Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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In other words, there i~ a factor of ✓3 = 1.732
included in the two equations ( 17) that is not obvious
at all t(] human intuition, and mmt indeed he taken
into account.
The application of this result to the Statistical Drake
equation is discussed in the next ~ccti(]n.
3.1 STEP 5: A NL1MERICAL EXAMPLE
OF THE STATISTICAL DRAKE
EQUATION WITH llNIFORM
DISTRIBUTIONS FOR THE DRAKE
RANDOM VARIABLES D;
The first variable Ns in the classical Drake
equation (I) is the number uf stars in uur Galaxy.
Nobody knows how many they arc exactly(!). Only
statistical estimate~ can be made by a~tronomers, and
they oscillate (say) mound n mean value of 350
billions (if this value is indeed c:onect!). This being
the situation, we nssume that our uniformly
distributed random variable Ns has a mean value of
350 billions minu~ or plus a standard deviation of
(say) one billion (we don't care whether this number
is ~cientifically the be~t estimate a~ of August 2008:
we just want to set up a numerical example of our
Stafotical Drake equation). In other words, we now
assume that one has:
{
( uniforrn_D 1 ) = 350 • 10 9
O"untk1rn1_D1 =]. l0 9 •
(18)
Therefore, according to equation~ ( 17) the lower and
upper limit of our uniform distribution for the
rand(]m variable Ns=IJ, arc, respectively
la.v, =(unifon11_D 1)-J3aur1it<>nll_D, =348.3-10'!
r:; ') (19)
h,v, =(unifmm_D,)+-v3au,,i,illm_D, =351.7-10
Similarly we proceed for all the other six random
variables in the Statistical Drake equation(]).
For instance, we assume that the fraction of stars
that have planet~ is 50%, i.e. 50/100, and this will he
the mean value of the random variable fp=lh. We
also assume that the relevant standard deviation will
be JO%. i. e. that u 1P = l(]Jl[K) Therefore, the
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relevant lower and upper limits for the uniform
distributiun of/jJ=D'- turn out lo be
i(I.fl' : (uniform_D, )- ✓3 O"uniln1m_D,
lb 11, - ( urntomi_D 2 ) + ✓3 a un,k>rm_D,
= 0.327
= 0.673 (20)
The next Drake random variable 1s the number
ne of "Earth-type" planets in a given star system.
Taking example from the Solar Sy~tem, since only
the Earth is truly "Earth-type", the mean value of ne
is clearly I. but the standard deviation 1s not zero if
we a~wme that Mars also may be regarded as Earth-
type. Since there are thus two Earth-type planets in
the Solar System, we mu~t assume a standard
deviation of I/ J3 = 0.577 to compensate the fJ
appearing in ( 17) in order to !inally yield tv..o "Earth-
typc" planets (Earth and Mars) for the upper limit of
the random variable ne. In other words. we as~ume
thnt
j(1 1", =(unifom1_D_,)- ✓3a""'k"m_D, =0 l2 l)
1b,1,, =(uniform_D,)+ ✓3aun,k>rm_D, =2
The next four Drake random variables have even
more "arbitrarily" assumed values that we simply
assume for the sake of making up a numerical
example of our Statistical Drake equation with
uniform entry d1stribut1ons. So, we really make no
assumption about the a.~tronomy, or the biology, or
the .~ociology of the Drake equation: we ju.~t care
about its mathematics.
All (Jur as~umcd entries arc given in Table 1.
Please notice that, had we a~~umcd all the
standard dcviati(]ns to equal zem in Table 1, then our
Statistical Drake equation (]) would have obviously
reduced to the cla~sical Drake equation (I). and the
resulting number of civiliLations in the Galaxy would
have turned out to be 3500:
(22)
This is the important deterministic number that we
will use in the sc4ucl of this paper for compari~on
with our .~tati.~tical result~ on the mean value of N,
i.e. (N). This will be explained in Section~ 3.3 and 5.
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