Mathematical and statistical applications for HP Prime

Sch ausse , D. G. (2025). Ma hema ical and s a is ical applica ions o HP P ime. DOI:10.31235/os .io/ s8a6_ 1
1
Ma hema ical and s a is ical applica ions
o HP P ime
Die ma G. Sch ausse o cid.o g/0000-0002-4924-8280
Co espondence: die ma .sch aus[email p o ec ed]
Ka l-F anzens Uni e si y, G az, Aus ia
Abs ac
Applica ions o HP P ime CAS, Use unc ions and Applica ions, an
o e iew o he me hods and hei o igins is gi en.
1. In oduc ion
Ma hema ical and s a is ical applica ions HP_P ime_MATH
1
o (1) he
Compu e Algeb a Sys em CAS, by means o he Pascal based HP P ime
P og amming Language (HP PPL), (2) he HP P ime Use unc ions and
(3) HP P ime Applica ions (s. HP Inc., 2017), including me hods o (1)
co ela ion, (2) exposu e, (3) in eg a ion, (4) dis ibu ion, (5) p oba-
bili y, (6) combina o ics, (7) esampling and (8) complex plane cal-
cula ions (Sch ausse , 2025a).
The desc ip ion o he unde lying algo i hms and unc ions is delibe-
a ely omi ed, as hese a e p esen ed and discussed in de ail in
Sch ausse (2025b). Ins ead, an o e iew o he implemen ed me-
hods is gi en and addi ionally hei his o ical de elopmen is ou lined
(c. . Tab. 1).
2. Func ions
2.1. Co ela ion
To measu e he deg ee o a linea ela ion be ween a iables, Ka l
Pea son (1904) was de eloping s a is ical p ocedu es o biome y
including he co ela ion and eg ession coe icien s based on he
wo ks o B a ais (1844) and Gal on (1877) who in oduced he symbol
𝑟, on he hen designa ion o he e m e e sion.
The me hodological appa a us o ac o analysis as a u he and
b oade concep , based on mul iple eg ession and ma ix calcula ion
was i s discussed by Cha les Edwa d Spea man (1904), la e he
ini ial de eloped ook place by Louis Leon Thu s one (1931, 1934,
1935; s. also Ca ell, 1966).
The ollowing unc ions o co ela ion- and eg ession- echniques a e
implemen ed, c. . also Sch ausse (2025b):
(1) Pea son p oduc -momen co ela ion coe icien 𝑟𝑥𝑦, see
Pea son (1904, 1905).
(2) Spea man’s 𝜌, being equi alen o he p oduc momen
co ela ion when ank alues a e p esen (s. Spea man, 1904).
(3) Kendall’s au 𝜏𝑎, i.e. wi hou adjus men o ies (s. Kendall,
1938).
(4) Some s’ 𝐷, o bina y da a [0,1] (s. Some s, 1962).
(5) Poin bise ial co ela ion coe icien 𝑟𝑝𝑏 o also poin bise al.
(6) Bise ial co ela ion coe icien 𝑟𝑏𝑖𝑠, Pea son (1909), s. Ta e
(1955), also called bise al.
(7) Rank bise ial co ela ion coe icien 𝑟𝑏𝑖𝑠𝑅 o ank bise al,
co esponding o he e ec size o he Mann–Whi ney 𝑈- es
(Mann & Whi ney, 1947).
(8) Phi coe icien 𝛷, Yule (1912).
(9) Te acho ic co ela ion 𝑟𝑡𝑒𝑡, Pea son (1900a), E e i (1910,
1912), s. e.g. B own (1977), Digby (1983), also Bone and
P ice (2005) o Long e al. (2009), p oposed app oxima e
algo i hm.
(10) Pa ial co ela ion 𝑟𝑥𝑦⋅𝑧.
(11) Fishe 𝑍- ans o ma ion, Fishe (1915).
(12) Fishe 𝑍 di e ence, also Cohen’s 𝑞 (Cohen, 1988, p. 110).
(13) A e aged Fishe 𝑍.
(14) Coe icien o mul iple co ela ion 𝑅𝑐,12, o 𝑅
𝑐,12
2 see Olkin
and P a (1958), wi h he e ec size o mul iple eg ession
𝑓2 (Cohen, 1988, p. 410).
1
h ps://gi hub.com/Sch ausse /HP_P ime_MATH
2.2. Exposu e
To de e mine he app op ia e ime-ape u e-speed combina ion o
gi en ligh alues on a loga i hmic scale (c. . Allb igh , 1991; Ma sden
& Weins ein, 1985; Howie, 2001 and Sobo , 2021), ollowing unc ions
a e included o he calcula ion o (1) exposu e alues 𝐸𝑣, whe e 𝐸𝑣 =
log(𝑇𝑣⋅𝐴𝑣2)
log(2), (2) ape u e 𝐴𝑣 o ime 𝑇𝑣 o speed 𝑆 wi h gi en 𝐸𝑣, (3)
ape u e 𝐴𝑣 shi om ime 𝑇𝑣 o speed 𝑆 in s eps 𝑘 and (4) speed 𝑆
in loga i hmic 𝐼𝑆𝑂° o a i hme ic 𝐼𝑆𝑂 alues.
2.3. Func ions o in eg a ion o 𝛑 and 𝜞
Go ied Wilhelm Leibniz (1684, 1686, 1693) along wi h Si Isaac
New on (1687, 1713, 1726) a e conside ed he disco e e s o
di e en ial and in eg al calculus. Acco ding o cu en consensus,
bo h de eloped he me hods independen ly o each o he , see he so-
called Leibniz-New on calculus con o e sy (c. . Cajo i, 1919; Cassi e ,
1943; Rosen hal, 1951; Sch ade , 1962; Kosso sky, 2020).
New on began wo king on a geome ic o m o calculus ( he me hod
o luxions and luen s) in 1666, published in 1687 (c. . Roe o, 2005),
ye , i was Leibniz who in oduced he symbols ∫and ∂.
He e, he unc ions a e p ima ily in ended o display and calcula e π
and 𝛤 wi hin he coo dina e sys em:
(1) Ci cula unc ion o π, whe e Weie s aß (1894, p. 53)
desc ibes π
2=∫1
1−𝑥2
∞
0𝑑𝑥, which may be less heu is ic (s.
Sch ausse , 2025b).
(2) Sphe ical unc ions o π, o sou ce codes o olume in eg als
o he sphe e see Sch ausse (2024d).
(3) Gamma unc ion 𝛤, mean o ex end he ac o ial o non-
in ege a gumen s, was i s conside ed by Daniel Be noulli
and Ch is ian Goldbach (Be noulli, 1729), la e Leonha d Eule
(1738) and Johann Ca l F ied ich Gauss (s. Remme , 1998),
i s ables we e gi en by Jahnke and Emde (1909, 1933, 1938,
1945), Knoll (1939) and Jahnke e al. (1966).
2.4. Dis ibu ion unc ions
The disco e y o he no mal dis ibu ion is a ibu ed o Ab aham de
Moi e (1738), la e Gauss (1809) desc ibed he a i hme ic mean as an
es ima o in con ex wi h he no mal law o e o s. Benea h he
no mal dis ibu ion, Gauss (1823) also in oduces se e al impo an
s a is ical concep s, such as he me hods o leas squa es and o
maximum likelihood.
The 𝑡-dis ibu ion i s de i ed as a pos e io dis ibu ion by Lü o h
(1876), appea ing la e as Pea son Type IV (Pea son, 1895), howe e
ge s i s name as S uden ’s 𝑡-dis ibu ion om William Sealy Gosse
(1908), who published i using he pseudonym S uden , hough i was
ac ually h ough he ex ensi e wo ks o Si Ronald Aylme Fishe ha
he dis ibu ion became well known.
The 𝜒2-dis ibu ion was i s desc ibed by F ied ich Robe Helme
(1876) and independen ly edisco e ed by Pea son (1900b) in con ex
wi h he goodness o i pa adigm, whe e he de eloped he 𝜒2- es
wi h compu ed able o alues, published by Elde on (1902), s. u he
Pea son (1914) o Placke (1983).
Fishe (1918, 1921, 1925) in oduced he e m a iance and p oposed
i s o mal analysis, as well as he 𝐹-dis ibu ion (Fishe , 1924; s. also
Snedeco , 1934 and Sche é, 1959). The me hods became widely
known om Me hods o Resea ch Wo ke s (Fishe , 1925, 1954, 1973,
2017).
Following unc ions o he mos ele an me hods a e a ailable:
C ea i e Commons
A ibu ion 4.0 In e na ional
Sch ausse , D. G. (2025). Ma hema ical and s a is ical applica ions o HP P ime. DOI:10.31235/os .io/ s8a6_ 1
2
(1) S anda dizing, i.e. 𝑧- alues and 𝜁- alues.
(2) Quan i y p opo ion o 𝑎 a 𝑁 o 𝑛≥𝑝.
(3) Weigh ed a i hme ic mean 𝑥󰇘.
(4) Geome ic mean 𝑥󰇗, o he weigh ed geome ic mean 𝑥󰇗󰇘 s.
Siegel (1942).
(5) Ha monic mean 𝑥.
(6) Coe icien o a ia ion 𝜔.
(7) Mean dispe sion 𝑑, Sch ausse (2022a, p. 33).
(8) S anda d no mal dis ibu ion 𝑓(𝑥 = 𝑧), de Moi e (1738),
Gauss (1809, 1823).
(9) Bi a ia e no mal dis ibu ion 𝑓(𝑧1,𝑧2).
(10) S uden ’s 𝑡-dis ibu ion 𝑓(𝑥 = 𝑡), Lü o h (1876), Pea son
(1895), Gosse (1908).
(11) 𝜒2-dis ibu ion 𝑓(𝑥 = 𝜒2), Helme (1876), Pea son (1900b,
1914), Elde on (1902), Placke (1983).
(12) 𝐹-dis ibu ion 𝑓(𝑥 = 𝐹), Fishe (1924), Snedeco (1934),
Sche é (1959).
(13) Thi d s anda dized momen , skewness 𝛼3.
(14) Fou h s anda dized momen , excess ku osis 𝛼4.
(15) Es ima ed s anda d e o o mean 𝜎𝑥, con idence in e al 𝐶𝐼𝑝.
Neyman (1937) in oduced he con idence in e al in o
s a is ical hypo hesis es ing s. Fishe ’s null hypo hesis
es ing, he Neyman–Pea son lemma (Neyman & Pea son,
1933; Lehmann, 1993).
(16) S anda d e o o p edic ion 𝜎𝑦
𝑥 , con idence in e al 𝐶𝐼𝑝.
(17) E ec size 𝜖, Cohen’s 𝑑 (Cohen, 1977, 1988, p. 20, p. 49, 1992),
Bo ens ein e al. (1997), Bo ens ein e al. (2001).
(18) Op imal e ec size 𝜖𝑝.
(19) Op imal alpha le el.
(20) Va iance di e ence 𝑡- es o pai ed samples (𝑥1|𝑥2).
(21) Pai ed 2-sample 𝑡- es .
(22) Unpai ed 2-sample 𝑡- es .
(23) One-sample 𝑡- es .
(24) 𝜒2- es o independence.
(25) 2 × 2 𝜒2- es o independence, o Ya es’s co ec ion o
con inui y see Ya es (1934).
(26) McNema ’s 𝜒2- es o pai ed 2 × 2 con ingency ables wi h
dicho omous ai , McNema (1947).
2.5. P obabili y
Since un il he Renaissance a p obable opinion was me ely con i med
by an au ho i y and hence he e was no u he concep o induc i e
e idence (see Hacking, 1975; Hald, 2003, p. 31), an objec i e
ep esen a ion o p obabili y as such was i s discussed by An oine
A nauld and Pie e Nicole (1662, 1682, 1693; c. . also A nauld e al.,
1970; an E a, 1997; Dessì & Albu y, 1997 o Finocchia o, 1997).
The binomial dis ibu ion is p ima ily a ibu able o de Moi e (1711,
1718, 1738) and Jacob Be noulli (1713), see also Schneide (2005a, b).
Al hough no included as unc ion, due o i s conside abili y in his
con ex , he con igu a ion equency analysis, CFA should be
men ioned pa icula ly (c. . K au h, 1973; K au h & Liene , 1993).
An accoun o he sys ema ics and logic o dependen p obabili ies
wi hin he amewo k o Bayes’ heo em (Bayes & P ice, 1763; c. .
S igle , 2018) can be ound in Sch ausse (2024c).
The a guably mos impo an me hods ega ding he calcula ion o
p obabili y pa ame e s a e implemen ed as ollows:
(1) A csine ans o ma ion, Cohen’s ℎ (Cohen, 1988, p. 181).
(2) Addi i e p obabili y o independen e en s 𝑢𝑝(∪𝑛𝐴), which
co esponding o he geome ic dis ibu ion 𝑓(𝑋 ≤ 𝑟|𝑝).
(3) Geome ic dis ibu ion 𝑓(𝑋 ≤ 𝑟|𝑝), co esponding o he
addi i e p obabili y 𝑢𝑝(∪𝑛𝐴).
(4) Nega i e binomial dis ibu ion 𝑓(𝑋 ≤ 𝑟|𝑟,𝑝), wi h 𝑘 = 1 i
co esponds o he geome ic dis ibu ion 𝑓(𝑋 ≤ 𝑟|𝑝) and he
addi i e p obabili y 𝑢𝑝(∪𝑛𝐴).
(5) Exac binomial es .
(6) Exac hype geome ic 2 × 2 es , he so-called Fishe Exac es
(Fishe , 1922; Ag es i, 1992).
2.6. Combina o ics
A e Ge sonides’ pionee ing wo k om 1321 dealing wi h
a i hme ical ope a ions and combina o ics (s. Ab aham Ba Hiyya
Sa aso da, 1450; Rabino i ch, 1970), he me hods, being a undamen-
al pa o p obabili y calcula ions, a e mainly based on Blaise Pascal
(1665), Be noulli (1713) and Eule (1753), c. . E ingshausen (1826).
See u he Syl es e (1904, 1908, 1909, 1912) and MacMahon (1915,
1916), gi ing undamen al con ibu ions o ma ix- heo y and combi-
na o ics.
The ollowing unc ions o gene a e pe mu a ion and a ia ion ma i-
ces a e a ailable, p ima ily o suppo he esampling p ocedu es des-
c ibed below:
(1) Pe mu a ion ma ix 𝑷𝒏, wi h 𝑛 elemen s o 𝑘 = 1 class.
(2) Va ia ion ma ix 𝒘𝑽𝟐
𝒎 o he dependen 2 sample design, wi h
𝑛 = 2 elemen s o class 𝑚.
(3) Va ia ion ma ix 𝒘𝑽𝒏
𝒎, wi h 𝑛 elemen s o class 𝑚.
(4) Pe mu a ion ma ix 𝒘𝑷𝒏
(𝒌𝒎,𝒌𝒏−𝒎), wi h 𝑛 elemen s o class 𝑚.
2.7. Resampling
Pe mu a ion o andomiza ion es s we e i s men ioned by Fishe
(1935), based on expe imen s in ag icul u e (Fishe , 1926; Neyman,
1923). In his con ex see Pi man (1937a, b, 1938), Fishe (1966, 1971,
es.), especially Eugene Sinclai Edging on (1964, 1980, 1987, 2011) o
Edging on and Onghena (2007).
The boo s ap me hod was in oduced by B adley E on (1979, 1981,
1982) as a u he de elopmen (Quenouille, 1949; Me opolis &
Ulam, 1949), o so wa e solu ions see e.g. Solomon (1982), Dallal
(1986, 1988), Peladeau (1993), Woo and Peladeau (1994), Meh a e
al. (2014), also Sch ausse (2024d).
Following unc ions we e de eloped:
(1) Pe mu a ion es P o 2 pai ed samples (𝑥1|𝑥2). Random
sampling model, sys ema ic pe mu a ion, 𝑝- alue no
andomized, a ia ion ma ix 𝒘𝑽𝟐
𝒎 equi ed, s. Scambo
(1997), Scambo and Sch ausse (2022, p. 7), espec i ely.
(2) Randomized pe mu a ion es mP o 2 pai ed samples
(𝑥1|𝑥2). Random sampling model, 𝑝- alue no andomized.
(3) Pe mu a ion es P o 2 independen samples (𝑥|𝑔). Random
sampling model, sys ema ic pe mu a ion, 𝑝- alue no
andomized, pe mu a ion ma ix 𝒘𝑷𝒏
(𝒌𝒎,𝒌𝒏−𝒎) equi ed, see
Sch ausse (1996, 1998b, 2022b, p. 2).
(4) Randomized pe mu a ion es mP o 2 independen samples
(𝑥|𝑔). Random sampling model, 𝑝- alue no andomized.
(5) Boo s ap es B o 2 independen samples (𝑥|𝑔), c. .
Quenouille (1949), E on (1979, 1981, 1982).
2.8. Complex plane
I was he I alian ma hema ician Ge olamo Ca dano (1545a, b) who
i s concei ed he e m imagina y, o he u he his o ical
de elopmen o imagina y o complex numbe s see René Desca es
(1664, 2012, es.) and Gauss (1828, 1832), c. . also Wi inge (1927).
He e inally ealized a e (1) he geome ic ep esen a ion o complex
numbe s 𝑧 in he complex plane, he A gand diag am (s. A gand, 1813,
1874, es.) and (2) he g aph o he complex unc ion, whe e 𝑧 = ℜ +
ℑ.
A his poin , one should ecall he de ini ional impo ance o geo-
me y and igonome y in con ex wi h he calcula ion o complex
numbe s i sel , whe e |𝑧| is calcula ed acco ding o Py hago as (c. .
Ra dol , 1482, p oposi io 46) by |𝑧|= 𝑟 = √𝑥2+ 𝑦2.
A e he undamen al change in ma hema ics om geome ic o
algeb aic ep esen a ion ook place in he 16 h cen u y (c. . Hea h,
1908a, b, c; Bochne , 1978; Anglin & Lambek, 1995; Male , 2006 o
Al en e al., 2014), he o igins o igonome ic se ies o angen s and
sine can be seen ollowing ea ly a emp s (s. Jyes hade a, 1530;
Whish, 1834; Gup a, 1974 o Di aka an, 2007) du ing he Eu opean
ein en ion in he wo ks o G ego y (1671, 1668a, b), Leibniz (1682,
2012), New on (1669, 1711) and B ook Taylo (1715, 1717) wi h he
de ini ion o he Taylo se ies o sine, whe e sin⁡𝑥 = ∑(−1)𝑛
(2⋅𝑛+1)!
∞
𝑛=0 ⋅
𝑥2⋅𝑛+1 (c. . G ego y & Collins, 1939; Boye , 1968, p. 422 .; Feigen-
baum, 1985).
Sch ausse , D. G. (2025). Ma hema ical and s a is ical applica ions o HP P ime. DOI:10.31235/os .io/ s8a6_ 1
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Finally, Eule (1748a, b) es ablished he analy ic ea men o
igonome ic unc ions, de ining hem in ela ion wi h complex
exponen ial unc ions by e𝒊𝑥 =cos 𝑥 + 𝒊sin⁡𝑥, whe e e = ∑1
𝑛!
∞
𝑛=0 and
hus laid he ounda ion o mode n ma hema ical analysis (c. . Finkel,
1897; Wal e , 1982; Koyama & Ku okawa, 2005; Calinge , 2016 and
Sch ausse , 2024b).
3. Conclusion
In addi ion o he sou ce codes o he unc ions, aw da a se s a e
p o ided o co ela ion- as well as esampling-me hods. CAS p o-
g ams, HP P ime Use unc ions and unc ions o HP P ime Appli-
ca ions in compa ison o co esponding SCHRAUSSER-MAT unc ions
(Sch ausse , 2022a) a e displayed in Sch ausse (2025b). Fu he mo e,
he applica ion Funk ionWin o a p ecise calcula ion o p obabili y
dis ibu ions can addi ionally be conside ed (Sch ausse , 2023c) as
well as he au ho ’s u he so wa e applica ions o ma hema ical
and s a is ical analyses (Sch ausse , 2023a, b, d).
On ma hema ical s a is ical me hods in gene al see e.g. Cox and
Hinkley (1974), Bo z and Webe (2005), Lehmann and Romano (2008)
o Bo z and Schus e (2010), Sch ausse (2024a) p o ides a com-
p ehensi e o e iew o he mos impo an dis ibu ion unc ions and
co esponding algo i hms.
Fo calculus and heo y o unc ions see e.g. Meybe g and Vachenaue
(2001a, b) o Remme and Schumache (2002), on complex numbe s
in he complex plane see e.g. Bu ckel (2021) and Vince (2021), in-
oducing wo ks on esampling me hods a e gi en by e.g. Good (2006)
o Beasley and Rodge s (2009).
Fo he his o y o s a is ical in e ence in gene al see e.g. S igle (1986)
and Hald (1990, 1998, 2003, 2007), he his o ical ounda ions o
ma hema ics a e hema ized and discussed in e.g. Su e (1887), Hea h
(1921a, b), Boye (1968), Neugebaue (1969), Ewald (1996a, b), Ka z
(2009) o Me zbach and Boye (2011), c. . Tab. 1.
Table 1. Timeline (yea ) o ini ial wo k on he me hods, co esponding au ho s wi h o igin and ield o expe ise.
Yea
n
Name
O igin
om
o
Field
n
Me hod
Wo k
1290
1
Rabbi Le i ben Ge shon
F ance
1288
1344
Theologian
1300
1310
1320
1330
1
Combina o ics
1321
1340
:
:
1500
1510
2
Ge olamo Ca dano
I aly
1501
1576
Polyma h
1520
1530
1540
1550
2
"imagina y"
1545
1560
1570
1580
1590
1600
3
René Desca es
F ance
1596
1650
Philosophe
1610
1620
4
An oine A nauld
F ance
1618
1698
Theologian
5
Blaise Pascal
F ance
1623
1662
Philosophe
1630
4
Pie e Nicole
F ance
1625
1685
Theologian
1640
6
Si Isaac New on
England
1643
1727
Polyma h
1650
7
Go ied Wilhelm Leibniz
Ge many
1646
1716
Polyma h
1660
8
Jacob Be noulli
Swi ze land
1655
1705
Ma hema ician
4
P obabili y
1662
1670
9
Ab aham de Moi e
F ance
1667
1754
Ma hema ician
3
Complex numbe s
1664
5
Combina o ics
1665
1680
1690
10
B ook Taylo
England
1685
1731
Ma hema ician
6,7
Calculus
1684
1700
11
Daniel Be noulli
Swi ze land
1700
1782
Ma hema ician
12
Re . Thomas Bayes
England
1701
1761
Theologian
13
Leonha d Eule
Swi ze land
1707
1783
Ma hema ician
1710
9,8
Binomial dis ibu ion
1711
1720
10
Taylo se ies o sine
1715
1730
11
Gamma
1729
1740
9
No mal dis ibu ion
1738
1750
13
Complex exponen ial unc ions
1748
1760
12
Bayes' heo em
1763
1770
14
Jean-Robe A gand
Swi ze land
1768
1822
Polyma h
1780
15
Johann Ca l F ied ich Gauss
Ge many
1777
1855
Ma hema ician
1790
1800
1810
15
Es ima o o mean
1809
1820
14
A gand diag am
1813
1830
16
Si F ancis Gal on
England
1822
1911
An h opology
1840
1850
17
F ied ich Robe Helme
Ge many
1843
1917
Geodesy, ma hema ics
18
Jacob Lü o h
Ge many
1844
1910
Ma hema ics
1860
19
Ka l Pea son
England
1857
1936
Biology, ma hema ics
1870
17,18
𝑡-, 𝜒2-dis ibu ion
1876
1880
16
Re e sion
1877
1890
20
Louis Leon Thu s one
USA
1887
1955
Psychophysics
1900
21
Si Ronald Aylme Fishe
England
1890
1962
Biology, ma hema ics
1910
19
Co ela ion
1904
1920
1930
22
Jacob Cohen
USA
1923
1998
Psychology, s a is ics
21
𝐹-dis ibu ion
1924
1940
20
Fac o analysis
1931
21
Pe mu a ion es
1935
23
B adley E on
USA
1938
S a is ics
1950
1960
1970
1980
23
Boo s ap
1979
1990
22
E ec size
1988
2000
Sch ausse , D. G. (2025). Ma hema ical and s a is ical applica ions o HP P ime. DOI:10.31235/os .io/ s8a6_ 1
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