy
TOTAL NUMBER of ~100 million ~4590
Standard Deviation of the Habitable Planets in the Galaxy ET Civilizations in the Galaxy
TOTAL NUMBER of ~200 million ~11195
Mean Value of the Nearest Habitable Planet Nearest ET Civilization
DISTANCE of ~88 light years ~2670 light years
Standard Deviation of the Nearest Habitable Planet Nearest ET Civilization
DISTANCE of ~40 light years ~1309 light years
235
The Statistical Fermi Paradox
()
ss
exp col s s
s
scol
vD
vD,t,v k
D
vt
=+⋅ (30)
This is the expression for the expansion speed of the empire
throughout the Galaxy.
An immediate consequence of (30) is the Galaxy coloniza-
tion time, denoted TGalaxy, i.e. the overall time that our expand-
ing empire will need to colonize the whole Galaxy. If a civiliza-
tion starts conquering the Galaxy from the outskirts (more or
less like ours), the largest possible amount of time is clearly
given by
22
Galaxy Galaxy
s
scol
Galaxy
exp ss
RR
D
vt
TvkvD
+⋅
==
(31)
It would take less if the conquerors lived nearby the centre
of the Galaxy, so, to be conservative,
21
Galaxy col
Galaxy
ss
Rt
TkvD

=⋅+


or
22
Galaxy Galaxy
col
Galaxy
s
s
RR
t
TkDkv
=⋅+
(32)
Two positive constants a and b are introduced
2200000
220
001
Galaxy
Galaxy
ss
ss
R
a light years
k
R
b million light years
kv
for v . c.
=≈
=≈
(33)
Then, the Galaxy colonization time TGalaxy is
col
Galaxy
t
Tab
D
=⋅ + (34)
18. THE CLASSICAL FERMI PARADOX (1950)
Consider three different numerical test cases of (34) for com-
parison purposes.
1. First case
1
2
01
5
150
ss
col
k
v.c
D
ly
tyr.
=
=
=
=
(35)
Then, (30) and (34) yield, respectively:
3747 0 0125
8
exp
Galaxy
km
v.c
s
T million year.
==
(36)
2. Second, suppose that
1
2
001
5
1000
ss
col
k
v.c
D
ly
tyr.
=
=
=
=
(37)
Then, (30) and (34) yield, respectively:
500 0 001
60
exp
Galaxy
km
v.c
s
T million year.
==
(38)
3. Third, the Human case. i.e. that the habitable planets are
just those planets habitable by humans. Thus, we must apply
the classical Dole equation (18) of the “Habitable Planets for
Man”, which shows that the average distance between planets
habitable by Humans is 84 light years, and not just 5 light
years, as in (35) and (37). In other words, assuming:
1
2
001
84
1000
ss
col
k
v.c
D
ly
tyr.
=
=
=
=
(39)
Then, (30) and (34) yield, respectively:
1339 0 004
22
exp
Galaxy
km
v.c
s
T million year.
==
(40)
So, about 22 million years would be the overall time neces-
sary for Humankind to colonize (unopposed) the whole Milky
Way had Humans spaceships capable of traveling at 1% of the
speed of light and was the average colonization time for every
new planet about 1000 years.
The basic difference between the Humanity expansion model
and the two previous models is the difference in the average
distance among habitable extrasolar planets, It is interesting to
take the limit of (34) for the distance D increasing more and
more, i.e. D → ∞. This yields
21
Gxy
Galaxy
D
s
s
R
lim T kv
αλα
→∞ =⋅ (41)
meaning that the spaceship speed vss plays an increasing role in
the Galaxy colonization when the average distance D increases.
If ETs of a certain “race” can live on fewer planets only, then
they must have much faster spaceships to colonize the Galaxy
than ETs that can live on a variety of planets.
In conclusion, from the above three examples we see that the
time of colonization of the whole Galaxy seems to be of the
order of some tens million year: just a blink compared the
Galaxy age of about 10 billion years, and that is the Fermi
paradox, of course.
236
Claudio Maccone
19. THE STATISTICAL CORAL MODEL
OF GALACTIC COLONIZATION
The goal of this paper is to derive the statistical generalization
of the classical Fermi paradox described in the previous sec-
tion.
Consider first the statistical expansion speed (30) of the
empire, that is rewritten here in the form
()
ss
exp col
s
scol
vD
VD,T k
D
vT
=+⋅ (42)
All random variables are denoted by capitals, and ordinary
(deterministic) variables are denoted by lower-case letters. Thus,
in (42), Vexp, D and Tcol, are random while vss is a real positive,
known parameter.
The (average) spaceship speed, moving from one star to the
next one, is entirely under human (or ET’s) control, and so can
be regarded just as a sheer number rather than a random vari-
able.
On the contrary, the colonization time Tcol is a random
variable, inasmuch as it is unknown what difficulties will be
faced in colonising a new planet, or the time required to de-
velop facilities on this planet in preparation for the next jump.
Finally, D, the (average) distance in between any two nearby
Habitable Planets follows the Maccone distribution. Its prob-
ability density function (pdf) is
2
3
3
2
2
31
2
C
ln d
D
f(d) e
d
µ
σ
πσ








=⋅ ⋅ (43)
For the best choice of the probability density function for
the new (positive) random variable Tcol (which yields the amount
of time needed to colonize a new planet), a lognormal pdf is
adopted because this can be thought of as the product of many
positive random variables. Thus, it isassumed that
2
2
2
11
2
col
t
ln yr
T
yr
f(t) e
yr t
µ
σ
πσ







=⋅⋅ ⋅ (44)
The log-normal pdf can be fully specified so that we know in
advance both its mean value and its standard deviation. Using
Table 1:
2
2
2
2
21
N
Nee
ee e
σ
µ
σ
µσ
σ
=
=−
(45)
Solving this system produces:
2
22
2
21
N
N
N
ln
N
ln .
N
µ
σ
σ
σ


=


+



=+


(46)
Equation (46) is now treated for the lognormally-distributed
new random variable Tcol. Thus, Nmust be replaced by the
mean value of the colonization time,
µ
t_col, and
σ
N must be
replaced by the standard deviation of the colonization time,
σ
t_col Thus, for the colonization time, (46) becomes
2
22
2
21
t_col
t_col t_col
t_col
t_col
ln
ln .
µ
µµσ
σ
σµ


=


+



=+


(47)
Suppose that the mean colonization time equals 1000 years,
with a standard deviation of ±500 yr. Then replacing these two
values (previously divided by yr to make things dimensionally
correct) into (47), yields
2
22
2
2
6 7961835
1 0 4723807
t_col
t_col t_col
t_col
t_col
ln .
ln .
µ
µµσ
σ
σµ


==


+



=+=


(48)
and the relevant log-normal distribution (44) is thus perfectly
determined. This log-normal pdf of the time (in years) needed
to colonize a new planet if one assumes a mean value of 1000
years plus or minus a standard deviation of 500 years is plotted
in the following Fig. 6. In our opinion, much of History on
Earth, such as the colonization of America, is similar.
20. FINDING THE PROBABILITY DISTRIBUTION
OF THE OVERALL TIME NEEDED TO
COLONIZE THE WHOLE GALAXY
In this section the probability distribution (i.e. the probability
density function, or pdf) of the overall time needed to colonize
the Galaxy is calculated. This is given by the positive random
variable TGalaxy defined by (34) where the variable Tcol is now a
positive random variable lognormally distributed as described
in the last section, while D is the random variable yielding the
average distance between any two nearby “Habitable Planets
for Man” and given by the Maccone distribution (43). There-
fore:
col
Galaxy
T
Tab
D
=⋅ + (49)
where all capitals denote random variables while a and b are
the two positive constants defined by (33). So the pdf of the
quotient of two random variables defined by the fraction
lognormal
Maccone
col
T
D
=(50)
must be determined. Standard textbooks about Probability
Theory (e.g. [6]) define the random variable Z, quotient of the
two random variables X and Y
X
ZY
=(51)
237
The Statistical Fermi Paradox
with its pdf given by the integral
() ()
ZXY
f
zyfyz,ydy
−∞
=⋅
(52)
The function fXY
(…, …) is the joint pdf of the two random
variables X and Y. Now, the two random variables
col
X
T
YD
=
=
(53)
are statistically independent of each other. Thus, their joint pdf
fXY (…, …) is the product of the two pdfs, i.e. the lognormal one
(44) and the Maccone one (43), i.e.
() () ()
2
23
3
22
22
11 31
22
col col
t_col D
t_col D
TD T D
tC
ln ln
yr d
t_col D
ft,dftfd
ee
td
µµ
σσ
πσ πσ

 


 

 


 

=⋅
=⋅ ⋅ ⋅
(54)
Rearranging, this becomes
()
2
23
3
22
22
3
2
t_col D
t_col D
col
tC
ln ln
yr d
TD
t_col D
ft,d e e
td
µµ
σσ
πσ σ














=
(55)
This is the joint pdf that must be introduced into the integral
(52). Notice, however, that the integral actually ranges from 0
to infinity only, since both t and d do so. The modulus affecting
y in (52) thus disappears also, and this leaves the computation
of the definite integral
() ( )
0
col col
TTD
D
f
zyfzy,ydy
=⋅
(56)
That is
()
2
23
3
22
22
0
3
2
11
col
t_col D
t_col D
T
t_col D
D
zy C
ln ln
yr y
fz
ye e dy
zy y
µµ
σσ
πσ σ














=
⋅⋅⋅ ⋅
(57)
The integral can be reduced to the Gauss integral, i.e. to the
normalization condition of the ordinary Gaussian or normal
curve, but many steps are required to perform the integration
with respect to y. Using Macsyma, the outcome is the function
of z shown in Eq. (58). As one can see, this function of z is a
complicated mix of exponentials in z through the natural log of
z squared, times a power of z at the denominator, times many
other constants, like the dimensional yr = year.
()
()
()
()
22
_
_
22
22 _
_
_
22
_
22
__
22
_
3 3log( )
39
29
9
39 9log9log
9
9399log
922
_
() 3 2
2
9
Dtcol
Dtcol
Dtcol
Dtcol
col
Dtcol
Dtcol
DtcolDtcol
Dtcol
Cyrz
T
D
Cz
C
Dtcol
fz C e
yr
z
µµ
µµ
σσ
σσ
µµ
σσ
σσ µµ
σσ
π
σσ
++
++
+
+−
+
+−−+
+
÷
×+
(58)
Though the pdf (58) is difficult to handle by hand, it can be
easily handled by Macsyma. Thus, one can prove that it fulfills
indeed the normalization condition
()
0
1
col
T
D
f
zdz
=
(59)
A similar calculation then shows that the mean value of the
quotient of random variables Tcol/D reads
Fig. 6 The lognormal distribution of
the time needed to colonize each new
planet in our statistical extension of
the coral Galactic expansion model,
assuming that, for instance, it takes
1000 years plus or minus 500 years
to colonize that planet.
238
Claudio Maccone
()
22
69 18
18
0
DDt_col t_col
col col
TT
DD
yr
zf z dz e C
σµσ µ
µ
++ +
== ⋅
(60)
The corresponding variance is
22
22
__ _
22
2
293 9
2
1
DD
D t col t col t col
col
T
D
yr
ee
C
σσ
µσ µ σ
σ
++ + +


=−


(61)
Its square root is the relevant standard deviation:
2222_
_2
93 2
_
9
21
DDtcol
tcol
D
tcol
col
T
D
yr
ee
C
σµσ µ
σσ
σ
++ +
+
=⋅
(62)
In order to find the mode of the pdf (58), i.e. the abscissa of
its peak, the first derivative of (58) is calculated with respect to
z and then the resulting equation is set to zero. The two results
are the two abscissas of the minimum of (58), obviously at z =
0, and of the maximum (i.e. the peak, or mode) at
22
__
1
93
mode
DDtcol tcol
yr
ze C
σµσ µ
−+ − +
=⋅
(63)
Finally, the two inflexion points of (58), that is the one
before and the one after the peak (or mode), are found as the
two roots of a quadratic algebraic equation in log(z) that is
found after equalling to zero the second derivative of (58) with
respect to z. Thus, it is found that the abscissas of such two
inflexion points of (58) read, respectively
22 22 2 2
__ __
inflexion_1
9 9 36 3 6 27 18
18
DtcolDtcol DD tcol tcol
z
yr
eC
σσ σσ σµ σ µ
+ ⋅ + ++−+
=
=⋅
(64)
22 22 2 2
__ __
inflexion_2
9 9 36 3 6 27 18
18
=
D tcol D tcol D D tcol tcol
z
yr
eC
σσ σσ σµ σ µ
+⋅++− −
=⋅
(65)
Recall that, if the pdf of a random variable X exists and the
pdf of a new, linearly-transformed random variable a X + b is
required, where a and b are constants, then the two pdfs are
related to each other by
()
1
aX b X
xb
fx f
aa
+

=⋅

 (66)
Constant a defined by (33) is positive, and so no absolute
value is needed. Thus, (49) and (66) yield
()
1
Galaxy col
TT
D
tb
ft f
aa

=⋅ 
 (67)
Thus the pdf of the random variable TGalaxy is obtained from
(58) by letting (58) undergo the transformation given by (67).
In other words, the pdf of the overall time needed to colonize
the whole Galaxy is given by equation (58):
()
()
2
2
_
39
_
22
22 _
_
_
22
_
22
__
(3 )3log
29
9
3 9 9 log 9log
9
9399
1
() 3 2
2
Dtcol
Dtcol
Dtcol
Dtcol
Galaxy
Dtcol
Dtcol
DtcolDtcol
tb
Cyr a
T
tb
Ca
ft C e
a
yr
tb
a
µµ
µµ
σσ
σσ
µµ
σσ
σσ µµ
π
+


++



+
+

+−


+
+−−+
=× 
÷

×

()
22
_
log
922
_
9
Dtcol
C
Dtcol
σσ
σσ
+
+
(68)
Note that this probability distribution holds only for positive
values of the time that also are larger than the constant b
defined by the second equation in (33). This is of course
requested to avoid imaginaries that would otherwise be brought
in by the real power of (t-b) at the denominator. In other words,
for values of t ranging between zero and b, the above pdf is
equal to zero.
Thus, one can prove that it fulfills indeed the normalization
condition
()
1
Galaxy
T
b
f
tdt
=
(69)
The mean value of the time TGalaxy needed to colonize the whole
Galaxy is found using the fact that the mean value operator is a
linear operator, so the requested mean value is found immediately
by letting (60) undergo the linear transformation (67).
22
__
69 18
18
D D tcol tcol
Galaxy
T
yr
ae b
C
σµσ µ
µ
++ +
=⋅+
(70)
Note that this equation is dimensionally correct since
a = 2 Galaxy/k = 4 RGalaxy has the dimension of a length. Constant,
b, given by (33), has the dimension of a time and depends only
on the speed vss of the pure interstellar flight to hop between
planets.
The corresponding variance is:
22
22
__ _
22
2
22
93 9
2
1
DD
D t col t col t col
Galaxy
T
yr
ae e C
σσ
µσ µ σ
σ
++ + +


=−


(71)
and the corresponding standard deviation:
239
The Statistical Fermi Paradox
2222_
_2
93 2
_
9
21
DDtcol
tcol
Dtcol
Galaxy
T
yr
ae e C
σµσ µ
σσ
σ
++ +
+
=⋅
(72)
The mode of the pdf (68), i.e. the two results are the two
abscissas of the minimum of (68), obviously at z = 0, and of the
maximum (i.e. the peak, or mode) at
22
__
Galaxy
1
93
mode_T
DD t col t col yr
z
ae b
C
σµσ µ
−+ − +
=⋅+
(73)
Finally, the two inflexion points of (68), that is the one
before and the one after the peak (or mode), are found as the
two roots of a quadratic algebraic equation in log(z). Thus, it is
found that the abscissas of such two inflexion points of (68)
are, respectively
22 22 2 2
__ __
inflexion_1
9 9 36 3 6 27 18
18
D tcol D tcol D D tcol tcol
z
yr
ae b
C
σσ σσ σ µ σ µ
+⋅++++ −
=
=⋅+
(74)
22 22 2 2
__ __
inflexion_2
9 9 36 3 6 27 18
18
D tcol D tcol D D tcol tcol
z
yr
ae b
C
σσ σσ σ µ σ µ
+⋅++− −
=
=⋅+
(75)
21. CONCLUSIONS
The classical Fermi paradox has been extended to encompass
Statistics and Probability. Statistical equations that are related
to both the Statistical Drake equation and the Statistical Dole
1. G. Benford, J. Benford and D. Benford, “Searching for Cost Optimized
Interstellar Beacons”, Submitted October 2008, http://arxiv.org/abs/
0810.3966. (Date Accessed 6 November 2010)
2. C. Sagan, “Cosmos”, Random House, New York, 1983. See in particular
the pages 298-302.
3. J. Bennett and S. Shostak, “Life in the Universe”, second edition,
Pearson–Addison Wesley, San Francisco, 2007. See in particular page
404.
4. C. Maccone, “The Statistical Drake Equation”, 59th International
equation for habitable planets for Man have been considered. A
numerical code should be written to allow our results to be
applied to cases of practical interest.
This approach appears to pave the way to future, more
profound investigations intended not only to associate “error
bars” to each factor in the Drake and Dole equations, but
especially to increase the number of factors themselves. In fact,
this seems to be the only way to incorporate into these equa-
tions more and more new scientific information as soon as it
becomes available. In the long run, the Statistical results might
become a huge computer code, growing in size and especially
in the depth of the scientific information it contains. It would
thus be Humanity’s first “Encyclopaedia Galactica.”
Unfortunately, to extend the Drake and Dole equation to
Statistics, it was necessary to use a mathematical apparatus that
is more sophisticated than just the simple product of numbers.
When this author had the honour and privilege to first
present his results at the SETI Institute on April 11th, 2008, in
front of an audience also including Professor Frank Drake, he
felt he had to add these words: “My apologies, Frank, for
disrupting the beautiful simplicity of your equation.”
ACKNOWLEDGEMENTS
The author is grateful to Drs. Hal Puthoff and Eric Davis of the
Institute for Advanced Study at Austin for the reading the
author’s paper. Thanks are also due to Professor Frank Drake
and Drs. Jill Tarter, Seth Shostak, and Doug Vakoch of the
SETI Institute, as well as Dr. H. Paul Shuch, for their apprecia-
tion of the statistical work described in this paper. Finally,
special thanks go to Marc Millis and Paul Gilster of the Tau
Zero Foundation for realizing the importance of our statistical
extension of the Drake and Dole equations and helping to have
our results popularized within the international scientific com-
munity.
REFERENCES
Astronautical Congress, Glasgow, Scotland, UK, 29 September-3
October 2008. Paper No. IAC-08-A4.1.4
5. S.H. Dole, “Habitable planets for Man”, first edition, 1964, © 1964 by
the RAND Corporation, Library of Congress Catalogue Card Number
64-15992. See in particular page 82, i.e. the beginning of Chapter 5,
entitled “Probability of Occurrence of Habitable Planets”.
6. A. Papoulis and S. Unnikrishna Pillai, “Probability, Random Variables
and Stochastic Processes”, Fourth Edition published by Tata-McGraw-
Hill, New Delhi, 2002. See in particular pages 186-187.
(Received 19 October 2009; 21 September 2010)
* * *
... He extended the Drake equation so as to embrace Statistics in his 2008 paper (Maccone 2008). This paper was later published in Acta Astronautica (Maccone 2010a), and more mathematical consequences were derived in Maccone (2010b) and Maccone (2011a). ...
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In this paper we propose a new mathematical model capable of merging Darwinian Evolution, Human History and SETI into a single mathematical scheme: 1) Darwinian Evolution over the last 3.5 billion years is defined as one particular realization of a certain stochastic process called Geometric Brownian Motion (GBM). This GBM yields the fluctuations in time of the number of species living on Earth. Its mean value curve is an increasing exponential curve, i.e. the exponential growth of Evolution. 2) In 2008 this author provided the statistical generalization of the Drake equation yielding the number N of communicating ET civilizations in the Galaxy. N was shown to follow the lognormal probability distribution. 3) We call "b-lognormals" those lognormals starting at any positive time b ("birth") larger than zero. Then the exponential growth curve becomes the geometric locus of the peaks of a one-parameter family of b-lognormals: this is our way to re-define Cladistics. 4) b-lognormals may be also be interpreted as the lifespan of any living being (a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization). Applying this new mathematical apparatus to Human History, leads to the discovery of the exponential progress between Ancient Greece and the current USA as the envelope of all b-lognormals of Western Civilizations over a period of 2500 years. 5) We then invoke Shannon's Information Theory. The b-lognormals' entropy turns out to be the index of "development level" reached by each historic civilization. We thus get a numerical estimate of the entropy difference between any two civilizations, like the Aztec-Spaniard difference in 1519. 6) In conclusion, we have derived a mathematical scheme capable of estimating how much more advanced than Humans an Alien Civilization will be when the SETI scientists will detect the first hints about ETs.
... is called geometric Brownian motion (GBM), and is widely used in financial mathematics, where it is the 'underlying process' of the stock values (Black-Scholes models (1973), or Black-Scholes-Merton models, with the Nobel prize in Economics awarded in 1997 to Sholes and Merton only since Black had unfortunately passed away in 1995). This author used the GBM in his previous mathematical models of evolution and SETI (Maccone 2010a(Maccone , 2010b(Maccone , 2011a(Maccone , 2011b(Maccone , 2012(Maccone , 2013, since it was assumed that the growth of the number of ET civilizations in the Galaxy, or, alternatively, the number of living species on Earth over the last 3.5 billion years, grew exponentially (Malthusian growth). Notice also that, upon equating the two righthand sides of (3) and (9), we find that e M GBM (t) e σ 2 GBM 2 (t−ts) = N 0 e μ GBM (t−ts) . ...
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In a series of recent papers and in a book, this author put forward a mathematical model capable of embracing the search for extra-terrestrial intelligence (SETI), Darwinian Evolution and Human History into a single, unified statistical picture, concisely called Evo-SETI. The relevant mathematical tools are: (1) Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (Black-Sholes models). (2) The probability distributions known as b-lognormals, i.e. lognormals starting at a certain positive instant b>0 rather than at the origin. In the framework of Darwinian Evolution, the resulting mathematical construction was shown to be what evolutionary biologists call Cladistics. (3) The (Shannon) entropy of such b-lognormals is then seen to represent the 'degree of progress' reached by each living organism or by each big set of living organisms, like historic human civilizations. (4) All these results also match with SETI in that the statistical Drake equation (generalization of the ordinary Drake equation to encompass statistics) leads just to the lognormal distribution as the probability distribution for the number of extra-terrestrial civilizations existing in the Galaxy. (5) The well-known 'Molecular Clock of Evolution', namely the 'constant rate of Evolution at the molecular level' as shown by Kimura's Neutral Theory of Molecular Evolution, identifies with growth rate of the entropy of our Evo-SETI model. (6) Furthermore, we apply our Evo-SETI model to lognormal stochastic processes other than GBMs. For instance, we provide two models for the mass extinctions that occurred in the past. (7) Finally, we show that the Markov & Korotayev model for Darwinian Evolution identifies with an Evo-SETI model.
Chapter
Darwinian Evolution over the last 3.5 billion years is defined as one particular realization of a certain stochastic process called Geometric Brownian Motion (GBM). This GBM yields the fluctuations in time of the number of species living on Earth. Its mean value curve is an increasing exponential curve, i.e. the exponential growth of Evolution.
Chapter
In a paper (Maccone, Orig. Life Evol. Biosph. (OLEB) 41:609–619, 2011, [15]) and in a book (Maccone, Mathematical SETI, 2012, [17]), this author proposed a new mathematical model capable of merging SETI and Darwinian Evolution into a single mathematical scheme. This model is based on exponentials and lognormal probability distributions, called “b-lognormals” if they start at any positive time b (“birth”) larger than zero. Indeed: (1) Darwinian evolution theory may be regarded as a part of SETI theory in that the factor fl in the Drake equation represents the fraction of planets suitable for life on which life actually arose, as it happened on Earth. (2) In 2008 (Maccone, The Statistical Drake Equation, 2008, [9]) this author firstly provided a statistical generalization of the Drake equation where the number N of communicating ET civilizations in the Galaxy was shown to follow the lognormal probability distribution. This fact is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution if the number of factors is increased at will, i.e. it approaches infinity. (3) Also, in Maccone (Orig. Life Evol. Biosph. (OLEB) 41:609–619, 2011, [15]), it was shown that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of b-lognormal distributions constrained between the time axis and the exponential growth curve. This was a brand-new result. And one more new and far-reaching idea was to define Darwinian Evolution as a particular realization of a stochastic process called Geometric Brownian Motion (GBM) having the above exponential as its own mean value curve. (4) The b-lognormals may be also be interpreted as the lifespan of any living being, let it be a cell, or an animal, a plant, a human, or even the historic lifetime of any civilization. In Maccone (Mathematical SETI, 2012, [17, Chapters 6, 7, 8 and 11]), as well as in the present paper, we give important exact equations yielding the b-lognormal when its birth time, senility-time (descending inflexion point) and death time (where the tangent at senility intercepts the time axis) are known. These also are brand-new results. In particular, the σ = 1 b-lognormals are shown to be related to the golden ratio, so famous in the arts and in architecture, and these special b-lognormals we call “golden b-lognormals”. (5) Applying this new mathematical apparatus to Human History leads to the discovery of the exponential trend of progress between Ancient Greece and the current USA Empire as the envelope of the b-lognormals of all Western Civilizations over a period of 2500 years. (6) We then invoke Shannon’s Information Theory. The entropy of the obtained b-lognormals turns out to be the index of “development level” reached by each historic civilization. As a consequence, we get a numerical estimate of the entropy difference (i.e. the difference in the evolution levels) between any two civilizations. In particular, this was the case when Spaniards first met with Aztecs in 1519, and we find the relevant entropy difference between Spaniards an Aztecs to be 3.84 bits/individual over a period of about 50 centuries of technological difference. In a similar calculation, the entropy difference between the first living organism on Earth (RNA?) and Humans turns out to equal 25.57 bits/individual over a period of 3.5 billion years of Darwinian Evolution. (7) Finally, we extrapolate our exponentials into the future, which is of course arbitrary, but is the best Humans can do before they get in touch with any alien civilization. The results are appalling: the entropy difference between aliens 1 million years more advanced than Humans is of the order of 1000 bits/individual, while 10,000 bits/individual would be requested to any Civilization wishing to colonize the whole Galaxy (Fermi Paradox). (8) In conclusion, we have derived a mathematical model capable of estimating how much more advanced than humans an alien civilization will be when SETI succeeds.
Chapter
In a series of papers (Ref. Maccone, The Statistical Drake Equation, 2008) through (Maccone, A Mathematical Model for Evolution and SETI, 2011.) and (Maccone, Int J Astrobiol 12(3):218–245, 2013) through (Maccone, Acta Astronautica, 317–344, 2014) and in a book (Ref. Maccone, Mathematical SETI, 2012), this author suggested a new mathematical theory capable of merging Darwinian Evolution and SETI into a unified statistical framework. In this new vision, Darwinian Evolution, as it unfolded on Earth over the last 3.5 billion years, is defined as just one particular realization of a certain lognormal stochastic process in the number of living species on Earth, whose mean value increased in time exponentially. SETI also may be brought into this vision since the number of communicating civilizations in the Galaxy is given by a lognormal distribution (Statistical Drake Equation). Now, in this paper we further elaborate on all that particularly with regard to two important topics: (1) The introduction of the general lognormal stochastic process L(t)L\left( t \right) whose mean value may be an arbitrary continuous function of the time, m(t)m\left( t \right), rather than just the exponential mGBM(t)=N0eμtm_{{{\text{GBM}}}} \left( t \right) = N_{0} \,e^{\mu \,t} typical of the Geometric Brownian Motion (GBM). This is a considerable generalization of the GBM-based theory used in ref. (Maccone, The Statistical Drake Equation, 2008) through (Maccone, Acta Astronautica, 317–344, 2014). (2) The particular application of the general stochastic process L(t)L\left( t \right) to the understanding of Mass Extinctions like the K/Pg one that marked the dinosaurs’ end 65 million years ago. We first model this Mass Extinction as a decreasing Geometric Brownian Motion (GBM) extending from the asteroid’s impact time all through the ensuing “nuclear winter”. However, this model has a flaw: the “final value” of the GBM cannot have a horizontal tangent, as requested to enable the recovery of life again after this “final extinction value”. (3) That flaw, however, is removed if the rapidly decreasing mean value function of L(t)L\left( t \right) is the left branch of a parabola extending from the asteroid’s impact time all through the ensuing “nuclear winter” and up to the time when the number of living species on Earth started growing up again, as we show mathematically in Sect. 3. In conclusion, we have uncovered an important generalization of the GBM into the general lognormal stochastic process L(t)L\left( t \right), paving the way to a better, future understanding the evolution of life on Exoplanets on the basis of what Evolution unfolded on Earth in the last 3.5 billion years. That will be the goal of further research papers in the future.
Chapter
Geometric Brownian motion (GBM), the stochastic process representing evolution as the stochastic increase of the number of species living on Earth over the last 3.5 billion years. This GBM is well known in the mathematics of finances (Black–Sholes models). Its main features are that its probability density function (pdf) is a lognormal pdf, and its mean value is either an increasing or, more rarely, decreasing exponential function of the time.
Chapter
In a series of recent papers (Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9]) this author gave the equations of his mathematical model of Evolution and SETI, simply called “Evo-SETI”.
Chapter
Mathematical formulae showing that the Statistical Drake Equation (namely the statistical extension of the classical Drake Equation typical of SETI) can be regarded as the “frozen in time” part of GBM. This makes SETI a subset of our Big History Theory based on GBMs: just as the GBM is the “movie” unfolding in time, so the Statistical Drake Equation is its “still picture”, static in time, and the GBM is the time-extension of the Drake Equation.
Chapter
Ray Kurzweil’s famous 2006 book “The Singularity Is Near” predicted that the Singularity (i.e. computers taking over humans) would occur around the year 2045. In this chapter we prove that Kurzweil’s prediction is in agreement with the “Evo-SETI (Evolution and SETI)” mathematical model that this author has developed over the last ten years in a series of mathematical papers published in both Acta Astronautica and the International Journal of Astrobiology.
Chapter
In 2013, MIT astrophysicist Sara Seager introduced what is now called the Seager Equation (Refs. https://en.wikipedia.org/wiki/Sara_Seager#Seager_equation; P. Gilster, Astrobiology: Enter the Seager Equation, Centauri Dreams, 11 September 2013, https://www.centauri-dreams.org/?p=28976): it expresses the number N of exoplanets with detectable signs of life as the product of six factors: Ns = the number of stars observed, fQ = the fraction of stars that are quiet, fHZ = the fraction of stars with rocky planets in the Habitable Zone, fO = the fraction of those planets that can be observed, fL = the fraction that have life, fS = the fraction on which life produces a detectable signature gas. This we call the “classical Seager equation”. Now suppose that each input of that equation is a positive random variable, rather than a sheer positive number. As such, each input random variable has a positive mean value and a positive variance that we assume to be numerically known by scientists. This we call the “Statistical Seager Equation”. Taking the logs of both sides of the Statistical Seager Equation, the latter is converted into an equation of the type log(N) = SUM of independent random variables. Let us now consider the possibility that, in the future, the number of physical inputs considered by Seager when she proposed her equation will actually increase, since scientists will know more and more details about the astrophysics of exoplanets. In the limit for an infinite number of inputs, i.e. an infinite number of independent input random variables, the Central Limit Theorem (CLT) of Statistics applies to the Statistical Seager Equation. Thus, the probability density function (pdf) of the output random variable log(N) will approach a Gaussian (normal) distribution in the limit, whatever the distribution of the input random variables might possibly be. But if log(N) approaches the normal distribution, then N approaches the lognormal distribution, whose mean value is the sum of the input mean values and whose variance is the sum of the input variances. This is just what this author realized back in 2008 when he transformed the Classical Drake Equation into the Statistical Drake Equation (Refs. C. Maccone. 2008, The Statistical Drake Equation, Paper #IAC-08-A4.1.4 presented on 1st October, 2008, at the 59th International Astronautical Congress (IAC), Glasgow, Scotland, UK, 29 September–3 October, 2008; Maccone in Acta Astronaut. 67:1366–1383, 2010). This discovery led to much more related work in the following years (Refs. Maccone in J. Br. Interplanet. Soc. 63:222–239, 2010; Maccone in Acta Astronaut. 68:63–75, 2011; C. Maccone, A mathematical model for evolution and SETI, Orig. Life Evolut. Biosph. (OLEB) 41 (2011) 609–619. Available online 3rd December, 2011; C. Maccone, 2012, Mathematical SETI, A 724-pages book published by Praxis–Springer in the fall of 2012. ISBN-10: 3642274366; ISBN-13: 978–3,642,274,367, edition: 2012; Maccone in Int. J. Astrobiol. 12:218–245, 2013; C. Maccone, Evolution and history in a new “Mathematical SETI” model, Acta Astronaut. 93 (2014) 317–344. Available online since 13 August 2013; Maccone in Acta Astronaut. 101:67–80, 2014; Maccone in Int. J. Astrobiol. 13:290–309, 2014). In this paper we study the lognormal properties of the Statistical Seager Equation relating them to the present and future knowledge for exoplanets searches from both the ground and space.
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What would SETI beacon transmitters be like if built by civilizations that had a variety of motives but cared about cost? In a companion paper, we presented how, for fixed power density in the far field, a cost-optimum interstellar beacon system could be built. Here, we consider how we should search for a beacon if it were produced by a civilization similar to ours. High-power transmitters could be built for a wide variety of motives other than the need for two-way communication; this would include beacons built to be seen over thousands of light-years. Extraterrestrial beacon builders would likely have to contend with economic pressures just as their terrestrial counterparts do. Cost, spectral lines near 1 GHz, and interstellar scintillation favor radiating frequencies substantially above the classic "water hole." Therefore, the transmission strategy for a distant, cost-conscious beacon would be a rapid scan of the galactic plane with the intent to cover the angular space. Such pulses would be infrequent events for the receiver. Such beacons built by distant, advanced, wealthy societies would have very different characteristics from what SETI researchers seek. Future searches should pay special attention to areas along the galactic disk where SETI searches have seen coherent signals that have not recurred on the limited listening time intervals we have used. We will need to wait for recurring events that may arrive in intermittent bursts. Several new SETI search strategies have emerged from these ideas. We propose a new test for beacons that is based on the Life Plane hypotheses.
The Statistical Drake Equation
  • C Maccone
C. Maccone, "The Statistical Drake Equation", 59th International REFERENCES Astronautical Congress, Glasgow, Scotland, UK, 29 September-3 October 2008. Paper No. IAC-08-A4.1.4
© 1964 by the RAND Corporation, Library of Congress Catalogue Card Number 64-15992
  • S H Dole
S.H. Dole, "Habitable planets for Man", first edition, 1964, © 1964 by the RAND Corporation, Library of Congress Catalogue Card Number 64-15992. See in particular page 82, i.e. the beginning of Chapter 5, entitled "Probability of Occurrence of Habitable Planets".