AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

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UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥
Table 2. Summary of the Properties of the Probability Distribution That Applies
to the Random Variable ET_Distance Yielding the (average} Distance Between
Any Two Neighboring 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 qalaxy volume.
Probability distribution
Unnamed
Probability density function
_(•{6 R[ata '.I l,Gola')]-µ J
1
3
I
2u2
fET_Distana,(r) =-;. ·Ji; CT ·e
Numerical constant C related to the Milky
Way size
C =3 6 R 8alruy h Gala.,y "" 28,845 light years
Mean value
µ
u2
(Er_Distance) =C e-3 e18
Variance
2
2
2
-
2
2
a
-3p 9 [
a9 -l
CTET_Distance - C e
e
e
I
Standard deviation
2
_ji_
er ✓ a
-
3
18
9 _
CTET_Distance - C e
e
e
l
All the moments, i.e. k-th moment
k2,u2
- k!!.
(Er_Dis tan eek) =c k e
3 e
18
Mode (= abscissa of the log normal peak)
_ji_
er
9
rnnde =rpeak = Ce 3 e
Value of the Mode Peak
Peak Value of fET_Distanu:(r) =
a'
!!.
3
-
·e 3 . e 18
=fET_Distana,Cr,rnde) = cfi; CT
Median ( = fifty-fifty probability value for N)
_ji_
~dian =m =Ce 3
Skewness
,,.,
5,,.2
,,., ]
e-µ [ e 2 - 3e 18 +2e 6
__!!i_ =
3
3
(K4 )2
8a2
5a2
4a2
2
a
2a2 ]2
C3 [ e9 -4e- 9 -3e9
+12 e3 - 6e- 9
Kurtosis
4 a 2
a'
2 a 2
-
-
K4
_
9 +2e 3 +3e 9 -6
(K2)2 - e
Expression of µin terms of the lower (ai)
and upper (bi) limits of the Drake uniform
input random variables Di
µ= I (r;) = I b; [ln (b;)- 1]- a;[ln(a;)- 1]
i=I
;~1
b; - a;
Expression of CT 2 in terms of the lower (ai)
and upper (bi) limits of the Drake uniform
input random variables Di
7
7
a;b; [ln (b;)- ln(a;)f
CT 2 = Io} =Ii
(b; - a; )2
i=I
i=I
UNCLASSIFIED/ J'FQA QFFICiIPL. !Iii 011! Y
20