Ilija
Ba ukčić
Theo iae
causali a is
p incipia
ma hema ica
Ilija Ba ukčić
Theo iae causali a is
p incipia ma hema ica
–Causali y –
Decembe 22, 2024
P in ed by Books on Demand,No de s ed ,
Ge many
His o y o publica ion:
Die Kausali ä
Fi s Edi ion published 1989 (Fi s Ge man Edi ion) ISBN 3-9802216-0-1
Second Edi ion published1997 (Second Ge manEdi ion) ISBN3-9802216-4-4
Causali y.New S a is icalMe hods.
Thi d Edi ion published 2005 (Fi s EnglishEdi ion, BoD) ISBN 3-8334-3645-X
Fou h Edi ion published2006 (Sec. EnglishEd., BoD) ISBN 3-8334-3645-X
Fi h Edi ion published2008 (Thi d EnglishEdi ion, Lulu)
Causali y Volume I:ISBN 978-1-4092-2952 -0
Causali y Volume II:ISBN 978 -1-4092 -2954 -4
Fi h Edi ion published2008, 13 h Re ision o he 5 h Edi ion, May24 h, 2009
Fi h Edi ion published2008, 14 h Re ision o he 5 h Edi ion, June 14 h, 2009
Fi h Edi ion published2008, 15 h Re ision published Ap il 5 h, 2010.
Fi h Edi ion published2008, 16 h Re ision published May24 h, 2010.
Fi h Edi ion published2008, 17 h Re ision published Augus 21s , 2010.
Fi h Edi ion published2008, 18 h Re ision published Dec. 30 h, 2010.
Fi h Edi ion published 2008, 19 h Re ision published May, 1s , 2011. Lulu.com
©Decembe 22, 2024 by Ilija Ba ukčić, Ho ands asse, Je e , Ge many.
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Theo iae causali a is p incipia ma hema ica.
Fi s Edi ion (Ap il 23,2017; ISBN: 9783744815932)
Second Edi ion (Augus 2, 2021; ISBN: 9783754331347 )
Thi d Edi ion (Decembe 22,2024; ISBN: 9783769322385 )
Imagec edi : NASA,ESA,CSA,STScI, Uni edS a es o Ame ica.
This imagecap u ed by Webb’sMIRI (Mid-In a edIns umen ) and eleased
Sep embe 18, 2024 10:00AM(EDT), e eals he supe massi eblackholea he
cen e o he la ge spi algalaxy on he igh .The blackholed aws in mucho
he su ounding dus , c ea ingdis inc i elanes.Addi ionally, i exhibi s Webb’s
cha ac e is ic diff ac ion spikes,which esul om he emi ed ligh in e ac ing wi h
he elescope’s s uc u e.
Con en s
Pa I Na u eand causa ion
1Reali y and causali y ................................ 3
Causali y ...........................................3
An i-Causali y....................................... 4
2E olu ion and causa ion .............................7
E olu ion and human b ain............................7
Ei he adap o become ex inc .........................8
3P oo me hods and causa ion .........................9
Induc ion and expe imen ..............................10
Deduc ion and logical allacies.........................12
P oo me hods andhuman knowledge................... 13
P oo by Though Expe imen s.................... 16
P oo by Coun e example .........................16
P oo by modus ponens........................... 18
P oo by modus ponens con aposi i us ............. 21
P oo by modus sine .............................22
P oo by modus ollens ........................... 23
P oo by modus in e sus..........................24
P oo by o he me hods ........................... 26
Pa II Science and causa ion
xiii
xx Con en s
Boole and nega ion ..............................495
Ma xand Engels andnega ion..................... 496
Lexnega ionis....................................... 497
Lexnega ionisgene alis...............................497
Pa VI Causa ion
26 Mono-Causali y ..................................... 507
In oduc ion ......................................... 508
Basics .............................................. 509
The diffe ence be ween cause and no -cause......... 509
The diffe ence be ween cause and effec .............510
The asymme y o he causal ela ion ...............512
Theo ems o mono-causali y ........................... 517
The iden i yo cause and effec .................... 517
The coincidence o cause and effec .......... 517
Causa aequa effec um ..................... 519
Da id Hume’s p oblemo induc ion ......... 525
One cause, one effec I..................... 526
The diffe ence be ween cause and effec .............529
The con adic ion be ween cause and effec .......... 531
One cause, one effec II .................... 532
Causal Rela ionship kand samplesize........ 535
The con adic ion be weencause and effec ... 536
Causa ion and co ela ion .................. 542
27 Mul i-Causali y ..................................... 543
In oduc ion ......................................... 544
Theo ems o mul i-causali y ........................... 545
One cause and achain o effec s ................... 545
One cause and manyeffec s....................... 548
Manycauses and one effec ....................... 551
Manycauses and manyeffec s..................... 555
Manycauses and andachain o effec s.............. 559
Achain o causes and one effec ................... 564
Achain o causes and manyeffec s ................. 568
Achain o causes and achain o effec s.............573
Gene al ela i i y and causali y ......................... 578
Pa VII Lawo Na u e
Con en s xxi
28 Law’so na u e ..................................... 585
Inde e minism....................................... 586
De e minism........................................587
The lawo na u e ela ionship g........................589
29 P oo o God’sexis ence .............................593
Science and ideology .................................594
P oo /Disp oo o he exis ence o God .................. 595
Gene al ela i i y and he exis ence o God...............598
No ice .................................................. 605
Re e ences ..........................................612
Au ho Index ...........................................629
Subjec Index ...........................................633
Volume I
The gene al heo y o
causali y
Pa I
Na u eand causa ion
Pa II
Science and causa ion
Pa III
Basic Defini ions
220 9Dis ibu ions
The Va iance o he Binomial Dis ibu ion
Ka lPea son p esen ed he defini ion o he s anda d de ia ion o he
binomial dis ibu ion as ollows:
“... he mean squa e e o o any binomial dis ibu ion ... is iden ical wi h
he alue√npq”(see Pea son,1895,p.351)
wi h he consequence ha σ(X)2=N·p(X)·(1−p(X)).
Nume us Legis Na u alis 51. In s a kcon as oKa lPea son’s
pe spec i e, he a iance o heBinomial dis ibu ion is gi en as:
σ(X)2=E(X2)−(E(X))2=N2·p(X)·(1−p(X)) (464)
whe e E(X)is heexpec ed alue o X,and E(X2)is heexpec ed alue
o hesqua eo X.
P oo by Di ec P oo . I holds ha 1 =1and E(X)=E(X).Fo
aBinomial andom a iable Xwi h pa ame e s N(numbe o ials)
and p(X)(p obabili yo success), he expec ed alue o Xis:
E(X)=X·p(X)=N·p(X)(465)
Di iding his equa ion by p(X), andunde he condi ions o he Binomial
dis ibu ion, we find ha :
X=N(466)
Nex , we calcula e E(X2), he expec ed alue o X2.Fo aBinomial
dis ibu ion, hiscan be exp essed as:
E(X2)=X·X·p(X)=N·N·p(X)(467)
Subs i u ing hese exp essions in o he a iance o mula:
σ(X)2=E(X2)−(E(X))2(468)
=(N·N·p(X))−(N·p(X))2(469)
=N2·p(X)·(1−p(X)) (470)
In gene al, he a iance o he Binomial dis ibu ion is de e mined as:
σ(X)2=N2·p(X)·(1−p(X)) (471)
Quod e a demons andum.
9Dis ibu ions 221
The Cumula i eDis ibu ion Func ion In Gene al
The Cumula i eDis ibu ion Func ion (CDF) o a andom a iable X
is a unc ion ha gi es he p obabili y ha X akes a alue less hano
equal o agi en alue x.I is o mallydefined as:
FX(x)=p(X)=p(X≤x)(472)
whe e FX(x)is he alue o he CDF a x.The CDF is anon-dec easing
unc ionbecause p obabili ies canno dec easeasxinc eases. Fo a
con inuous andom a iable, he CDF is acon inuous unc ion. Fo a
disc e e andom a iable, he CDF isas ep unc ion. Fo con inuous
andom a iables, he CDFis ela ed o hep obabili y densi y unc ion
(PDF) X(x)as:
FX(x)=p(X)=p(X≤x)=
x
∫
−∞
X( )d (473)
Fo acon inuous andom a iable, he CDF is acon inuous unc ion.
Fo disc e e andom a iables, he CDF is ela ed o he p obabili y
mass unc ion(PMF) pX(x)as:
FX(x)=p(X)=p(X≤x)=
≤x
pX( )(474)
Fo adisc e e andom a iable, heCDF is as ep unc ion. The
CDFsa isfies he ollowing limi p ope ies:limx→−∞ FX(x)=
0,limx→+∞FX(x)=1. These eflec he bounds o he CDF as i co e s
he en i e angeo X.
Nume us Legis Na u alis 52. Unde hegi en condi ions, hep ob-
abili y mass unc ion sa isfies he ela ionship:
p(X<k)+p(X≥k)=p(X≤k)+p(X>k)=1(475)
P oo by Di ec P oo . To begin, i holds ha :
p(X=k)=p(X=k)(476)
222 9Dis ibu ions
This equali yisa undamen al p ope yo he p obabili y mass unc ion.
Fu he mo e, we obse e ha :
p(X=k)+p(X>k)=p(X=k)+p(X>k)(477)
This eflec s he addi i ena u e o p obabili ies o disjoin e en s. F om
his,i ollows ha :
p(X≥k)=p(X=k)+p(X>k)(478)
This exp ession decomposes he cumula i ep obabili y o X≥k.
Combining p obabili ies o allpossible ou comes:
p(X<k)+p(X≥k)=p(X<k)+p(X=k)+p(X>k)=1(479)
He e, he o al p obabili y o all disjoin e en ssums o1.The e o e,
we conclude:
p(X<k)+p(X≥k)=p(X≤k)+p(X>k)=1(480)
This demons a es he comple enesso he p obabili y dis ibu ion.
Quod e a demons andum.
F om hese ela ionships,wecan deduce he ollowing equi alences:
p(X≥k)=1−p(X<k)(481)
This equali y eflec s ha he p obabili y o Xbeing g ea e han o
equal o kcomplemen s he p obabili y o Xbeing s ic lyless han k.
Simila ly,weha e:
p(X≤k)=1−p(X>k)(482)
This exp esses he complemen a y ela ionshipbe ween X≤kand
X>k.
Nume usLegis Na u alis 53. In gene al i is
p(X≤(n−1)) =1−p(X>n−1)=1−p(X=n)=1−(pn)(483)
P oo by Di ec P oo . In gene al, i is
p(X≤x)+p(X>x)=1(484)
9Dis ibu ions 223
Unde condi ions x=(n−1),Equa ion 484 becomes (see Equa ion 479,
p. 222)
p(X≤(n−1))+p(X>(n−1)) =1(485)
In gene al, o adisc e e andom a iable whe e n ep esen s he sample
o size, he e a e no ou comes g ea e han n.The e o e, he p obabili y
o exceeding n−1(i.e., being s ic lyg ea e han n−1) is equi alen
o he p obabili yo aking he alue no in o he wo ds
p(X>n−1)=p(X=n)(486)
Equa ion485 becomes
p(X≤(n−1))+p(X=n)=1(487)
As ound be o e, i isp(X≥n)=p(X=n)=pn(see Equa ion 547,p.
235), Equa ion 487 becomes
p(X≤(n−1))+pn=1(488)
In gene al, i is
p(X≤(n−1)) =1−(pn)(489)
Quod e a demons andum.
224 9Dis ibu ions
Nume us Legis Na u alis 54. Unde hespecific condi ions o bi-
nomial dis ibu ion (see Equa ion 489,p.223),i is
p(X≥n)=p(X=n)=1−p(X≤(n−1)) =pn(490)
P oo by Di ec P oo . Unde he specific condi ion o hebi-
nomial dis ibu ion, he numbe o ials nis such ha n ep esen s he
maximum possiblenumbe o successes(i.e., he o al numbe o ials
in he expe imen ), which implies (X≤n)=(X≤n)always. Howe e ,
o he binomial dis ibu ion wi h pa ame e s n(numbe o ials) and p
(p obabili yo successinasingle ial), he ela ionship
p(X≥n)=p(X≥n)(491)
ansla es o mally o:
p(X≥n)=p(X=n)+p(X>n)(492)
In summa y, abinomial andom a iable Xhas no possibili y o ex-
ceeding n, he o al numbe o ials. In b ie ,by he defini ion o he
binomial dis ibu ion, i is
p(X>n)=0(493)
The e o e, he equali y
p(X≥n)=p(X=n)=pn(494)
holds unde hese specific condi ions.
Quod e a demons andum.
In gene al (see Equa ion 634,p.251), i is
p(X≥n)=p(X=n)=(p)n=(1−q)n≈e−E(X)(495)
Gi en Be noulli ials wi h n=6138 (see Ba ukčić,2023c), he ex-
pec edsuccesses a e k=6138,bu he obse edsuccesses a e k=6132.
This indica es ha in 6o 6138 ials, he e en didno occu as ex-
pec ed. He e, E(X=6)is he expec ed sho all (e en s ha did no
occu ). The p obabili y can be app oxima ed as:
p(X≥n)=p(X=n)≈e−E(X)≈e−6=0.002478752177 (496)
9Dis ibu ions 225
Nume us Legis Na u alis 55. In gene al, i is
p(X>0)=1−((1−p)n)(497)
P oo by Di ec P oo . In gene al, i is
p(X=0)+p(X>0)=1(498)
As ound be o e, i isp(X=0)=(1−p)n.Equa ion 498 becomes
(1−p)n+p(X>0)=1(499)
A he end, we ob ain
p(X>0)=1−((1−p)n)(500)
Quod e a demons andum.
Nume us Legis Na u alis 56. In gene al, i is
p(0<X<n)=1−(pn)−((1−p)n)(501)
P oo by Di ec P oo . In gene al, i is
p(X=0)+p(X=1)+p(X=2)+···+p(X=(n−1))+p(X=n)=1(502)
We define
p(0<X<n)=p(X=1)+p(X=2)+···+p(X=(n−1)) (503)
Equa ion503 becomes
p(X=0)+p(0<X<n)+p(X=n)=1(504)
Rea anging equa ion 504,i is
p(0<X<n)=1−p(X=0)−p(X=n)(505)
As ound be o e, i p(X=0)=(1−p)nis and p(X=n)=pn.
p(0<X<n)=1−(pn)−((1−p)n)(506)
Quod e a demons andum.
226 9Dis ibu ions
I is
p(0<X<n)=p(1≤X≤(n−1)) =1−(pn)−((1−p)n)(507)
The in e al de i ed in Equa ion 507 is an exac in e al, since he
same is based di ec ly on he binomial dis ibu ion a he han any
app oxima ion o he binomial dis ibu ion.
Je zyNeyman
(1894–1981)
His o ically, me hods o
calcula ing confidence in e als o binomial p opo ions simila o
p(1≤X≤(n−1)) =1−(pn)−((1−p)n)(508)
began eme ging in he1920s (see Wilson,1927), bu i wasin he ea ly
1930s (see Cloppe and Pea son,1934) ha he ounda ional concep s
o confidence in e als we esys ema icallyde eloped.Thede elopmen
o confidence in e al heo yisla gely a ibu ed o Je zyNeyman, (see
Neyman,1937)who, in 1937, p o ided he fi s comp ehensi eand
gene al accoun o he me hod. Using oday’s o mula o heno mal
app oxima ion, he successp obabili y pcan be es ima ed as p≈ˆp±
zαˆp(1−ˆp)
n,whe e ˆp≡ns
nis he p opo ion o successes in aBe noulli
ial p ocessand se esasanes ima o o pin he unde lying Be noulli
dis ibu ion. He e: ns:Numbe o successes obse ed, n:To al numbe
o ials, zα:Z-sco e co esponding o he desi ed confidence le el(e.g.,
zα=1.96 o 95% confidence).
9Dis ibu ions 227
The BinomialDis ibu ion Unde Ex emeCondi ions
The beha iou o he binomialdis ibu ion unde ex eme condi ions,
suchask=0o k=n−1o k=n,highligh si s sensi i i y o bounda y
alues. A k=n, he p obabili yco esponds o hesuccesso all ials,
a a e e en unless p≈1. Fo k=n−1, he dis ibu ion eflec s
scena ios whe eall bu one ialsucceed, o en showing asignifican
d op in p obabili y. Theseedgecases a e c i ical in unde s anding he
ail beha io o he binomialdis ibu ionand a e o en analyzed in
eliabili y and isk assessmen s.
The Binomial Dis ibu ion o k=0
Theo e ically, i is possible ha abinomial andom a iabledoes no
occu a all in nBe noulli ials.In he binomialdis ibu ion, he case
k=0 ep esen s he p obabili y ha no successes occu in nindepen-
den ials,eachwi h success p obabili y p.These ci cums ances a e
eflec ing he scena iowhe eall n ials esul in ailu e. This case is
pa icula ly ele an inapplica ions suchas isk assessmen o eliabil-
i y es ing, whe e he absence o apa icula ou comeca ies significan
impo ance. Wha consequences a ise om his heo e ical possibili y?
Nume us Legis Na u alis 57. Unde condi ions whe e k=0,i is
n
0=1(509)
P oo by Di ec P oo .
n
k=n!
k!(n−k)!=n!
0!(n−0)!=n!
1·n!=n!
n!=1(510)
Quod e a demons andum.
Nume us Legis Na u alis 58. Unde condi ions whe e k=0,i is
(1−p)n−k=(1−p)n(511)
P oo by Di ec P oo .
(1−p)n−k=(1−p)n−0=(1−p)n(512)
Quod e a demons andum.
228 9Dis ibu ions
Nume us Legis Na u alis 59. Unde condi ions whe e k=0,i is
pk=1(513)
P oo by Di ec P oo .
pk=p0=1(514)
Quod e a demons andum.
In he case k=0, he e en o in e es has no occu ed and we a e
con on ed by condi ions unde which he e en is deemed unlikelyo
p e en ed by ci cums ances. As known, heabsence o an e en can
o en signi y as muchasi s p esence, depending upon hecon ex . In
p obabilis ic e ms, (k=0)in i es eflec ions wha ac o s con ibu ed
o he null esul , andhow does his absence eshape ou unde s anding
o he sys em unde obse a ion? The non-occu ence o some hing
o en emain silen ye p o oundlyconsequen ial.
Nume us LegisNa u alis 60. Unde condi ions whe e k=0, he
numbe o successes equals he o al numbe o ials. I is
p(X=0)=(1−p)n(515)
P oo by Di ec P oo . Unde hese ci cums ances, i is
p(X=0)=n
k=0p0(1−p)n−0(516)
As p oo ed be o e, i is
n
0=n!
0!(n−0)!=1,p0=1,und (1−p)n−0=(1−p)n(517)
Equa ion 516 simplifies o:
p(X=0)=n
k=0p0(1−p)n−0=1·p0·(1−p)n−0=(1−p)n(518)
Quod e a demons andum.
9Dis ibu ions 229
The Gene al Rule o Th ee
The ule o h eecan be gene alized as ollows.
Nume us Legis Na u alis 61. The gene alized o m o heRuleo
Th ee ex ends hees ima ion o he uppe limi o hep obabili y o
a e e en s o a bi a y confidence le els α.The gene alized o m o he
Rule o Th ee is gi en as:
p≤−ln(α)
n(519)
whe e:
•p:The uppe bound on hee en p obabili y.
•α:The desi edconfidence le el (e.g., o 95% confidence, α=
0.05).
•n:The numbe o independen ials.
P oo by Di ec P oo . The Poisson dis ibu ion is used o model
he p obabili y o obse ing agi en numbe o e en sinafixed in e al
when he e en s occu independen ly. The p obabili y mass unc ion
(PMF) o he Poisson dis ibu ion is:
p(X=k)=λke−λ
k!,(520)
whe e λis he expec ed numbe o e en s, kis he numbe o obse ed
e en s, and eis he baseo he na u al loga i hm. Unde condi ions,
whe e k=0, i is
p(X=(k=0)) =λke−λ
k!=λ0e−λ
0! =e−λ(521)
The p obabili y o obse ing ze o e en s when he expec ed numbe o
e en s is λ,isgi en as
p(X=0)=e−λ(522)
The gene al ule o h ee p o idesasimpleapp oxima ion o he uppe
bound on λwhen no e en sa e obse ed(X=0) wi h 100 ·(1−α)%
confidence. To de i e his,wesol e:
236 9Dis ibu ions
The Binomial Hypo hesisTes
The binomial es isanexac s a is ical p ocedu e whe e he es s a is-
ic ollows he binomial dis ibu ion. This es isused oe alua ehy-
po heses abou cha ac e is ics ha ha e exac ly wopossible ou comes
(dicho omous ai s,e.g., ‘Success’o ‘Failu e’, ‘Yes’o ‘No’). The es
de e mines whe he he obse ed equency dis ibu ion aligns wi h an
expec ed o heo e ical equency dis ibu ion π0.Typically, he bino-
mial es helps decide whe he de ia ions be ween obse edand ex-
pec ed esul sa e due o chance o indica e asignifican changein he
unde lying p obabili ies. In ahypo hesis es , i is necessa y o decide
whe he aclaim is ue o no . The binomial es isbased on he binomial
dis ibu ion and is used ode e mine whe he an obse ed alue o suc-
cesses significan lyexceeds an expec ed alue o successesunde H0.A
small p- alue indica es sufficien e idence o ejec H0in a ou o HA.
The igh ailed es and he le ailed es a e examples o one- ailed
es s. The al e na e hypo hesis(HA)de e mines whe he a igh ailed
es o ale - ailed es is gi en.
A igh ailed es (some imes called an uppe es )isa es whe e he
al e na i e hypo hesis HAs a emen con ains ag ea e han(>)symbol.
The hypo heses o he igh - ailed binomial es a e:
H0:p≤p(X≤n−1) s.HA:p>p(X≤n−1)(548)
Decision Rule:
I p- alue igh ailed <α, hen ejec H0:p≤p(X≤n−1),whe e α
is he significance le el.
Ale - ailed es (some imes called alowe es ) con ains aless han
(<)symbol and s a es ha he ue alue o he pa ame e specified in
he null hypo hesisisless han he nullhypo hesis claims.
H0:p≥p(X≤n−1) s.HA:p<p(X≤n−1)(549)
Decision Rule:
I p- aluele ailed <α, hen ejec H0:p≥p(X≤n−1),whe e α
is he significance le el.
9Dis ibu ions 237
Righ -Tailed Binomial Tes inGene al
The igh - ailed binomial es e alua es whe he pis g ea e han a
specific alue π0.
Hypo heses o aRigh -Tailed Tes :
•H0:p≤π0
The obse ed p obabili y pis less han o equal o he heo e ical
p obabili y π0.
•HA:p>π
0
Theobse edp obabili y pis g ea e han he heo e ical p obabili y
π0.
whichisequi alen wi h: H0:π0≥p s.HA:π0<p.
Tes S a is ic:
X∼Binomial(n,π
0).(550)
p-Value:The p obabili yo obse ing Xo ag ea e esul unde H0
is:
p- alue igh ailed =p(X≥k)=
i=n
i=kn
iπi
0(1−π0)n−i(551)
This calcula ion in ol essumming o e hep obabili ies o all possible
ou comes k=x,x+1,···,n.
Decision Rule igh - ailed es :I p- alue igh <α, hen ejec
H0:p≤π0,whe e αis he significance le el.
In a igh - ailed es ,we ocus on finding alues ha a e g ea e han
o equal o he obse ed esul .Ap- alue in a igh - ailed es helps us
unde s and howlikely i is o seeanou come a leas asex emeas he
one we obse ed, assuming he null hypo hesis is ue. I he p- alue
is e y small, i means he esul we obse edisunlikely ohappen by
chance, and we migh hink he null hypo hesis is w ong. I he p- alue
is la ge,i sugges s he obse ed esul is no ha su p ising, so he null
hypo hesis could be co ec .
238 9Dis ibu ions
Al e na i e igh - ailed p-Value
Nume us Legis Na u alis 68. In gene al, he igh - ailed p- alue,
p(X≥k),isequal o 1minus hecumula i ep obabili y up o k−1,
whichisp(X≤(k−1)).Weob ain:
p(X≥k)=1−p(X≤(k−1)) (552)
P oo by Di ec P oo . Fo any disc e e andom a iable X, he
o al p obabili ymus sum o 1:
p(X≤∞)=1(553)
This means ha he p obabili y ha X akes any alue is 1, and i is spli
be ween he wo egions:
•p(X≤(k−1)):The p obabili y ha Xis less han o equal ok−1.
•p(X≥k):The p obabili y ha Xis g ea e han o equal o k.
Since hese wo e en s a emu ually exclusi e(no o e lap), he o al
p obabili y is hesum o he wo p obabili ies:
p(X≤(k−1))+p(X≥k)=1(554)
Rea anging he abo e equa ion, we ge (seeEqua ion 603,p.246):
p(X≥k)=1−p(X≤(k−1)) (555)
Quod e a demons andum.
Fo he case whe e k=n, he p- alue becomes (see Equa ion 604,
246):
p(X≥n)=p(X=n)=1−p(X≤n−1)=1−(1−πn
0)=πn
0≈e−E(X)(556)
In gene al, he eisnopossibili y o mo e han nsuccesses. Thisis
because when k=n, he onlypossible ou come o Xis exac ly n,
and p(X>n)=0. Thus, he p obabili y o X≥nis he p obabili y
o obse ing exac ly nsuccesses, whichcan be calcula ed using he
complemen o p(X≤n−1).
9Dis ibu ions 239
Righ - ailed p-Value o k=0
Le Xbe abinomial andom a iable wi h pa ame e s n(numbe o
ials) and p(p obabili yo success).
Nume us Legis Na u alis 69. TheHypo heses o aRigh -Tailed Tes
a e: H0:p≤π0 s.HA:p>π
0.Fo a igh - ailed es whe e he
obse ed alue k=0, he p- alue is hep obabili y o obse ing ko
any alue g ea e han kunde henullhypo hesis and gi en as
p- alue igh ailed =p(X≥0)=1(557)
P oo by Di ec P oo . The p- alue o a igh - ailed es isgi en
by:
p- alue igh ailed =p(X≥k)=
i=n
i=kn
iπi
0(1−π0)n−i(558)
Subs i u ing k=0in o he o mula:
p- alue igh ailed =p(X≥0)=
i=n
i=0n
iπi
0(1−π0)n−i(559)
Since he summa ion includesall possible ou comes o abinomial an-
dom a iable, i e alua es o:
p- alue igh ailed =p(X≥0)=1(560)
Quod e a demons andum.
The possibili y oapp oxima e he binomial dis ibu ion wi h simple
o mulas offe ssignifican compu a ional and concep ual ad an ages.
Fo la ge n,di ec lycalcula ing binomial p obabili ies in ol es com-
pu ing ac o ials, which g ow exponen iallyand can quicklybecome
compu a ionally expensi eo imp ac ical.App oxima ions,suchas he
no mal o Poisson dis ibu ions,simpli ycalcula ions while main ain-
ing ahigh deg ee o accu acy unde ce ain condi ions.
240 9Dis ibu ions
Righ - ailed p-Value o k=1
Nume us LegisNa u alis 70. Fo aBinomial dis ibu ion X∼
Binomial(n,p), he cumula i e dis ibu ion unc ion (CDF) o p(X≥1)
can be de i edas ollows:
p(X≥1)=1−p(X=0)(561)
P oo by Di ec P oo . The binomial p obabili y mass unc ion
(PMF)isgi en by:
p(X=k)=n
kpk(1−p)n−k(562)
Fo k=0, i is:
p(X=0)=n
0p0(1−p)n=1·(1−p)n=(1−p)n(563)
In gene al, i is
p(X=0)+p(X≥1)=1(564)
The complemen ule gi es:
p(X≥1)=1−p(X=0)(565)
Subs i u e p(X=0)in o he complemen ule:
p(X≥1)=1−(1−p)n(566)
Quod e a demons andum.
Fo aBe noullidis ibu ion, i is n=1, and he equa ion becomes:
p(X≥1)=1−(1−p)n=1−(1−p)1=p(567)
Example.
Fo n=5and p=0.6, i is:
p(X≥1)=1−(1−0.6)5=1−(0.4)5=1−0.01024 =0.98976.
Thismeans he es a98.976% chanceo obse inga leas one success
in 5 ialswi h asuccessp obabili y o 0.6 pe ial.
9Dis ibu ions 241
Righ - ailed p-Value o k=n−1
The Hypo heses o aRigh -TailedTes o k=n−1a e:
H0:p≤π0 s.HA:p>π
0(568)
Nume us Legis Na u alis 71. Fo a igh - ailed es whe e he ob-
se ed alue is k=n−1, he p- alue is hep obabili y o obse ing
k=n−1o any alue g ea e han k=n−1unde henull hypo hesis
and gi en as
p(X≥n−1)=p(X=n−1)+p(X=n)=n·(1−π0)
π0
+1·πn
0(569)
P oo by Di ec P oo . The p obabili y o exac ly X=n−1
e en s is:
p(X=n−1)=n
n−1·πn−1
0·(1−π0)=n·πn−1
0·(1−π0)(570)
The p obabili y o exac ly X=ne en s is:
p(X=n)=n
n·πn
0·(1−π0)0=πn
0.(571)
By defini ion, he igh - ailed p- alueis:
p- alue igh ailed =p(X≥n−1)=p(X=n−1)+p(X=n)(572)
Subs i u ing he p obabili ies, he exac igh - ailed p- alue o X≥n−1
is gi en as:
p- alue igh ailed =p(X≥n−1)=p(X=n−1)+p(X=n)(573)
=n·πn−1
0·(1−π0)+πn
0(574)
=n·(1−π0)
π0·πn
0+πn
0(575)
=n·(1−π0)
π0+1·πn
0(576)
Quod e a demons andum.
242 9Dis ibu ions
In an in es iga ion, 6e en s occu ed ou o n=6138 ials, whe e 0
e en s (see Ba ukčić, 2023c)we e expec ed. The p- alue (seeEqua ion
576)isgi en as:
6138 ·(1−1−6
6138 )
1−6
6138 +1·1−6
6138 6138
=0.01731493293 (577)
The null hypo hesis(H0)is ejec ed. In he popula ion, π0>
0.9990224829 (p- alue ( igh - ailed) =0.0173). The exclusion ela-
ionship is:
pEXCL =1−6
6138=0.9990224829 (578)
The 99.9999% confidence bound (see Equa ion 531), whe e α=
0.000001 and n=6138,isgi en by:
1−−ln(α)
n≤p≤1(579)
o as:
0.9977491837 ≤0.99902248 ≤1(580)
Righ - ailed p-Value o k=n
The Hypo heses o aRigh -Tailed Tes o k=na e:
H0:p≤π0 s.HA:p>π
0(581)
Nume us Legis Na u alis 72. Fo a igh - ailed es whe e he ob-
se ed alue is k=n, he p- alue is hep obabili y o obse ing k=n
o any alue g ea e han k=nunde henull hypo hesis and is gi en
as
p- alue igh ailed =p(X≥n)=p(X=n)=πn
0(582)
P oo by Di ec P oo . The p obabili y o exac ly X=ne en s
is:
p(X=n)=n
n·πn
0·(1−π0)0=πn
0(583)
9Dis ibu ions 243
Since he e a e no alues g ea e han nin abinomial dis ibu ion, he
igh - ailed p- alue issolely:
p- alue igh ailed =p(X≥n)=p(X=n)(584)
Subs i u ing he p obabili y, he exac igh - ailed p- alue o X≥nis:
p- alue igh ailed =p(X≥n)=πn
0(585)
Quod e a demons andum.
In an in es iga ion, 6e en s occu ed ou o n=6138 ials, whe e 0
e en s (see Ba ukčić, 2023c)we e expec ed. Thep- alue (seeEqua ion
585) is gi en as:
p(X≥n)=p(X=n)=πn
0=1−6
6138 6138
=0.01731493293 (586)
The null hypo hesis(H0)is ejec ed. In he popula ion, π0>
0.9990224829 (p- alue ( igh - ailed) =0.0173).
Le -Tailed Binomial Tes inGene al
The le - ailed binomial es e alua es whe he pis smalle han aspe-
cific alue π0.
The hypo heses o ale - ailed es a e:
•H0:p≥π0
The obse ed p obabili y pis g ea e han o equal o he heo e ical
p obabili y π0.
•HA:p<π
0
The obse edp obabili y pis smalle han he heo e ical p obabil-
i y π0.
whichisequi alen wi h: H0:π0≤p s.HA:π0>p.
Tes S a is ic:
X∼Binomial(n,π
0),(587)
whe e Xis he numbe o successes in n ials.
p-Value:The p- aluele ailed o ale ailed binomial es is he
p obabili y o obse ing Xo asmalle esul unde H0and is calcula ed
as:
244 9Dis ibu ions
p- aluele ailed =p(X≤k)=
i=k
i=0n
iπi
0(1−π0)n−i(588)
Decision Rule le ailed:I p- aluele ailed <α, hen ejec H0:
p≥π0,whe e αis he significance le el.
In e ed Pe spec i e
Aglass can alwaysbepe cei ed as ei he hal ullo hal emp y.
Thus, he hypo hesescan equi alen ly be exp essedas:
•H0:1−p≥1−π0,
•HA:1−p<1−π0
The binomial dis ibu ion is disc e e, he exac o mula o p(X≥k)
is:
p(X≥k)=
i=n
i=kn
iπi
0(1−π0)n−i(589)
o equi alen ly:
p(X≥k)=1−p(X<k)=1−p(X≤k−1)(590)
Le - ailed p-Value o k=0
The Hypo heses o aLe -TailedTes o k=0a e:
H0:p≥π0 s.HA:p<π
0(591)
Nume us Legis Na u alis 73. Fo ale - ailed es whe e he ob-
se ed alue is k=0, he p- alue is hep obabili y o obse ing k=0
o any alue smalle han k=0unde henull hypo hesis, and is gi en
as:
p- aluele ailed =p(X≤0)=p(X=0)=(1−π0)n(592)
P oo by Di ec P oo . In gene al, he p- aluele ailed is defined
as:
p- aluele ailed =p(X≤k)=
i=k
i=0n
iπi
0(1−π0)n−i(593)
9Dis ibu ions 245
The summa ion o ale - ailed p- alue, p(X≤k),onlyincludes e ms
om i=0up oi=k.This means i sums o e all ou comes i om 0
(nosuccesses) up o i=k(exac lyksuccesses). The e o e, o he case
k=0, we ob ain he p- alue as:
p- aluele ailed =p(X≤k)=
i=0
i=0n
0π0
0(1−π0)n−0(594)
The p obabili y o exac ly X=0e en s is:
p(X=0)=n
0·π0
0·(1−π0)n=(1−π0)n(595)
Since he e a e no alues smalle han 0inabinomial dis ibu ion, he
exac le - ailed p- alue o X≤0issimply:
p- aluele ailed =p(X≤0)=p(X=0)=(1−π0)n=1−n·π0
nn(596)
Quod e a demons andum.
Now, aking he limi as n→+∞,we ha e:
p- aluele ailed =p(X≤0)=limn→∞ 1−n·π0
nn=e−n·π0≈e−E(X)(597)
Le - ailed p-Value o k=n−1
The Hypo heses o aLe -TailedTes o k=n−1a e:
H0:p≥π0 s.HA:p<π
0(598)
Nume us Legis Na u alis 74. Fo ale - ailed es whe e he ob-
se ed alue is k=n−1, he p- alue is hep obabili y o obse ing
k=n−1o any alue smalle han k=n−1unde henull hypo hesis,
and is gi en as:
p- aluele ailed =p(X≤(k=n−1)) =
i=n−1
i=0n
iπi
0(1−π0)n−i=1−πn
0(599)
252 9Dis ibu ions
Equa ion 585,p.243)i is:
p(X≥n)=πn
0(640)
Based on Equa ion 639 and Equa ion 640,i isequally(see Equa ion
495 on page224):
p(X≥n)=p(X=n)=1−p(X≤n−1)=(p)n=(1−q)n≈e−E(X)(641)
App oxima ion o k=n−1
Nume us LegisNa u alis 80. In gene al, i is
p(X=n−1)=n·z2·e−E(X)
P oo by Di ec P oo . The binomial dis ibu ion is gi en by:
p(X=k)=n
kpk(1−p)n−k,
whe e nis he numbe o ials, kis he numbe o successes, and pis
he p obabili yo success.
Subs i u ing k=n−1, we ha e:
p(X=n−1)=n
n−1pn−1(1−p)n−(n−1).
Simpli y he binomial coefficien ,weob ain:
n
n−1=n!
(n−1)!(n−(n−1))!=n!
(n−1)!·1! =n.
As nex , simpli y he powe o (1−p),i is:
(1−p)n−(n−1)=(1−p)1=1−p.
Fu he mo e, combine he esul s:
p(X=n−1)=n·pn−1·(1−p).
Rew i ing he exp ession:
9Dis ibu ions 253
p(X=n−1)=n·pn
p·(1−p),
and:
p(X=n−1)=n·1−p
p·pn.
Define zas:
z2=1−p
p
whe e:
•pis he p obabili yo success,wi h 0 <p<1,
•1−p ep esen s he p obabili yo ailu e.
Using he defini ion o z2,we ob ain:
p(X=n−1)=n·z2·pn.
Exp essing p=1−q, he equa ion simplifies as:
p(X=n−1)=n·z2·(1−q)n.
Fu he , ew i e:
p(X=n−1)=n·z2·1−n·q
nn
.
Define E(X)=n·q,and in he limi as n→∞,weapp oxima e:
p(X=n−1)≈n·z2·e−E(X).
Quod e a demons andum.
Example.
Conside ossing acoin n=100 imes wi h p=0.5. I is 100·e−50 =
1.92874985 ×10−20 and 0.5100 =7.88860905 ×10−31
Conclusion: -The cumula i ep obabili y up o k=n−1is e y
close o 1, as p=0.5makes he binomialdis ibu ion symme ic.-The
p obabili y o X=n−1(using he app oxima ion) is ex emelysmall
due o he apid dec ease o e−E(X).
254 9Dis ibu ions
P-Value App oxima ion o k=n
In he ollowing, he H0(null hypo hesis) s a es ha he p obabili y
pis less han o equal o he cumula i ep obabili y p(X≤n−1),
sugges ingnosignifican diffe ence om he expec ed ou come. HA
(al e na i e hypo hesis)asse s ha he p obabili y pis g ea e han he
cumula i ep obabili y p(X≤n−1),indica ing ameaning ul de ia ion
om he expec ed ou come. The hypo heses a e o mula ed as ollows:
H0:p≤p(X≤n−1) s.HA:p>p(X≤n−1)(642)
The es s a is ic is X, he obse ednumbe o successes.Unde he null
hypo hesis, he cumula i e p obabili y p(X≤n−1)is he h eshold o
p,and he es assesses whe he pexceeds his h eshold. The binomial
dis ibu ion o X,deno ed X∼Binomial(n,p),ischa ac e ized by n,
he numbe o ials, and p, he p obabili y o success in asingle ial.
The alue o p(X≤n−1)is calcula ed as:
p(X≤n−1)=
n−1
k=0n
kpk(1−p)n−k(643)
Fo ale - ailed es (H0:p≤p(X≤n−1)), he p- aluele ailed is:
p- aluele ailed =p(X≤xobs)=1−e−E(X)(644)
whe e xobs is he obse ednumbe o successes. Fo a igh - ailed es
(HA:p>p(X≤n−1)), hep- alue igh is (see Equa ion 495 on page
224):
p- alue igh ailed =p(X≥n)=p(X=n)=1−p(X≤xobs)=(p)n=(1−q)n≈e−E(X)(645)
which ep esen s he p obabili y o obse ing xobs o mo e suc-
cesses. Example. Unde heseci cums ances and mo e accu a e, i is:
p- alue =(1−(1−pn)) =pn=1−6
6138 6138
=0.00247148903.
In 6ou o 6138 cases, an e en occu ed ha shouldno ha e oc-
cu ed (see Ba ukčić, 2023c). The igh - ailed p- alue igh is calcula ed
app oxima elyas: p- alue igh =e−6=0.002478752177.The igh -
ailed null hypo hesis: H0:p≤p(X≤n−1)is ejec ed, leading o he
accep ance o he al e na i ehypo hesis: HA:p>p(X≤n−1)wi h
p- alue igh ailed=0.002478752177.
9Dis ibu ions 255
S i ling’sapp oxima ion Poisson’s dis ibu ion
Le us assume ha he heo e ical p obabili yo obse ing n·α ailu es
is gi en by he Poisson dis ibu ion o mula:
p(X=n·α)=(n·α)n·αe−(n·α)
(n·α)!(646)
whe e nis he numbe o ials,and αis he significance le el. This
is he expec ednumbe o ailu es unde he nullhypo hesis is e y
small and gi en as λ=n·α.The eal o obse ednumbe o ailu es is
deno ed by k,while he p obabili yo obse ing exac ly k ailu es is:
p(X=k)=(n·α)ke−(n·α)
k!(647)
whichis he p obabili yo obse ing exac ly k ailu es unde he same
assump ions. These wo p obabili ies can be compa ed di ec ly:
p(X=n·α)<p(X=k), his sugges s ha he obse ed numbe o
ailu es kis mo e likely han he expec ednumbe n·α.
Decision Rule:
I p(X=(n·α)) <p(X=k), his sugges s ha he obse ednumbe
o ailu es kis mo e likely han he expec ednumbe n·α.This could
be e idence agains he null hypo hesis H0,sugges ing ha he obse ed
da a is significan lydiffe en om wha wasexpec ed.
I p(X=n·α)>p(X=k), his sugges s ha he obse ednumbe
o ailu es kis less likely han he expec ednumbe n·α,meaning he
obse edda a isclose o he expec ed da a unde H0.
In hypo hesis es ing, hiscompa ison helps de e mine whe he he
obse ed da a isconsis en wi h he null hypo hesis H0,o i i sugges s
asignifican de ia ion ha may wa an ejec ing H0.
Nume us Legis Na u alis 81. S i ling’sapp oxima ion p o ides an
es ima ion o he Poisson dis ibu ion, exp essedas:
p(X=k)≈ 1
2π(n·(1−p))≈1
√2π·λ
(648)
P oo by Di ec P oo . We a e gi en he Poisson dis ibu ion
p obabili y mass unc ion:
256 9Dis ibu ions
p(X=n·(1−p))=(n·(1−p))n·(1−p)e−(n·(1−p))
(n·(1−p))!(649)
while pisex emelyclose o1.Fo la ge n·(1−p),weuse S i ling’s
app oxima ion (see S i ling,1730) o ac o ials:
n!≈√2πnn
en(650)
Subs i u ing hisin o (n·(1−p))!, we ge :
(n·(1−p))!≈2π(n·(1−p))n·(1−p)
en·(1−p)(651)
Subs i u eS i ling’sapp oxima ion in o heo iginal Poisson o mula:
p(X=n·(1−p))≈ (n·(1−p))n·(1−p)e−(n·(1−p))
2π(n·(1−p))n·(1−p)
en·(1−p)(652)
The e ms (n·(1−p))n·(1−p)cancel ou , lea ing:
p(X=n·(1−p))≈ 1
2π(n·(1−p))·e−(n·(1−p))+(n·(1−p))=1
2π(n·(1−p))(653)
Fo la ge λ=n·(1−p), he Poisson dis ibu ion p obabili y mass
unc ion can be app oxima edas:
p(X=k)≈ 1
2π(n·(1−p))≈1
√2π·λ
(654)
Quod e a demons andum.
Example. Se n=5500 and (1−p)=α=0.01.Wecalcula e
he app oxima ion as: 1
2π(5500·0.01)
=0.05379336612.I isn·α=
5500 ·0.01 =55.Asnex , we use he Poisson dis ibu ion o mula,
known as: p(X=n·α)=(n·α)n·αe−(n·α)
(n·α)!and subs i u e n·α=55
in o he o mula, yielding: (55)55e−55
(55)!=0.05371192363.This shows ha
he app oxima ion is e y close o he exac alue compu ed using he
Poisson dis ibu ion.
9Dis ibu ions 257
P-Value and significance le el α
Se he significance le el α o:
α=0.05.
Based on he significance le eland hesamplesize n,wede i e he
heo e ical p obabili yo asinglee en o k=nas ollows.
Nume us Legis Na u alis 82. Fo k=n,i holds ha :
p heo e ical =eln(α)
n
P oo by Di ec P oo . In gene al, o k=n:
p- alue igh - ailed =p(X≤k)=
x=n
x=nn
npn(1−p)n−n=pn.
Fu he mo e, he ela ionshipbe ween hecalcula ed p- alue and he
significance le elis: p- alue igh - ailed =pn
heo e ical =α. Taking he na -
u al loga i hm on bo hsides gi es:
n·ln(p heo e ical)=ln(α).
Sol ing o ln(p heo e ical),we ob ain:
ln(p heo e ical)=ln(α)
n.
Exponen ia ing bo h sides yields:
p heo e ical =eln(α)
n(655)
Quod e a demons andum.
Example. Unde condi ions,whe e α=0.000001 and n=6138 ,i
is:
eln(0.000001)
6138 =0.9977517149 (656)
whichclosely app oxima es(seeEqua ion 531, p. 231):
1−−ln (0.000001)
6138 =0.9977491837 (657)
258 9Dis ibu ions
The Binomial Hypo hesisTes :Examples
ExampleI.P-Values o aFai Coin
Example A
We a econside ing a ai coin (p=0.5),wi h n=100 osses and
k=51 obse edheads. The binomialdis ibu ion’s p obabili y mass
unc ion(PMF) is gi enby:
p(X=k)=n
kpk(1−p)n−k(658)
whe e: n=100 is he numbe o ials, k=51 is he numbe o
successes (heads), p=0.5is he ( heo e ical) p obabili y o success(a
head) in asingle ial.
Le -Tailed Tes The le - ailed hypo hesis is:
H0:p≥p(X≤51) s.HA:p<p(X≤51)(659)
Fo ale - ailed es ,wecalcula e he cumula i ep obabili y o obse -
ing k≤51,meaning he p obabili y o ob aining 51 o ewe heads:
pX ≤k)=
x=k
x=0n
xpx(1−p)n−x=
k
x=0
p(X=x)(660)
The p- aluele - ailed is: p- aluele - ailed =p(X≤51)=0.618. H0canno
be ejec ed.
Righ -Tailed Tes The igh - ailed hypo hesis is:
H0:p≤p(X≤51) s.HA:p>p(X≤51)(661)
Fo a igh - ailed es ,wecalcula e he p obabili y o obse ing k≥51,
meaning 51 o mo eheads:
p(X≥k)=
x=n
x=kn
xpx(1−p)n−x=
n
x=k
p(X=x)(662)
This can alsobeexp essedusing he complemen a ycumula i ep ob-
abili y:
p(X≥k)=1−p(X<k)=1−p(X≤k−1)(663)
9Dis ibu ions 259
The p- alue igh - ailed is: p- alue igh - ailed =p(X≥51)=0.460. H0
canno be ejec ed. These esul s eflec hep obabili ies o obse ing
ou comes as ex eme (o mo eex eme) as k=51 unde he null
hypo hesis o a ai coin.
Example B
We a e conside ing a ai coin (p=0.5),wi h n=100 osses and
k=100 obse edheads.The binomial dis ibu ion’s p obabili y mass
unc ion (PMF) o n=kis gi en by:
p(X=(k=n)) =n
kpk(1−p)n−k=n
npn(1−p)n−n=pn(664)
whe e:
•n=100 is he numbe o ials,
•k=100 is he numbe o successes (heads),
•p=0.5is he heo e ical p obabili yo success(ahead) in asingle
ial.
Le -Tailed Tes
The p- alue o he le - ailed es is he p obabili y o obse ing X≤k.
Since k=100 (all heads), he le - ailed p- alue is:
p(X≤100)=p(X=100)
The p obabili y p(X=100)is gi en by he PMF:
p(X=100)=100
100(0.5)100(1−0.5)0
p(X=100)=1·(0.5)100
Righ -Tailed Tes
The p- alue o he igh - ailed es is hep obabili y o obse ing
X≥k.Since k=100, he igh - ailed p- alue is:
p(X≥100)=p(X=100)
260 9Dis ibu ions
F om he p e ious calcula ion:
p(X=100)=(0.5)100
Fo n=100, k=100,and p=0.5, he p obabili y is:
p(X=100)=(0.5)100 ≈7.89 ×10−31.
The e o e:
•Le - ailed p- alue: p(X≤100)=p(X=100)≈7.89 ×10−31,
•Righ - ailed p- alue: p(X≥100)=p(X=100)≈7.89 ×10−31.
Gene al Case o n=k=1
Fo n=k=1, he PMF simplifies o: p(X=1)=1
1p1(1−p)0=p.
Thus, o k=n=1, i is
•Le - ailed p- alue: p(X≤1)=p(X=1)=p,
•Righ - ailed p- alue: p(X≥1)=p(X=1)=p.
Hume’s p oblem o induc ion
Howp obable isi ha e en s obse edn−1 imes will hold in
gene al e en o he nex ins ance (n)? Thisques ion lies a he hea o
Hume’s p oblem o induc ion (see Hume,1739,Book 1, pa III, sec ion
6, pp. 157-167), which challenges he logical jus ifica ion o induc i e
easoning. The p- alue canbeause ul ool o assess he ex en o which
suchinduc i e in e ences a e suppo ed by empi ical e idence, hough
i canno ullygua an ee ha he e en will occu a he ial naswell.
In gene al, o k=n,i is
p(X≥n)=p(X=n)=1−p(X<n)(665)
=1−p(X≤(n−1)) (666)
=1−(1−p(X=n))) (667)
9Dis ibu ions 261
Example II
An in es iga ionhas been pe o med. The obse edda aincludes:
•k=8successes,
•2 ailu es,
•To al ials: n=10.
The significance le el is α=0.05.
Righ -Tailed p-Value
Suppose we a e es ing he ollowing igh - ailed hypo heses:
•H0:π=0.4( he popula ion success p obabili yis0.4),
•HA:π>0.4( he popula ion successp obabili y is g ea e han
0.4).
P obabili ies o π=0.4,n=10
The binomialdis ibu ion o π=0.4,n=10 is gi en as:
p(X=i)=n
iπi(1−π)n−i(668)
The p obabili ies o p(X=k) om k=0 ok=10 o he gi en
binomialdis ibu ion (n=10,π=0.4):
p(X=0)=10
00.40(1−0.4)10−0=0.0060466176 (669)
p(X=1)=10
10.41(1−0.4)10−1=0.0403107840 (670)
p(X=2)=10
20.42(1−0.4)10−2=0.1209323520 (671)
p(X=3)=10
30.43(1−0.4)10−3=0.2149908480 (672)
268 9Dis ibu ions
Z(X)is ypically defined(see Kelley, 1924)as:
Z(X)=X−µ
σ,(702)
whe e µ=E(X)and σ=σ(X)deno e he expec ed alue and s anda d
de ia iono he andom a iable X, espec i ely. Howe e , applying
afixedglobal mean and s anda dde ia ionac ossall ials,isno
alwaysand in e e ysingle ins ance ullyjus ified. Unde condi ions
we e E(XR, )and σ(XR, ) a y a eachindi idual un o he expe imen
, eachindi idual ealiza ion XR, o he uno he expe imen has
aunique mean E(XR, )and σ(XR, )s anda d de ia ion. The s anda d
no mal a iable Z(X) need oaccoun o hese ial-specific pa ame e s
unde hese ci cums ances. Fo aspecific, single uno ial wi h a
unique ealiza ion XR, , he o mula o he s anda d no mal a iable o
asingle uno anexpe imen can be w i en as:
Z(XR, )=XR, −E(XR, )
σ(XR, )=
EXR,
2
EXR, ·EXR, (703)
whe e:
•XR, is he obse ed ealiza ion o he andom a iable Xwi hin he
pa icula condi ions o ial ,
•E(XR, )is he expec ed alue o mean o Xas de i ed specifically
wi hin ial ,
•σ(XR, ) ep esen s he s anda d de ia ion o X o ial ,encapsu-
la ing i s unique a iabili y.
The exp ession XR, −E(XR, )
σ(XR, )s anda dizes each XR, o as anda d no mal
a iable (i.e., Z-sco e).I XR, ollows ano mal dis ibu ion, hen each
e min he summa ion will ollowas anda d no mal dis ibu ion oo.
Agg ega ing ac oss n ials, we ob ain s anda d no mal a iable o
he whole sample/popula ion, Z(XR),as:
Z(XR)=
n
=1
Z(XR, )=
n
=1XR, −E(XR, )
σ(XR, )=X−µ
σ(704)
9Dis ibu ions 269
The chi-squa edis ibu ion is ounda ional in s a is ical hypo hesis
es ing and a iance es ima ion. Specifically,when Z1,Z2,...,Zna e
independen s anda dno mal a iables, hesum o hei squa es,
Z2
1+Z2
2+···+Z2
n, ollows achi-squa edis ibu ion wi h ndeg ees
o eedom. Thus,achi-squa edis ibu ion wi h ndeg ees o eedom,
deno ed χ2
n,is o mally defined as he sum o he squa es o nindepen-
den s anda d no mal a iables(see Sachs,1992,p.213).
χ2
n=
n
=1
Z2
=
n
=1XR, −E(XR, )
σ(XR, )2
=
n
=1
EXR,
EXR,
(705)
In quan um heo y, EXR, is some imes he expec a ion alue
o he local hidden a iable. The ndis ibu ion, commonlyknown as
he S uden s -dis ibu ion,isap obabili ydis ibu ion o en used in
s a is ics, especially o smalle sample sizes o when he popula ion
a iance is unknown and defined (see Sachs,1992,p.213) as:
n=Z
χ2
n
n
(706)
This dis ibu ion ispa ame e ized by i s deg ees o eedom,de-
no ed as n,whichaffec s he shape o hedis ibu ion. As n→∞,
he -dis ibu ion app oaches he s anda d no mal dis ibu ion, which
is use ul in hypo hesis es ing andconfidence in e ales ima ion. The
-dis ibu ionispa icula ly use ul o cases whe e he sample mean is
used o es ima e apopula ion mean, and hepopula ion s anda d de i-
a ion is unknown. The o mula o he -dis ibu ionwi h ndeg ees o
eedomis:
n=
¯
X−µ
S
√n
(707)
whe e ¯
Xis he samplemean, µis he popula ion mean, Sis he
sample s anda dde ia ion, and nis hesample size.The sample s anda d
de ia ion Sis gi en by:
270 9Dis ibu ions
S=
1
n−1
n
i=1(Xi−¯
X)2(708)
whe e:
•nis he sample size,
•Xi ep esen s eachindi idual obse a ion in he sample,
•¯
X=1
n
n
i=1
Xiis hesamplemean.
This dis ibu ion waso iginallyde eloped by William SealyGosse
(see Gosse ,1908b)unde he pseudonym S uden in 1908. The -
dis ibu ion playsacen al ole in a ious s a is ical es s,suchas he
- es o compa ing samplemeans.
Some hingand i s owno he
The ollowing heo emiden ifies ha E(UR, )is akind o acoun e -
pa oE(UR, ),highligh ing an opposi e ela ionshipbe ween hese
woen i ies i.e expec a ion alues. Specifically,knowledgeo Zand
E(UR, ) he expec ed alue o UR, p o ides insigh in o E(UR, ),which
is some imes ea ed as an indica o o an unde lying, possiblylocal hid-
den pa ame e wi hin ace ainsys em. This pa ame e can be iewed
as ep esen ing akind o An i-UR, o an al e na i eexp ession o UR, ,
sugges ing ha he beha iou o one expec a ion inhe en lyde e mines
he o he .Inab oade scien ific sense, his ela ionship e ealsas uc-
u ed duali ybe ween hese wo en i ies,sugges ing ha knowledgeo
one aspec leads oanimplici unde s anding o i s complemen .
Nume us Legis Na u alis 83. In gene al, i holds ha
E(UR, )=z(UR, )2·E(UR, )(709)
P oo by Di ec P oo . S a ing om Axiom 1, whichs a es ha
+1=+1(710)
we ex end hisiden i y u he in o de o ob ain
UR, =UR, (711)
and also,
9Dis ibu ions 271
UR, −E(UR, )=UR, −E(UR, )(712)
Nex , by no malizing he mean de ia ion, we a i e a
UR, −E(UR, )
σ(UR, )=UR, −E(UR, )
σ(UR, )(713)
leading us o define z(UR, )as
z(UR, )≡UR, −E(UR, )
σ(UR, )(714)
and hence,
z(UR, )2=U2
R, ·(1−p(UR, ))2
σ(UR, )2(715)
Equa ion715 simplifies as:
z(UR, )2=U2
R, ·(1−p(UR, ))2
U2
R, ·p(UR, )·(1−p(UR, ))(716)
Finally, z(UR, )2is gi en as
z(UR, )2=(1−p(UR, ))
p(UR, )(717)
In gene al, i is
(1−p(UR, )) =z(UR, )2·p(UR, )(718)
and 1
z(UR, )2=p(UR, )
(1−p(UR, ))(719)
The ac ion 1−p(UR, )
p(UR, )(see Equa ion 717)can be spli as:
z(UR, )2=1−p(UR, )
p(UR, )=1
p(UR, )−p(UR, )
p(UR, )=1
p(UR, )−1(720)
I is z(UR, )2+1·p(UR, )=1(721)
The p obabili yo asingle e en is gi en as:
272 9Dis ibu ions
p(UR, )=1
z(UR, )2+1(722)
Equa ion 722 ensu es ha p(UR, )>0 o all eal alues o z(UR, ).The
denomina o z(UR, )2+1isalwaysg ea e han o equal o 1, as squa es
o eal numbe s a e non-nega i e. Based on z-sco e, p(UR, )canno be
ze o, ensu ing he p obabili y emains s ic ly posi i e. Th ough u he
manipula ion o Equa ion 715,wees ablish equally ha
z(UR, )2=E(UR, )2
σ(UR, )2(723)
and also ha
z(UR, )2=E(UR, )2
E(UR, )·E(UR, )(724)
This esul isob ained consis en lyand we conclude ha
z(UR, )2=E(UR, )
E(UR, )(725)
and he e o e, i is gene ally ue ha
E(UR, )=z(UR, )2·E(UR, )(726)
Quod e a demons andum.
The equa ion E(UR, )=z(UR, )2·E(UR, ) esembles Eins eins mass-
ene gy equi alence ER, =c2·mR, ,asbo h exp ess ap opo ional
ela ionshipbe weenene gyand ano he sys emcha ac e is ic,media ed
by asqua ed pa ame e . While Eins einsequa ion ela es ene gy o mass
wi h he cons an c2, he o me equa ion shows how E(UR, )dependson
E(UR, ),wi h z(UR, )2ac ing as asys em-specific scaling ac o .Bo h
equa ions showap opo ional scaling o pa ame e s.S ill, heydiffe
in scope, wi hEins ein’sequa ion ep esen ing a undamen al physical
law, and he second equa ion ep esen ing ap obabilis ic ela ionship
specific oasys em.
9Dis ibu ions 273
Va iance and s anda dno mal a iable
The s a egic significance o his p oo lies in i s abili y o link he
a iance o a andom a iable σ(UR, )2wi h he z- alue z(UR, ),ano -
malized measu e o en assumed o eadilycalculable. By es ablishing a
ela ionship be ween he expec a ion alues o UR, and UR, , he p oo
enables he calcula ion o a iance h ough hez-sco e,which simplifies
hep ocess o de e mining a iabili y in si ua ions whe e di ec a i-
ance calcula ion maybecomplexo in easible. This app oachle e ages
hepowe o expec a ion alues and hei s a is ical ela ionships, o -
e ing a obus ool o analyzing hesp ead o unce ain y o andom
a iables in p obabilis ic sys ems. In p ac ical e ms, he z- alue se es
as ab idge ocompu e a iance, u ning abs ac s a is ical concep s
in o angible,compu able quan i ies.
Nume usLegis Na u alis 84. In gene al, he ollowing holds:
σ(UR, )2=E(UR, )·E(UR, )=z(UR, )2·E(UR, )·E(UR, )(727)
P oo by Di ec P oo . S a ing wi hAxiom 1, whichs a es:
+1=+1(728)
i ollows ha his iden i y holds ue in all ins ances.Consequen ly, we
addi ionallyes ablish he equi alence:
E(UR, )=E(UR, )(729)
Re e ing o equa ion 726 om ea lie ,wesubs i u e i in o he equa ion
abo e o yield:
E(UR, )=z(UR, )2·E(UR, )(730)
Mul iplyingbo hsides o equa ion 730 by E(UR, ),wede i e he ol-
lowing exp ession:
σ(UR, )2=E(UR, )·E(UR, )=z(UR, )2·E(UR, )·E(UR, )(731)
whichisequi alen wi h he ela ionship:
σ(UR, )2=E(UR, )·E(UR, )=z(UR, )2·E(UR, )·E(UR, )(732)
Quod e a demons andum.
274 9Dis ibu ions
No mal andan ino mal dis ibu ion
Ilija Ba ukčić
(1.10.1961)
The p obabili ydensi y unc ion o he no mal dis ibu ion illus a ed
by Figu e 11 is exp essed as:
p(X)=1
σ√2π
e−1
2x−µ
σ2
=1
σ√2π
e−1
2Ex
E(x),x∈R(733)
whe e µis he mean (cen e )o he dis ibu ion, σis he s anda d
de ia ion (sp ead) o he dis ibu ion, xis he andom a iable, xis he
an i andom a iable.
Fig. 11: S anda d No mal Dis ibu ion.
x
p(X)
+∞
−∞
The la ge σbecomes, he fla e he cu esp og ession ( esul ing in
ab oade cu e and alowe peak). The e m 1
√2πwasin oduced by
Ca lF ied ich Gaussin he ea ly19 h cen u yaspa o his wo k on
p obabili y heo yand s a is ics. This e mo igina es om he p ocess
o no malizing he dis ibu ion, ensu ing ha he o al a ea unde he
cu e equals1,whichisa undamen al equi emen o any p obabili y
dis ibu ion. The p esence o πconnec s he p obabili y heo y o he
geome y(see Ba ukčić, 2023e). Assuming ha he ma hema ical
o mula o he no mal dis ibu ion is co ec ,whichhas no been e ified
a his poin ,weob ain he p obabili y densi y unc ion o he an ino mal
dis ibu ion exp essed as:
p(X)=1−p(X)=1−1
σ√2π
e−1
2x−µ
σ2
,x∈R(734)
9Dis ibu ions 275
Fig. 12: No mal and An i No mal Dis ibu ion.
x
p(X)
+∞
−∞
An i no mal dis ibu ion
No mal dis ibu ion
The ela ionshipbe ween he no mal dis ibu ion and he an i-no mal
dis ibu ion is illus a edin he p e ious figu e pu ely o cla i yand
be e unde s anding whilea he same imese ing as a empla e o
o he dis ibu ions (Poisson and an i Poisson dis ibu ion e ce e a (see
Ba ukčić,2022c).
Unde ce ainassump ions, hecausal ela ionship kcanbeexp essed
using he s anda d no mal a iable Z.
Nume us Legis Na u alis 85. The s anda dno mal a iable Zand
hesample size na e ela edas:
n
i=1UR, −EUR,
σUR, ·WR, −EWR,
σWR,
n
=z2
n(735)
P oo by Di ec P oo .
The p oo begins by assuming he iden i y
UR, =WR, (736)
be ween hese wo en i ies despi e anyunde lying dis inc ions.Since
UR, =WR, , hei expec ed alues a e alsoequal:
EUR, =EWR, (737)
I ollows ha he expec ed squa es o UR, and WR, a e equal:
EUR, 2=EWR, 2(738)
276 9Dis ibu ions
Since UR, =WR, , hei squa ed aluesa e equal:
U2
R, =W2
R, (739)
The expec ed alues o he squa ed e ms a ealsoequal:
EU2
R, =EW2
R, (740)
Sub ac ing EUR, om each a iable yields:
UR, −EUR, =WR, −EUR, (741)
Gi en E(UR, )=E(WR, ), he abo e exp ession implies:
UR, −EUR, =WR, −EWR, (742)
Di iding Equa ion 742 by σUR, i is
UR, −EUR,
σUR, =WR, −EWR,
σUR, (743)
i is σUR, =σWR, .Equa ion 743 is ea anged as
UR, −EUR,
σUR, =WR, −EWR,
σWR, (744)
I is
zUR, =UR, −EUR,
σUR, (745)
and
zWR, =WR, −EWR,
σWR, (746)
Equa ion 744 becomes
zUR, =zWR, (747)
o
zUR, 2=zWR, 2(748)
Based on Equa ion 747 we ea angeEqua ion 748 as:
9Dis ibu ions 277
zUR, ·zWR, =zWR, 2(749)
Unde condi ions o n uns o an expe imen , we ob ain
n
i=1zUR, ·zWR, =
n
i=1zWR, 2=z2(750)
Based on Equa ion 744,we ob ain
n
i=1UR, −EUR,
σUR, ·WR, −EWR,
σWR, =z2(751)
Di iding 751 by n, i is
n
i=1UR, −EUR,
σUR, ·WR, −EWR,
σWR,
n
=z2
n(752)
The e a e ci cums ances,whe e hecausal ela ionshipkis gi en as:
k=z2
n
=
n
i=1UR, −EUR,
σUR, ·WR, −EWR,
σWR,
n(753)
Quod e a demons andum.
Gi en he p obabili ydensi y unc ion p(Z)o he s anda d no mal
dis ibu ion:
p(Z)=1
√2π
e−Z2
2,z∈R(754)
we ea ange his equa ion as:
√2π·p(z)=e−z2
2(755)
Take he na u al loga i hm, ln:
ln(√2π·p(z)) =ln e−z2
2(756)
Simpli y equa ionas:
284 9Dis ibu ions
Chi-Squa eDis ibu ionand Causal Rela ionship k
The ela ionshipbe ween he chi-squa ed dis ibu ion (seePea son,
1900b)and he causal ela ionship k e eals undamen al insigh s abou
dependency s uc u es be ween andom e en s.Thisconnec ion allows
k o measu e he ex en o whichobse edoccu ences can be ea ed
as causally ela ed, hus playing api o al ole in hypo hesis es ing and
causal in e ence.
Nume us LegisNa u alis 88. In gene al, he ela ionship be ween
hechi-squa ed dis ibu ion and hecausal ela ionship kis gi en by:
kUR, ∩WR, =2
χ2
EUR ·EWR (782)
P oo by Di ec P oo . We ini ia e he p oo wi h Axiom 1,
conside ed sel -e iden and equi ing no addi ional jus ifica ion:
1=1(783)
Mul iplyEqua ion 783 by he causal ela ionship kUR, ∩WR, ,
yielding:
kUR, ∩WR, =kUR, ∩WR, (784)
We can exp ess Equa ion 784 in de ail as:
kUR, ∩WR, =UR, ·WR, ·pUR, ,WR, −pUR, ·pWR,
U2
R, ·pUR, ·1−pUR, ·W2
R, ·pWR, ·1−pWR,
=σUR, ,WR,
σUR, ·σWR,
(785)
Simpli ying Equa ion 785 u he ,weob ain:
kUR, ∩WR, =EUR, ,WR, −EUR, ·EWR,
E(UR )·EUR ·E(WR )·EWR (786)
Squa ing Equa ion 786,we ha e:
kUR, ∩WR, 2=EUR, ,WR, −EUR, ·EWR, 2
E(UR )·EUR ·E(WR )·EWR (787)
Since he chi-squa ed e misdefined as
9Dis ibu ions 285
χ2=EUR, ·WR, −EUR, ·EWR, 2
EUR, ·EWR, (788)
we can subs i u e his in o Equa ion 787:
kUR, ∩WR, 2=χ2
EUR ·EWR (789)
The ela ionshipbe ween he causal ela ionshipand he chisqua e
dis ibu ion is gi en as:
kUR, ∩WR, =2
χ2
EUR ·EWR (790)
Quod e a demons andum.
Ka lPea son
(1857–1936)
Ka lPea son defined (see Pea son,1900b,1904) he phi coefficien
φo a2×2con ingency able as:
φ=χ2
n(791)
whe e χ2is he chi-squa ed s a is ic and nis he o al sample size. Unde
he condi ion ha EUR ·EWR =n,i ollows ha
kUR, ∩WR, =φ(792)
;howe e , his equi alence does no hold uni e sally.
286 9Dis ibu ions
Empi ical Cumula i e Dis ibu ion Func ion (ECDF)
T adi ional s a is ical me hods, such as he - es and o he me hods,
equi e assump ions abou he dis ibu ions o he da a(e.g., no mali y).
Se ious p oblem a ise equen ly in s a is ical analysis,especiallywhen
he exac dis ibu ion o he da a is unknown o whenno mali y is no
gi en o su e. Dis ibu ion- ee es s,alsoknown as non-pa ame ic
es s, a e s a is ical me hods ha do no elyonassump ions abou
heunde lying dis ibu ion o he da a. Exampleso dis ibu ion- ee
es sinclude he Mann-Whi neyU es ( o compa ing woindependen
samples), he Wilcoxonsigned- ank es ( o pai ed samples), and he
K uskal-Wallis es ( o compa ing mo e han woindependen g oups).
Howe e , me hods basedonempi ical cumula i edis ibu ion unc ions
wi hou assumingany specificunde lying dis ibu ion o he da a could
be o use oo. The Empi ical Cumula i eDis ibu ion Func ion (ECDF)
as suchisanon-pa ame ic es ima o o he cumula i edis ibu ion
unc ion(CDF) o asample o da a. An ECDF migh p o ide away (see
D ion, 1952) oes ima e he cumula i ep obabili y o agi en alue
based on heobse edda a.
In gene al, gi enase o nobse a ions, x1,x2,...,xn, he ECDF is
defined as he p opo ion o he sampleless han o equal oapa icula
alue x.Ma hema ically, he ECDF a apoin xis:
Fn(x)=1
n
n
=1
I(x ≤x),
whe e:
•Fn(x)is he empi ical CDF a x,
•xis afixed h eshold(no ied o indi idual ials),
•x a e heobse edou comes o he andom a iables X
•I(x ≤x)is an indica o unc ion whichcheckswhe he eachob-
se edou come x is less han o equal o he fixed h eshold x.I
x ≤x, hen I(x ≤x)=1, and I(x ≤x)=0o he wise.
•nis he o al numbe o obse a ions,
The ECDF is especially use ul whenaspecificunde lying dis ibu ion
is no assumed and when he cumula i ep obabili y is calcula ed om
he da a a hand. Rules o using he Empi ical Cumula i eDis ibu ion
Func ion can be ound in seconda yli e a u e.
9Dis ibu ions 287
Righ ailed es
In gene al, i isapp op ia e o dis inguish among he expec ed alue
o he andom a iable X, he singleexpec ed alue o he andom
a iable X a aspecific ial ,and he indi idual ou come x o he
andom a iable X a aspecific ial .To educe he complexi y o
ma hema ical no a ions and o simpli y ma hema ical exp essions,
in his en i epublica ion p(X )is equen lyused as asho hand o
p(X =x ).None heless,le Xdeno e adisc e e andom a iable defined
on afini e sample space, whe e he p obabili y p(X =x ) ep esen s
he likelihood o X aking he alue x a he ial =1,2,...,n.Unde
hese de e minis iccondi ions,each x is fixed o each wi h i s own
p(X =x ) o all .
Nume us Legis Na u alis 89. The igh - ailed pValue based on
empi ical cumula i e dis ibu ion unc ion is gi enas:
p- alue ( igh - ailed) =n−kobs +1
n(793)
P oo by Di ec P oo . We beginou p oo wi h he lawo
iden i y,a undamen alp inciple in logic and ma hema ics ha asse s:
1=1(794)
This equali y isuni e sally alid and o ms he basis o es ablishing
u he ela ionships and de i a ions in ou p oo . In he con ex o a
single ial ,wea e dealing wi hone specific ou come x o he single
andom a iable X .Ingene al, in asingle ial, he ou comeo he
andom a iable X is x ,whichisde e minedbyi s singlep obabili y
p(X =x )while heexpec ed alue o his single ial is defined
as E(X ).Building on he sel -e iden ounda ion as s a ed be o e, i
logically ollows ha i is equally
E(X )=E(X )(795)
Howe e , he expec ed alue o asingle ial E(X )which ep esen s he
ou come x weigh ed by he p obabili y p(X =x )is ma hema ically
gi en as:
E(X )=x ·p(X =x )(796)
288 9Dis ibu ions
unde he assump ion ha p(X =x )0. This app oachemphasizes
ha he expec ed alue is aweigh ed a e ageo possible ou comes, and
in hesingle-ou come de e minis iccase, he weigh is simply p(X =
x ).Inb ie , i is
p(X =x )=E(X )
x
(797)
In he con ex o asingle ial, he e is onlyone ou comeand we do no
sum o e possibleou comes.None heless, i is necessa y o conside
mul iple ials (o e n), he o al expec ed alue E(X)is he sum o
he single expec ed alues o eachindi idual ial. In o he wo ds, we
sum he singleexpec ed alues o e all ials.The o al expec ed alue
E(X)ac ossall n ials, whe e each ial has i s ownou come x and i s
associa ed p obabili y p(X =x )is gi en as:
E(X)=
n
=1
x ·p(X=x )(798)
=E(X1)+E(X2)+···+E(Xn)(799)
=
n
=1
E(X )(800)
whe enis hesampleo popula ionsize.Unde de e minis iccondi ions,
whe e p(X =x )=1, he expec a ion alue i sel educes o he
obse ed alue:
E(X)=
n
=1
x ·p(X =x )=
n
=1
x ·1=
n
=1
x (801)
When he expec ed alue o each ial, E(X ),isiden ical ac oss all
ials =1,...,n, he o al expec ed alue E(X)simplifies o:
E(X)=
n
=1
x ·p(X =x )=
n
=1
E(X )=n·E(X =1)(802)
Unde hesede e minis ic condi ions,whe e p(X =x )is cons an
o all ials and each ial has afixedde e minis ic ou comeX =1, he
expec a ion alue simplifies o:
9Dis ibu ions 289
E(X)=
n
=1
E(X )(803)
=
n
=1
x ·p(X =x )(804)
=
n
=1
1·p(X =x )(805)
=n·p(X =x )(806)
In he p esence o uni o mi yac oss ials,i ollows ha :
p(X =x )=E(X)
n(807)
In gene al, unde de e minis ic condi ions whe e p(X =x )=1 o all
ials and whe e X =1has afixedde e minis ic ou comea each ial,
we do expec ha kobs =n.Howe e ,due o e o s in measu emen o
o he sou ces o bias e ce e a, he obse ed o al kobs mayde ia e om
n. S ill, howcan we,unde he ci cums ances whe e kobs <nsuccesses
a e obse edou o n ials, assume wi h some p obabili y ha kobs =n?
In o he wo ds, howcan we calcula e he igh - ailed p- alue unde hese
condi ions? In gene al, he igh - ailedp- alue is defined as:
p- alue ( igh - ailed) =p(X ≥kobs)(808)
=
n
x =kobs
p(X =x )(809)
=1−p(X ≤(kobs −1)) (810)
whe e:
•kobs is heobse ed alue o he andom a iableX,
•X is he andom a iable o ial ,
•p(X =x )is he p obabili ymass unc ion (pm ) o X ,which
gi es he p obabili yo obse ing aspecific ou come x o ial ,
•The summa ion uns o e all ou comes om he obse ed alue
kobs o he maximum possible alue n, ep esen ing he numbe o
ials.
290 9Dis ibu ions
E(X)=
n
x =kobs
E(X )(811)
=
n
x =kobs
x ·p(X =x )(812)
=
n
x =kobs
1·1(813)
=
n
x =kobs
1(814)
The summa ion:
n
x =kobs
1(815)
is essen iallysumming he alue 1o e each alue o x om kobs o n.
So, i coun show many alues o x a e in he ange om kobs o n.
-The a iable x can akein ege alues, and we sum 1 o each alue
o x om kobs o n.
-The esul o he summa ion will simply be he o al coun o in ege s
om kobs o n,inclusi e.
The in ege s om kobs o n o m asequence:
kobs,kobs +1,kobs +2,...,n.
We need o coun howmany e ms he e a e in his sequence. The
sequence s a sa kobs and ends a n.The diffe ence be ween he fi s
e m kobs and he las e m nis n−kobs,bu since bo h endpoin s a e
included in he sequence, we need o add 1. The e o e, he o al numbe
o e ms in he sequence is gi en by (seeTheo em 90,Equa ion 822):
E(X)=
n
x =kobs
1=n−kobs +1(816)
This is because: n−kobs gi es he diffe ence be ween he endpoin s.
Thus a , adding 1accoun s o he inclusion o bo h kobs and nin he
sequence. The o al p obabili ymassmus sum o 1. Di iding by he
o alnumbe o ialsn, he no malized p obabili y is:
9Dis ibu ions 291
p- alue ( igh - ailed) =E(X)
n
=n−kobs +1
n(817)
Quod e a demons andum.
Example.
Mos o he ela ionships de eloped in his publica ion a e de e -
mined by he o mula p∗=kobs
n=1. In his case, we ob ain he igh -
ailed p- alue as ollows:
p- alue ( igh - ailed) =E(X)
n=n−kobs+1
n=1−p∗+1
n(818)
Fo la ge sample sizesn, he e m 1
napp oaches ze o and becomes
negligible,causing he igh - ailed p- alue o simpli y o:
p- alue( igh - ailed) =E(X)
n=n−kobs+1
n=1−p∗+1
n≈1−p∗(819)
Thedecision ulescan be summa ized as ollows:
Tes Type pV alue <α pV alue ≥α
Le -Tailed Tes HAaccep ed H0accep ed
Righ -Tailed Tes HAaccep ed H0accep ed
Le ailed es
To o mula e he le - ailed es analogous o he igh - ailed es de-
sc ibed be o e, he ocus shi s ocalcula ing he p obabili y ha he
obse ed alue kobs o ewe successes occu .The le - ailed p- alue is
defined as:
p- alue (le - ailed) =p(X≤kobs)=
kobs
x =1(p(X) =x )(820)
The o alp obabili yisno malized by he o al numbe o ials n,
esul ing in hele - ailed p- alue:
p- alue (le - ailed) =kobs
n(821)
292 9Dis ibu ions
The leng h o asequence
Nume us Legis Na u alis 90. The leng ho he sequence N om
=kobs,kobs +1,...,(kobs +u)=nis gi en by:
N=n−kobs +1(822)
P oo by Di ec P oo . We p o e he leng h o he sequence
=kobs,kobs +1,...,ns ep by s ep. The sequence consis s o he
indices:
=kobs,kobs +1,kobs +2,...,n(823)
This sequence o msana i hme ic p og ession wi h:
•Ini ial alue a1=kobs,
•Final alue aN=n,
•S ep size d=1.
The gene al o mula o he N- h e mo ana i hme ic p og ession is:
aN=a1+(N−1)·d,(824)
whe e: aNis he final e mo he sequence, Nis he numbe o e ms.
Subs i u ing he known alues: aN=n,a1=kobs,d=1,we
ob ain:
n=kobs +(N−1)·1.(825)
Simpli y:
n=kobs +N−1(826)
Rea anging oisola e N:
N=n−kobs +1(827)
Thus, he leng h o he sequence is:
N=n−kobs +1(828)
Quod e a demons andum.
Pa IV
Condi ionalismand causa ion
Pa VI
Causa ion
Pa VII
Lawo Na u e
612 No ice
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Au ho Index
A chimedes,o Sy acuse (c. 287–c. 212 B.
C. E.), 107, 110, 204
A is o le, o S agi a (ca. 384 BCE-ca. 322
BCE), 32, 204, 379, 380
The doc ine o ou causes, 32
Ba ,Ca lLudwig on(1836–1913), 97,
382, 383
Ba ukčić, Ilija (1961–), 58
Bayes, Re e end Thomas (1701–1761),74
Bell, John S ewa (1928–1990),87, 92, 93
Be keley, Bishop Geo ge (1685–1753),48,
56
Be noulli, Daniel (1700–1782), 138, 152
Be noulli, Jacob (1654–1705), 180, 217,
295
Be ns ein, Se gei(1880–1968),332
Bianchi, Luigi (1856–1928),195
Boh ,Niels Hen ik Da id (1885–1962),
90–92
Bombelli, Raffaele (1526–1572),107, 110,
153
Boole, Geo ge (1815–1864), 107
Bo n, Max (1882–1970), 90, 153
B a ais, Augus e (1811–1863),61
B ouwe ,Lui zen Egbe us Jan
(1881–1966), 369
B uno, Gio dano (1548–1600),33, 358, 519
Ca nap, Rudol (1891–1970), 78
Cauchy,Augus in-Louis(1789–1857),122
Chebyshe , Pa nu y (1821–1894),162, 332
Ch is offel, Elwin (1829–1900),195
Cloppe ,Cha les J. (1899–1983),226
Co nfield, Je ome (1912–1979),300, 305
Co es,Roge (1682–1716), 111
d’Holbach, Paul-Hen iThi y(1723–1789),
34, 56
causal chain, 35
cause, 34
effec , 34
Da win, Cha les Robe (1809–1882), 7, 8,
115
DeG oo , Mo is He man (1931–1989),
129, 172, 173,179
Di ac, Paul Ad ien Mau ice(1902–1984),
107, 110, 206
D ude, Paul (1863–1906), 107, 110
Ducasse, Cu John (1881–1969), 410
Edgewo h, F ancis Ysid o (1845–1926), 61
Eins ein, Albe (1879–1955), 13,78, 85,
92, 150, 155, 358, 359, 516, 599, 601,
603
Does he moon exis onlywhen youlook
a he same?, 49
ImmanuelKan , 48
Objec i e eali y,49, 54
Reichenbach, 78
Euclid, o Alexand ia (ca. 360-280 BCE),
124
Eule ,Leonha d (1707–1783), 111, 118,
120, 248
Feue bach, Ludwig And eas (1804–1872),
57
Finsle ,Paul (1896–1970), 142
Fishe ,Si Ronald Aylme (1890–1962),
174, 180, 183,267, 300, 306, 371
F anz, John E., 396
F ege, F ied ichLudwig Go lob
(1848–1925), 107, 409, 422
Gal on, Si F ancis (1822–1911), 61, 267
629
630 AUTHORINDEX
Gauss, Ca l F ied ichGauss (1777–1855),
174, 180, 267
Geyse ,Ge ha d Joseph An on Ma ia
(1869–1948), 544
Good, I ing John (1916–2009),79
Gosse , William Sealy(1876–1937),270
Hadama d, Jacques Salomon (1865–1963),
207
Haldane, John Bu don Sande son
(1892–1964), 61, 69, 70
Halley, Edmond (1656–1742),111
Hegel, Geo gWilhelmF ied ich
(1770–1831), 52, 55, 56, 116, 117,
510, 520, 522, 523
-Kan ’s scep icims, 52
Causali y,57
Necessi y,57
Heisenbe g, We ne Ka l(1901–1976),
90–92, 516, 586
Visi o Eins eininP ince on, 516
Helme , F ied ichRobe (1843–1917),
280, 391
Henle, F ied ichGus a Jakob
(1809–1885), 72
Hess, Ka l(1945-), 93
Hessen, Johannes (1889–1971), 36, 526,
532, 544
Hesslow, Ge mund (1949-), 82
Hey ing, A end (1898–1980), 369
Hill, Si Aus in B ad o d (1897–1991),72,
300
Hume, Da id (1711–1776),36, 40, 80, 411,
518, 524–526, 542
P oblem o induc ion, 454, 525
Huygens, Ch is iaan (1629–1695),162, 167
Jus ice Ma hews
U.S. Sup eme Cou 1884, 381, 394, 396
U.S. Sup eme Cou 1884: Hayes .
Michigan Cen al R.Co., 111 U.S.
228, 381, 394, 396
Kan , Immanuel (1724–1804),43, 52, 103,
147, 598
Kelley,T uman Lee (1884–1961), 267
Kh enniko ,And ei (1958-), 93
Knu h, Donald E. (1938–),123
Koch,Robe Hein ichHe mann
(1843–1910), 72
Kohl ausch, R.(1809–1858),107, 110
Kolmogo o , And ei Nikolae ich
(1903–1987), 83, 124, 140, 143, 155,
162, 361, 372, 405, 412, 426
Ko ch, Helmu (1926–1998), 38, 40, 544
K öbe ,Gün e (1933–2012),87
Langii, Iohannis Ch is iani,426
Laplace,Pie e-Simon Ma quis de
(1749–1827), 89, 267, 587
Leibniz, Go iedWilhelm (1646–1716),
56, 107, 115, 153, 512, 519
Le i-Ci i a,Tullio (1873–1941), 195
Lewis, Da id Kellogg (1941–2001), 41, 78
Lexis, Wilhelm (1837–1914), 267
Lib i, Guillaume (1803–1869), 123
Lo en z, Hend ik An oon (1853–1928), 134
Mackie, John Leslie (1917–1981), 410, 411
Ma ko , And eyAnd eye ich(1856–1922),
332
Ma x, Ka l(1818–1883), 52, 53,57, 116
Ma hews, Jus ice, 381, 394, 396
Me cie ,Cha les (1851–1930), 82
Mill, John S ua (1806-1873), 411
Mises,Richa d Edle on(1883–1953), 161
Moi e, Ab ahamde(1667–1754), 83,155,
157, 180, 267
Neumann, John on(1903–1957), 113, 148
New on, Si Isaac (1643–1727), 89, 118,
205
Neyman, Je zy(1894–1981), 226
Nicod, Jean Geo ge Pie e (1893–1924),
368
Nieuwenhuizen, Theodo us Ma ia, 93
Noo dho , Paul Jona hanPi , 42
Pacioli, Luca (1447–1517), 108, 111
Pais, Ab aham (1918–2000), 49
Peano, Giuseppe (1858–1932), 113
Pea l, Judea(1936-), 75,592
Coun e ac uals andquan um heo y, 76
Pea sons ill ules s a is ics, 69
Pea son, Egon S. (1895–1980), 226
Pea son, Ka l (1857–1936), 60,70, 174,
180, 280, 284,285, 295, 391, 542
Pei ce, Cha lesSan iagoSande s
(1839–1914), 267
Philo o Mega a,409
Pisa, Leona doo (1170–1240), 110
Planck, MaxKa lE ns Ludwig
(1858–1947), 88, 107, 110, 205, 588
Poinca é,Jules Hen i(1854–1912), 89
Polack, Fe nando P, 369
Poppe , Si Ka lRaimund (1902–1994), 13,
15, 17, 593
Py hago as, o Samos (ca. 570–ca. 495
BCE), 131
Pólya,Gyö gy(1887–1985), 267, 337
Que ele , Lambe Adolphe Jacques
(1796–1874), 267
Raed , Hans de, 93
Reco de, Robe (1510–1558), 108, 111
AUTHOR INDEX 631
Reichenbach, Hans (1891–1953), 78, 89,
92, 515, 587
Ricci-Cu bas o, G ego io(1853–1925),
195
Riemann, Be nha d (1826–1866), 195
Robins, James M., 74
Rolle, Michel (1652–1719), 108, 111
Ro a, Gian Ca lo, 124
Russell, Be and (1872–1970), 409, 422
Russell, Be and A hu William
(1872–1970), 90
Salmon, WesleyCha les (1925–2001),
79–81
Schlick, F ied ichAlbe Mo i z
(1882–1936),588
Sch ödinge ,E win Rudol Jose Alexande
(1887–1961),115, 588
Schu ,Issai (1875–1941), 207
Sheffe ,Hen yMau ice (1882–1964),367
Simpson, Thomas (1710–1761),174
Spinoza, Benedic us de (1632–1677),597
Spohn, Wol gang Kon ad, 42
S eno, Nicholas (1638–1686),138
S iehle ,Go ied (1924–2007), 58
S i ling, James,1692–1770, 255
Suppes, Pa ick(1922–2014),79, 80, 155
Ta ski, Al ed (1901–1983), 422
Thales o Mile us (ca. 624/623–ca.548/545
BCE), 127
Thompson, Ma y Elino e (1944-), ii
Toohey, John Joseph, 11
Ulyano , Vladimi Ilyich(1870–1924), 53
Uspensky,James Vic o (1883–1947), 145,
295, 383
Uyomo , A eni I ano ich(1928–2012),
515
Venn, John (1834–1923), 426
Voig , Woldema (1850–1919), 195
Wallisii, Iohannis (1616–1703), 117, 118
Webe ,W.E.(1804–1891), 107, 110
Weisio, Ch is iano, 426
Whi ehead, Al edNo h (1861–1947), 409
Whi wo h, WilliamAllen (1840–1905),
162, 167
Widmann, Johannes(1460–1498), 108, 111
Williamson, Jon, 42
Wilson, Edwin Bidwell(1879–1964), 226
W igh , Sewall G een (1889–1988), 592
Ya es, F ank (1902–1994), 373, 392, 415,
427
Yule, Geo ge Udny(1871–1951), 65, 70,
303, 305
Zesa ,Pa ickManuel, 507
