Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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(N)=e·,, e 2 '°'4589.559. (95)
In other words, there are 459() ET Civilizations in
the Galaxy according the Central Limit Theorem of
Statistics with the inputs of Table I. This number
459() is HIGHER than the 3500 foreseen by the
classical Drake equation working with sheer
1111mbers only, rather than with probability
distributions. Thus equation (95) IS GOOJJ FOR
NEWS FOR SETI, .fince it show.f that the expected
1111mber of Hn is HIGHER with an adequate
statfrtical treatment than ju.fl with the too .fimple
Drake sheer numbers of (1 ).
2) Variance of N. The variance of the lognormal
distribution i~ given by (62) and turn~ out to be a
huge number:
3) Standard deviation af lV. The ~tam.lard deviation
(If the lognormal distrihuti(]n i~ given hy (63) and
turns out to be:
(97)
Again, this is GOOD NEWS FOR SETI. In fact,
such a high standard deviation means that N may
range from ve,y· low values (zero, theoretically, and
one since H11ma11i1y exists) up to tens of 1l10usands
(4590+11195=15785 is (95)+(97)).
4) Mode of N. The mode (= peak abscissa) of the
lognormal di~tribution uf N is given by (81 ). and has
a surpri~ingly low numeric value:
(98)
Thi~ is well shown ltl Figure 4: the mode peak is very
pronounced and clo~e to the origin. but the right tail
is high. and thi~ means that the mean value of the
distribution 1s much higher than the mode:
4590>>250.
44
5) Median of N. The median(= fifty-fifty ab~cissa,
~plitting the pdf in twu exactly cqui-probablc parts)
of the lognonnal distribution of N 1s given by (89),
and has the numenc value:
In word~. a~~uming the input value~ listed in Table I,
we have exactly a 500t prnbability that the actual
value of N is lower than 1740, and 50% that it is
higher than 1740.
7. COMPARING THE CLT RESULTS
WITH THE NON-CLT RESULTS
The time is now ripe to compare the CLT-
based results about the lognormal distribution of N.
jui;! dei;cribed in Section 5, agaim! the Non-CLT-
based re~ulti; obtained numerically in Section 3.3
To do so in a simple, visual way, let us plot on
the same diagram two curves:
1) The numeric curves appearing in Hgure 2
and obtained after laboriou~ Fourier
tramform cakula!iom in !he complex
domain, and
2) The lognormal distribution (56) with
numeric panda given by (91) and (94)
respectively.
Vv'c ~cc that the two curve~ arc virtually coincident
for values of N larger than 1500. Thi.1· i.~ a
consequence of the law of large numbers, of which
the CLT is just one of the many facets.
Similarly it happens fur natural lug of N. i.e. the
random variable Y uf (5). that i~ plotted in Figure 5
both in its normal curve \-crsiun (thin cunc) and in
its numeric vcrsiun, obtained via Fourier tramfonns
nnd already ~hown in Figure 2.
The cm1clu.1·ion i.1' .1·imple: from now 011 we .~hall
di.1Tard forn'er the numeric calc11latir111x and we'll
.~tick only to the equation.\' derived by virtue of the
CLT, i.e. tr, the log11ormal (56) and if.I'
con.1·equence.1·.
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