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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 variable | N~~=~~numberofcommunicatingET civilizations in galaxy | N~~=~~numberofcommunicatingET civilizations in galaxy | N~~=~~numberofcommunicatingET civilizations in galaxy |
|---|---|---|---|
| Probability distribution | Lognormal | ||
Probability density function | J 1 f,.,,(n)=-·—e Ii_.j2;;tT_ | ~~- ~~(1]1(11) ~Ir 2a:! (n20) | |
| Mean value | cr _(N)=e 11e 1 _ | ||
| Variance | .) 1 .,~ “ .J a- - j-,lI/’1- ,a- 1 N-(; (; {. - | ||
| Standard deviation | rr~ 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 ModePeak | f( . 1 A‘“uude)=J2i _2ff _ | (]’~ -t(-:;- .e. .e - (j | |
| Median(=fifty-fiftyprobability valuefor N) | n:edian=Tn=e’U | ||
| Skewness | K ,,) 3=e cr- I2 f (KJ.:! | ,-6.u,- , _ (-. ( ~~t 1 : e·‘<r+3f’!WO’+61’/‘+6 | |
| Kurtosis | ~ = ’l {. (K2)- | ~: (. • (. | |
| 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 (hi-a/f |
| """ …“’_ | ”= w | ”~“0.109 __ | |||
|---|---|---|---|---|---|
| ~ | 50 | ~ 10 | |||
| tp:= | 100 | olP:=- 100 | |||
| 1 | |||||
| ne | := | 1 | pne:=ne | One~=-,fj | |
| 50 fl:=- 100 | pH:=t1 | 10 crtl:= -100 | |||
| _ ~O 11:=-100 | pfi:=fi | _ 10 oti:=-100 | |||
| . 10 tc:=- | ~U:c:=tc | . 10 atc:=- | |||
| 100 | 100 | ||||
| fL | := | 10000 1010 | pfl.:=t1. | ||
| x=3500 |
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| Random variable | 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. | ET_Distance between anytwoneighboring ETcivilizations in galaxy assuming they are UNIFORMLYdistributedthroughoutthe whole galaxy volume. |
|---|---|---|---|---|
| Probability distribution | Unnamed | |||
| Probability density function | rI { . 3 I -” .I_ETDist:m - r _2lfa _ | I • r • } 2(7! | ||
| Numerical constant C relatedtotheMilky Way size | c=;;6_Rl{/fllxr_h{ | lightyears | ||
| Mean value | )I G”~ (ET_DisLuncc)=Ce-:>_e_IR | |||
| Variance | 2 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-thmoment | k Jl k~.(Y2 (Ef_Distancel:)=_ck e 3_e lR | |||
| Mode(=abscissaofthelognormal peak) | ,II- (T~ rrwd.!==rpeak=C_e_J_e_ I) | |||
| ValueoftheMode Peak | Peak ValueoffE’U)isl:mu:(r)= . ) 1 Jl (T~ ==_.t_ETDiqlHl(’{ ·(>J·el C 7’(0” | |||
| Median(=fifty-fiftyprobability value forN) | median | - ,tI =m=_Ce_J | ||
| Skewness | K.” -‘-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_<) . (. | ||
| Kurtosis | 4(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:=ne | erne:= | -f3 | |
| .t1:= | 50 100 | .ufl:=fl | 10 aft:=-100 | ||
| _ ~O fi.:=-100 | _ 10 Oti:=— lOa | ||||
| ~ | ::0 | _ | 10 | ||
| tc:= | - 100 | pfc:=fc | Ofe:= | - 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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