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DIRD 25 DIRD An Introduction to the Statistical Drake Equation

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1.Introduction.......................................................................................................iv
2.TheKeyQuestion:HowFarareThey? •.....•....•...•.•.•••••..•.•.•..•.........•......••••..•••..•4
3.Computing_N_ByVirtueoftheDrakeEquation(1961)........................................7
4.TheDrakeEquationisOver-Simplified.............................................................10
5.TheStatisticalDrakeEquation•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••11
6.SolvingtheStatisticalDrakeEquationByVirtueoftheCentralLimitTheorem
(CLT)ofStatistics.a............................................................
11•••••••••••••••••••••••••••••••••13
7.AnExampleExplainingtheStatisticalDrakeEquation.....................................15
8.FindingtheProbabilityDistributionoftheEt-DistanceByVirtueoftheStatistical
Drake Equation ......................................................................................
11••••••••••••••••18
9.The"'DataEnrichmentPrinciple"astheBestCL TConsequenceUponthe
StatisticalDrakeEquation(AnyNumberofFactorsAllowed)...........................23
10.Conclusions...................III......................................................................................23
AppendixA:ProofofShannon's1948TheoremStatingThattheUniform
Distributionisthe"MostUncertain"OneOveraFiniteRangeof
Values.........•....................................................................•..............2S
AppendixB:OriginalTextoftheAuthor'sPaper#IAC-08-A4.1.4Entitledthe
StatisticalDrakeEquation...............................................................28
References ............................................III.III..III. III.111.111.III.................................II........III.........................III.....111."111.......III.......55

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Random variableN~~=~~numberofcommunicatingET
civilizations in galaxy		N~~=~~numberofcommunicatingET
civilizations in galaxy		N~~=~~numberofcommunicatingET
civilizations in galaxy
Probability distributionLognormal

Probability density function		J
1
f,.,,(n)=-·--e
Ii_.j2;;tT_		~~- ~~(1]1(11) ~Ir
2a:!

(n20)
Mean valuecr
_(N)=e 11e 1 _
Variance.)
1
.,~
"
.J
a- -
j-,lI/'1- ,a-
1
N-(;
(;
{.
-
Standard deviationrr~
p~~
(TN=I.Je -
e'- 1
All the moments, i.e.k-thmoment(N k ) =/JI£'
k!,(r~
1
Mode(=abscissaofthe lognormal peak)n
- n
- "I'l'-(;-
m,dc=
peak -
,
Valueofthe ModePeakf(
.
1
A'"uude)=J2i
_2ff _		(]'~
-t(-:;-
.e. .e -
(j
Median(=fifty-fiftyprobability valuefor
N)		n:edian=Tn=e'U
SkewnessK
,,)
3=e cr-
I2

f
(KJ.:!
,-6.u,-cr
,
_
(-.
(
~~t~r,' ~~~
1

:r
~(r-]
e·'<r+3f'!WO'+61'/'+6
Kurtosis~
=4tT+2
'l
{.

(K2)-		~:cr+3,2"!- 6
(.
•
(.
Expressionof_J.l_intermsofthe lower_(a;)
and upper(bi)_limitsofthe Drake
uniforminput random variables Dj		/.J=±(Yj)=±bi[In(bJ-l]-{E;[ln(a;)-J]
i-I
i-l
_bi _- (Ii
Expressionof(J'2intermsofthe lower_(ai)
and Lipper(b;)limitsofthe Drake
uniforminputrandom variables_Di		7
7
2L~L
(j=
O'l';=

i-I
i-I		l.'lib; [hl(bJ-hl(a; W

1
(hi-a/f

"""
..."'_		"=

w		"~"0.109
__
~50~
10
tp:=100olP:=-
100
1
ne:=1pne:=neOne~=-,fj
50
fl:=-
100		pH:=t110
crtl:= -100
_
~O
11:=-100		pfi:=fi_
10
oti:=-100
.
10
tc:=-		~U:c:=tc.
10
atc:=-
100100
fL:=10000
1010		pfl.:=t1.
x=3500

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Random variableET_Distance between anytwoneighboring
ETcivilizations in galaxy assuming they are
UNIFORMLYdistributedthroughoutthe
whole galaxy volume.		ET_Distance between anytwoneighboring
ETcivilizations in galaxy assuming they are
UNIFORMLYdistributedthroughoutthe
whole galaxy volume.		ET_Distance between anytwoneighboring
ETcivilizations in galaxy assuming they are
UNIFORMLYdistributedthroughoutthe
whole galaxy volume.		ET_Distance between anytwoneighboring
ETcivilizations in galaxy assuming they are
UNIFORMLYdistributedthroughoutthe
whole galaxy volume.
Probability distribution
Unnamed
Probability density functionrI {
.
3
I
-"
.I_ETDist:m(r)=.j2;;.e
-
r
_2lfa _		,~~~~R;',,,,,,,,"""''''1"I'
I
•
r
•
}
2(7!
Numerical constant C relatedtotheMilky
Way size		c=;;6_Rl{/fllxr_h{~(lI'l.~r~28,845lightyears
Mean value)I
G"~
(ET_DisLuncc)=Ce-:>_e_IR
Variance2
a'_[a'_J
2
·.2- J,lI
<.)
1J
O'I::Tl)j"wnm=C(?
e
e-I
Standa rd deviation'
__,lI_
(7-
a-
_
,3
)
_,q__
_a_ETDist:m<e- C(.
e
t.
1
All the moments, i.e.k-thmomentk Jl
k~.(Y2
(Ef_Distancel:)=_ck e
3_e
lR
Mode(=abscissaofthelognormal peak),II-
(T~
rrwd.!==rpeak=C_e_J_e_
I)
ValueoftheMode PeakPeak ValueoffE'U)isl:mu:(r)=
.
)
1
Jl
(T~
==_.t_ETDiqlHl('{(';nldc='J2;
·(>J·el
C
7'(0"
Median(=fifty-fiftyprobability value forN)median-
,tI
=m=_Ce_J
SkewnessK."
-'-3=
(K4h		(u'
'u'
a')
e-,lIt!:1.- 3_e_18
l2_e_6

3
(
Xu'
:1tT~
4a~
rr"'"')'
-
-
c:'
,l)-4e()-3e_t)+]2e ~-6e_<)
.
(.
Kurtosis4(T'~
u"
2(/
K
-
-
-
4=e_9+2e 3 +3e_9-6
(K2f
Expressionof_J.1_intermsofthe lower (ai)
and upper(bi)limitsofthe Drake uniform
inputrandom variablesDi		)1=±(Y;)=±bi[ln(bi)-I]=ai[ln(ai)-1]
i-I
i-I
hi
_Gi _
'1
Expressionof0'-intermsofthelower (ai)
and upper(bi)limitsoftheDrake uniform
inputrandom variablesDi		7
'1
2L22:
(j=
aT:,=
1
i-I
i-l		a/')i [In(hi)-111(Gi)f
(bi -aif

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p:"s:=Ks
10
afP:=100
1
ne:=1.~Ule:=neerne:=-f3
.t1:=50
100		.ufl:=fl10
aft:=-100
_
~O
fi.:=-100		_
10
Oti:=--
lOa
~::0_10
tc:=-
100		pfc:=fcOfe:=-
100
~ttL:=tL
~:=);s.!p·ne.tlfi.fc.tl.:\ =3500

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(In(yhlf
. (
)
1
1
2.,.2
INy,p,a=.r;::-
e
Y,,2lra
(y_2:0)		.(47)

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All the moments, i.e. k-thmomenl
Mode(=abscissuoflhelugnormu]peuk)1-(
-c;-
_n_In.~dc==_n_peak=e('
Vulueofthe Mode Peak,/
f
Ill-:;-
N(n.mtlc)=~·e··e-
"'2it'(j
Median(=fifty-fifty proh~tbi1ityvul1..1efortV)nedian= m=_el-t _
Skewness
Kurlosis
Expressionof_Ji_intermsofthelower_(ai)andupper
(hi)_1imitsofthe Drakeuniforminpulnmdom
variablesDj		___~(Y_~b,Dn{bJ)-lJ-(1;[ln((//)-J]
p-£....ii-£.....
i-I
i-l
hi-ai
Expressionof(1'2in termsofthe){)wer_(a,)undupper
(hi)Emitsof the Drake uniform input random
variables_Di		7
7
a
2 =Lo-:'·=LJ
i-I
i-I
til';[hl(b;)-In(a;}]2
(bj -£li }2

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