HP_Prime_MATH: Manual

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
1
HP_P ime_MATH: Manual
Die ma G. Sch ausse
Ka l-F anzens Uni e si y, G az, Aus ia
In oduc ion
Ma hema ical and s a is ical applica ions o HP P ime (s. HP Inc., 2017; Sch ausse , 2025).
Algo i hms a e p esen ed in con ex wi h he co esponding scope o applica ion (s.
Func ions). CAS p og ams (1), HP P ime Use unc ions (2) and unc ions o HP P ime
Applica ions (3) a e lis ed in alphabe ical o de (s. Sou ce Codes), o a compa ison o
co esponding SCHRAUSSER-MAT unc ions (Sch ausse , 2022a) see Table 2. 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 (s. Da a).
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 comp ehensi e o e iew o he mos impo an dis ibu ion unc ions and
co esponding algo i hms. In oducing wo ks on esampling me hods a e gi en by e.g. Good
(2006) o Beasley and Rodge s (2009), o 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).
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), 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).
Func ions
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.
Table 1. App op ia e co ela ion coe icien s; p oduc -momen o Pea son co ela ion 𝑟𝑥𝑦, Spea man’s
ank co ela ion coe icien ho 𝜌, bise ial (o bise al) coe icien s 𝑟𝑏𝑖𝑠, 𝑟𝑝𝑏𝑖𝑠, 𝑟𝑏𝑖𝑠𝑅 and phi coe icien
𝛷 a he co esponding scale le els, in e al i, o dinal o, and nominal n.
i
o
n
i
𝑟𝑥𝑦
o
𝜌 ¹
n
𝑟𝑝𝑏𝑖𝑠, 𝑟𝑏𝑖𝑠
𝑟𝑏𝑖𝑠𝑅
𝛷 ²
¹) also Kendall’s au 𝜏 (1938) o Some s’ 𝐷 (1962).
²) also e acho ic co ela ion 𝑟𝑡𝑒𝑡.
C ea i e Commons A ibu ion 4.0
In e na ional
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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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).
[KOR|IC_M] [ xy|RED| |TRW|pRW|pRWx] [E01] Pea son p oduc -momen
co ela ion coe icien 𝑟𝑥𝑦
B a ais (1844), Gal on (1877), Pea son 1904, 1905).
𝑟𝑥𝑦=𝜎𝑥𝑦
2
𝜎𝑥⋅𝜎𝑦,
𝜎𝑥𝑦
2=∑ (𝑥𝑖−𝑥)⋅(𝑦𝑖−𝑦)
𝑛𝑖=1 𝑛
wi h
𝑡(𝑑𝑓)=𝑟⋅√𝑛−2
√1−𝑟2
whe e
𝑟2 = coe icien o de e mina ion, edundancy 𝑑𝑒𝑡
𝜎𝑥𝑦
2 = co a iance o 𝑥 and 𝑦
𝑑𝑓 = 𝑛−2
[RHO] Spea man’s 𝜌
Equi alen o he p oduc momen co ela ion when ank alues a e p esen (s. Spea man,
1904).
𝑟𝑠=𝜌=1−6⋅∑𝑑𝑖2𝑛𝑖=1
𝑛⋅(𝑛2−2)
wi h
𝑡(𝑑𝑓)=𝜌⋅√𝑛−2
√1−𝜌2;𝑛≥30
whe e
𝑑𝑖 = ank di e ence o 𝑥𝑖 and 𝑦𝑖
𝑑𝑓 = 𝑛−2
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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[TAU] Kendall’s au 𝜏𝑎
Wi hou adjus men o ies (s. Kendall, 1938). 𝜏𝑎=1− 2⋅𝑛𝑑
0.5⋅𝑛⋅(𝑛−1),
wi h
𝑧=3⋅𝜏𝑎⋅√𝑛⋅(𝑛−1)
√2⋅(2⋅𝑛+5);𝑛>10
al e na i ely 𝑧= 𝑛𝑐−𝑛𝑑
√1
18⋅𝑛⋅(𝑛−1)⋅(2⋅𝑛+5)
whe e
𝑛 = o al numbe o pai s
𝑛𝑑 = numbe o disco dan pai s
𝑛𝑐 = numbe o conco dan pai s, wi h 𝑛𝑐=(𝑛2)−𝑛𝑑
[DELTA2] Some s’ 𝐷
Fo bina y da a [0,1] (s. Some s, 1962). 𝐷𝑌𝑋=𝑛1,1
𝑛−𝑛1,0
𝑛
whe e
𝑛 = o al numbe o pai s
𝑛1,1 = numbe o pai s wi h 𝑌=1,𝑋=1
𝑛1,0 = numbe o pai s wi h 𝑌=1,𝑋=0
[ pbis] Poin bise ial co ela ion coe icien 𝑟𝑝𝑏
Also poin bise al. 𝑟𝑝𝑏=𝑥1−𝑥0
𝜎𝑥⋅√𝑛1⋅𝑛2
𝑛2
wi h
𝑡(𝑑𝑓)=𝑟𝑝𝑏⋅√𝑛−2
√1−𝑟𝑝𝑏
2
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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whe e
𝑑𝑓 = 𝑛−2
[ bis|s bis|z bis|p bis] Bise ial co ela ion coe icien 𝑟𝑏𝑖𝑠
Pea son (1909), see e.g. Ta e (1955), also called bise al. 𝑟𝑏𝑖𝑠=𝑥1−𝑥0
𝜎𝑥⋅𝑛1⋅𝑛2
𝜗⋅𝑛2
wi h 𝑧=𝑟𝑏𝑖𝑠
𝜎𝑟𝑏𝑖𝑠,
𝜎𝑟𝑏𝑖𝑠=√𝑛1⋅𝑛2
𝜗⋅𝑛⋅√𝑛
whe e
𝜗= 1
√2⋅π⋅𝑒−𝐹(𝑝=𝑛0
𝑛)2
2
[ bisR|U_1|U_2|z bisR|p bisR] Rank bise ial co ela ion coe icien 𝑟𝑏𝑖𝑠𝑅
Also ank bise al co ela ion, co esponds o he e ec size o he Mann–Whi ney 𝑈 es
(Mann and Whi ney, 1947). 𝑟𝑏𝑖𝑠𝑅=2𝑛⋅(𝑖1−𝑖2)
wi h
𝑧= 𝑈−𝑛1⋅𝑛2
2
√𝑛1⋅𝑛2⋅(𝑛+1)
12
whe e
𝑈=𝑛1⋅𝑛2+𝑛12+𝑛1
2−∑𝑥𝑖
𝑛1
𝑖=1
[PHC] [PHI|xPHI|pPHI] Phi coe icien 𝛷
Yule (1912).
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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𝛷= 𝑎⋅𝑑−𝑏⋅𝑐
√(𝑎+𝑐)⋅(𝑏+𝑑)⋅(𝑎+𝑏)⋅(𝑐+𝑑)
wi h 𝜒(𝑑𝑓)
2=𝑛⋅𝛷2
whe e
𝑑𝑓=1
[PHC] [ e |s e |p e ] 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. 𝑟𝑡𝑒𝑡=cosπ
1+√𝑏⋅𝑐
𝑎⋅𝑑
wi h 𝑧=𝑟𝑡𝑒𝑡
𝜎𝑟𝑡𝑒𝑡,
𝜎𝑟𝑡𝑒𝑡=√𝑎+𝑏
𝑛⋅𝑎+𝑐
𝑛⋅𝑐+𝑑
𝑛⋅𝑏+𝑑
𝑛
𝑛⋅1
𝜗𝑥⋅𝜗𝑦
whe e
𝜗𝑥=1
√2⋅π⋅e−𝐹(𝑝=𝑐+𝑑
𝑛)2
2
𝜗𝑦=1
√2⋅π⋅e−𝐹(𝑝=𝑏+𝑑
𝑛)2
2
[PKR] [ xy_z|z xy_z|p xy_z| y_xz] Pa ial co ela ion 𝑟𝑥𝑦⋅𝑧
𝑟𝑥𝑦⋅𝑧=𝑟𝑥𝑦−𝑟𝑥𝑧⋅𝑟𝑦𝑧
√1−𝑟𝑥𝑧
2⋅√1−𝑟𝑦𝑧
2
wi h 𝑧=𝑍𝑟𝑥𝑦⋅𝑧⋅√𝑛−2
and semi pa ial co ela ion 𝑟𝑦(𝑥⋅𝑧)=𝑟𝑥𝑦−𝑟𝑥𝑧⋅𝑟𝑦𝑧
√1−𝑟𝑥𝑧
2,

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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[ZCo ] [Z | Z] Fishe 𝑍- ans o ma ion
Fishe (1915). 𝑍=12⋅ln1+𝑟
1−𝑟
wi h 𝑧= 𝑍
√1
𝑛−3
and
𝑟𝑍=𝑒2⋅𝑍−1
𝑒2⋅𝑍+1
[Z |p ] Fishe 𝑍 di e ence, Cohen’s 𝑞
Cohen (1988, p. 110). 𝜃=𝑑𝑍=𝑍𝑟1−𝑍𝑟2
wi h 𝑧= 𝑑𝑍
√1
𝑛1−3+1
𝑛2−3
[mZ|m ] A e aged Fishe 𝑍
𝑍=∑ (𝑛𝑖−3)
𝑘𝑖=1 ⋅𝑍𝑖
∑ (𝑛𝑖−3)
𝑘𝑖=1
[MCORR2] [MCORR|SCR|C 2|FMCORR|pMCORR] Coe icien o mul iple co ela ion
𝑅𝑐,12, Cohen’s 𝑓2
Fo 𝑅𝑐,12
2 see Olkin and P a (1958).
𝑅𝑐,12=√𝑟1𝑐
2+𝑟2𝑐
2−2⋅𝑟12⋅𝑟1𝑐⋅𝑟2𝑐
1−𝑟12
2,
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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𝑅𝑐,12
2=1− 𝑛−3
𝑛−𝑘−2⋅[(1−𝑅𝑐,12
2)+ 2
𝑛−𝑘⋅(1−𝑅𝑐,12
2)2];𝑘=2
wi h
𝑓2=𝑅𝑐,12
2
1−𝑅𝑐,12
2,
𝐹(3,𝑑𝑓2)=𝑅𝑐,12
2⋅(𝑛−4)
(1−𝑅𝑐,12
2)⋅3
whe e
𝑓2 = e ec size o mul iple eg ession (Cohen, 1988, p. 410)
𝑑𝑓2=𝑛−4
Exposu e unc ions
To calcula e he app op ia e ime-ape u e-speed combina ion o gi en ligh alues
on a loga i hmic scale, see e.g. Allb igh (1991), Ma sden and Weins ein (1985),
Howie (2001) and Sobo (2021).
[E |TE |AE ] [E02|E03] Exposu e alue 𝐸𝑣
𝐸𝑣=log2𝐴𝑣2
𝑇𝑣−1=log(𝑇𝑣⋅𝐴𝑣2)
log(2)
hence 𝑇𝑣=2𝐸𝑣
𝐴𝑣2,
𝐴𝑣=√2𝐸𝑣⋅𝑇𝑣
𝑇𝑣
wi h
𝑇𝑣 = ime alue wi h 𝑇𝑣=𝑠−1
𝐴𝑣= ape u e alue 𝑓
[A T ] Ape u e 𝐴𝑣 o ime 𝑇𝑣 wi h gi en 𝐸𝑣
𝐴𝑣𝑇𝑣=𝐴𝑣𝑇𝑣0⋅𝑎𝑇𝑣
wi h 𝑎𝑇𝑣=212⋅log2𝑇𝑣0
𝑇𝑣
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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=e12⋅log(𝑇𝑣0)
𝑇𝑣
whe e
𝑇𝑣 = ime alue wi h 𝑇𝑣=𝑠−1
𝑇𝑣0 = ini ial ime alue wi h 𝑇𝑣0=𝑠−1
𝐴𝑣= ape u e alue 𝑓
[A S] [E03] Ape u e 𝐴𝑣 o speed 𝑆 wi h gi en 𝐸𝑣
𝐴𝑣𝑆=𝐴𝑣𝑆0⋅𝑎𝑆
wi h 𝑎𝑆=212⋅log2𝑆
𝑆0
=e12⋅log𝑆
𝑆0
whe e
𝑆 = a i hme ic speed 𝐼𝑆𝑂
𝑆0 = ini ial a i hme ic speed 𝐼𝑆𝑂
𝐴𝑣= ape u e alue 𝑓
[A T k] Ape u e 𝐴𝑣 shi om ime 𝑇𝑣 in s eps 𝑘
𝑇𝑣𝑛−𝑘=𝑇𝑣𝑛⋅2𝑘,𝑇𝑣𝑛+𝑘=𝑇𝑣𝑛
2𝑘,
wi h 𝐴𝑣=𝐴𝑣0⋅√2𝑘
whe e
𝐴𝑣0 = ini ial ape u e alue
[A Sk] Ape u e 𝐴𝑣 shi om speed 𝑆 in s eps 𝑘
𝑆𝑛+𝑘=𝑆𝑛⋅√2𝑘,𝑆𝑛−𝑘=𝑆𝑛
√2𝑘,
wi h 𝐴𝑣=𝐴𝑣0⋅√2𝑘
whe e
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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𝐴𝑣0 = ini ial ape u e alue
[ISOL|ISOA] Speed 𝑆 in loga i hmic 𝐼𝑆𝑂° o a i hme ic 𝐼𝑆𝑂 con e sion
𝑆°=10⋅log10(𝑆)+1=10⋅log(𝑆)
log(10)+1,
𝑆=10𝑆°−1
10
Func ions o in eg a ion, 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.
[F01|F05] Ci cula unc ion, π
Weie s aß (1894, p. 53) desc ibes π2=∫1
1−𝑥2
∞
0𝑑𝑥, which may be less heu is ic.
𝑓(𝑥)=√1−(𝑥−𝑏
𝑎)2⋅𝑎+𝑐
wi h 𝐹(𝑥)=π2=∫𝑓
1
−1 (𝑥)𝑑𝑥;𝑎=1,𝑏=𝑐=0
[F01Z] Sphe ical unc ions, π
Fo Sou ce codes o olume in eg als o he sphe e see Sch ausse (2024).
𝑓1(𝑥,𝑦)=√(1−𝑥2)+(1−𝑦2)
wi h
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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𝑝 = p obabili y
𝑛 = numbe o cases
[CIXY] [CI ] [E01] S anda d e o o p edic ion 𝜎𝑦𝑥 , con idence in e al 𝐶𝐼𝑝
𝜎𝑦𝑥=𝜎𝑦⋅√1−𝑟2
wi h 𝐶𝐼𝑝=𝑦𝑥±𝑧(1−1−𝑝
2)⋅𝜎𝑦𝑥
𝑝 = p obabili y
𝑟 = co ela ion
𝑦 = p edic ed alue 𝑦
[EPSILON] [EFG|EFR] [E01] 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).
𝜖=𝑑=𝜇1−𝜇0
𝜎 ,
𝑑𝑣=𝑑
√1−𝑟
wi h 𝑥𝑐𝑟𝑖𝑡
𝛽=𝜇1±𝑡(𝑝𝑐𝑟𝑖𝑡,𝑑𝑓)⋅𝜎𝑥,
𝑡(𝑑𝑓)
𝛼=𝑥01−𝜇0
𝜎𝑥,
𝑡(𝑑𝑓)
𝛽=𝑥01−𝜇1
𝜎𝑥
whe e
𝑑𝑣 = 𝑑 o pai ed samples
𝑟 = co ela ion
Powe = 𝑝1−𝛽=1−𝑝𝛽
[EPSILON2] Op imal e ec size 𝜖𝑝
𝜖𝑝=√(2⋅𝑡(𝑝𝑐𝑟𝑖𝑡,𝑑𝑓))2
𝑛,

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[EPSILON2] Op imal alpha le el
𝑡(𝑜𝑝𝑡,𝑑𝑓)
𝛼=√𝜖2⋅𝑛
2
[TKV] [ TKV|pTKV] Va iance di e ence 𝑡- es
Fo pai ed samples (𝑥1|𝑥2). 𝜃=𝑑𝜎2=𝜎12−𝜎22
wi h
𝑡(𝑑𝑓)=𝑑𝜎2⋅√𝑛−2
2⋅√𝜎12⋅𝜎22⋅(1−𝑟2)
whe e
𝑑𝑓 = 𝑛−2
[TV_] [ TV|pTV] Pai ed 2-sample 𝑡- es
𝜃=𝑥𝑑=∑𝑥(𝑖,1)−𝑥(𝑖,2)
𝑛𝑖=1 𝑛
wi h 𝑡(𝑑𝑓)=𝑥𝑑
𝜎𝑥𝑑,
𝜎𝑥𝑑=√∑(𝑥(𝑖,1)−𝑥(𝑖,2))2
𝑛𝑖=1 −(∑𝑥(𝑖,1)−𝑥(𝑖,2)
𝑛𝑖=1 )2
𝑛
𝑛−1 ⋅1
√𝑛
whe e
𝑥𝑑 = mean o he di e ences o 𝑥1 and 𝑥2 alues
𝑑𝑓 = 𝑛−1
[TU_] [ TU_|pTU_| TUx|pTUx] Unpai ed 2-sample 𝑡- es
𝜃=𝑑𝑥=𝑥1−𝑥2
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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wi h 𝑡(𝑑𝑓)=𝑑𝑥
𝜎𝑑𝑥,
𝜎𝑑𝑥=√∑(𝑥(𝑖,1)−𝑥1)2
𝑛1
𝑖=1 +∑(𝑥(𝑖,2)−𝑥2)2
𝑛2
𝑖=1
𝑛−2 ⋅√1
𝑛1+1
𝑛2
whe e
𝑑𝑥 = di e ence o he means 𝑥1 and 𝑥2
𝑑𝑓 = 𝑛1+𝑛2=𝑛−2
[TT_] [ TT_|pTT_] One-sample 𝑡- es
𝜃=𝑑𝑥𝑦=𝑥−𝑦
wi h 𝑡(𝑑𝑓)=𝑑𝑥𝑦
√𝜎2
𝑛−1
whe e
𝑑𝑥𝑦 = di e ence be ween sample mean 𝑥 and es alue 𝑦
𝑑𝑓 = 𝑛−1
[ABT1] [x2F|p2F|zBN|pzBN] 𝜒2- es o independence
𝜒2=∑(𝑓𝑒𝑖−𝑓𝑏𝑖)2
𝑓𝑏𝑖
𝑛
𝑖=1
wi h
𝑧=𝑏−𝑏+𝑐
2
√𝑏+𝑐
4
[VFCH] [x4F|p4F|x4FY|p4FY|z4F|pz4F] 2 × 2 𝜒2- es o independence
Fo Ya es’s co ec ion o con inui y see Ya es (1934).
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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𝜒2=𝑁⋅(𝑎⋅𝑑−𝑏⋅𝑐)2
(𝑎+𝑏)⋅(𝑐+𝑑)⋅(𝑎+𝑐)⋅(𝑏+𝑑),
𝜒𝑌𝑎𝑡𝑒𝑠
2=𝑁⋅(|𝑎⋅𝑑−𝑏⋅𝑐|⋅𝑁2)2
(𝑎+𝑏)⋅(𝑐+𝑑)⋅(𝑎+𝑐)⋅(𝑏+𝑑);4<𝑓𝑒<7
wi h 𝑧= 𝑑−𝑁⋅𝑃𝑑
√𝑁⋅𝑃𝑑⋅(1−𝑃𝑑)−𝑁⋅(𝑁−1)⋅𝑃𝑑⋅(𝑃𝑑−𝑃𝑑)
whe e
𝑓𝑒 = expec ed equency
𝑃𝑑=(𝑑+𝑏)⋅(𝑐+𝑑)
𝑁2
𝑃𝑑=(𝑑+𝑏−1)⋅(𝑐+𝑑−1)
(𝑁−1)2
𝑑𝑓=1
[VFCH] [xMN|pMN|xMNY|pMNY] McNema ’s 𝜒2- es o pai ed 2 × 2 con ingency
ables wi h dicho omous ai
McNema (1947).
𝜒2=(𝑏−𝑐)2
𝑏+𝑐 ,
𝜒2=(|𝑏−𝑐|−12)2
𝑏+𝑐 ;20<(𝑏+𝑐)<30
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 and 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 and 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 and P ice, 1763; c. . S igle , 2018) can be ound in Sch ausse (2024c).
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
20
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.
[Ch|ChA|ChB] A csine ans o ma ion, Cohen’s ℎ
Cohen (1988, p. 181). 𝜃=𝑝1−𝑝2,
ℎ=2⋅sin−1√𝑝1−2⋅sin−1√𝑝2
wi h
𝑝1=sin(2⋅sin−1√𝑝2+ℎ
2)2,
𝑝2=−sin(−2⋅sin−1√𝑝1+ℎ
2)2
whe e
p obabili ies = 𝑝1, 𝑝2
[ABT1] [ADDP] [E01] Addi i e p obabili y o independen e en s 𝑢𝑝(∪𝑛𝐴)
Co esponds o he geome ic dis ibu ion 𝑓(𝑋≤𝑟|𝑝). 𝑢𝑝(∪𝑛𝐴)=1−(1−𝑝𝐴)𝑛
whe e
𝑛 = numbe o e en s 𝐴
𝑝𝐴 = p obabili y o e en 𝐴
[GMVTLG] Geome ic dis ibu ion 𝑓(𝑋≤𝑟|𝑝)
Co esponds o he addi i e p obabili y 𝑢𝑝(∪𝑛𝐴). 𝑓(𝑋=𝑟|𝑝)=𝑃𝑛=𝑝⋅𝑞𝑟
wi h
𝑓(𝑋≤𝑟|𝑝)=𝑝𝑛=∑𝑝⋅𝑞𝑖
𝑟
𝑖=0
whe e
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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𝑝 = p obabili y o e en
𝑟+1=𝑛 = numbe o e en s
[NBNMVTLG] [NBINOM] [E01] 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). 𝑓(𝑋=𝑟|𝑟,𝑝)=𝑃𝑛=(𝑘+𝑟−1)!
𝑟!⋅(𝑘−1)!⋅𝑝𝑘⋅𝑞𝑟
wi h
𝑓(𝑋≤𝑟|𝑟,𝑝)=𝑝𝑛=∑(𝑘+𝑖−1)!
𝑖!⋅(𝑘−1)!⋅𝑝𝑘⋅𝑞𝑖
𝑟
𝑖=0
whe e
𝑟+𝑘=𝑛 = numbe o e en s
𝑘 = numbe o successes
[ABT1] [BINOM|zBN|pzBN] [E01] Exac binomial es
𝑓(𝑋=𝑏|𝑏,𝑐)=𝑃0=(𝑏+𝑐)!
𝑏!⋅𝑐! ⋅2−𝑏⋅2−𝑐
wi h
𝑓(𝑋≤𝑏|𝑏,𝑐)=𝑝=𝑝𝑒𝑥𝑎𝑐𝑡1=∑ (𝑏+𝑐)!
𝑖!⋅(𝑏+𝑐−𝑖)!
𝑏
𝑖=0 ⋅2−𝑖⋅2−(𝑏+𝑐−𝑖);𝑝≤12,
𝑝𝑒𝑥𝑎𝑐𝑡1=(1−𝑝)+𝑃0;𝑝>12
also
𝑧=𝑏−𝑏+𝑐
2
√𝑏+𝑐
4
[FX_] [z4F|pz4F] Exac hype geome ic 2 × 2 es
Fishe Exac es (Fishe , 1922; Ag es i, 1992).
𝑓(𝑋=𝑎|𝑎,𝑏,𝑐,𝑑)=𝑃0=(𝑎+𝑏)!⋅(𝑐+𝑑)!⋅(𝑎+𝑐)!⋅(𝑏+𝑑)!
𝑁!⋅𝑎!⋅𝑏!⋅𝑐!⋅𝑑!

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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wi h
𝑓(𝑋≤𝑎|𝑎,𝑏,𝑐,𝑑)=𝑝𝑒𝑥𝑎𝑐𝑡1=∑𝑃
𝑎
𝑖=1 𝑖;𝑝≤12,
𝑓(𝑋≥𝑎|𝑎,𝑏,𝑐,𝑑)=𝑝𝑒𝑥𝑎𝑐𝑡1=∑𝑃
𝑛
𝑖=𝑎 𝑖;𝑝>12
whe e
𝑃𝑖=(𝑎+𝑏)!⋅(𝑐+𝑑)!⋅(𝑎+𝑐)!⋅(𝑏+𝑑)!
𝑁!⋅𝑖!⋅(𝑎+𝑏−𝑖)!⋅(𝑎+𝑐−𝑖)!⋅(2⋅𝑐+𝑑−𝑎−𝑖)!
also 𝑧= 𝑑−𝑁⋅𝑃𝑑
√𝑁⋅𝑃𝑑⋅(1−𝑃𝑑)−𝑁⋅(𝑁−1)⋅𝑃𝑑⋅(𝑃𝑑−𝑃𝑑)
whe e
𝑃𝑑=(𝑑+𝑏)⋅(𝑐+𝑑)
𝑁2
𝑃𝑑=(𝑑+𝑏−1)⋅(𝑐+𝑑−1)
(𝑁−1)2
𝑑𝑓=1
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 combina o ics.
The unc ions gene a e pe mu a ion and a ia ion ma ices p ima ily o suppo he
esampling p ocedu es desc ibed below (s. Resampling).
[PRM2] Pe mu a ion ma ix 𝑷𝒏
𝑛 elemen s o 𝑘=1 class.
𝐏𝐧=[𝑝1(𝑥1)⋯ 𝑝1(𝑥𝑛)
⋮ ⋱ ⋮
𝑝𝑃(𝑥1)⋯ 𝑝𝑃(𝑥𝑛)]
whe e
𝑃𝑛=𝑛!
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
23
[PRM5] Va ia ion ma ix 𝒘𝑽𝟐𝒎
Fo he dependen 2 sample design, 𝑛=2 elemen s o class 𝑚.
𝒘𝑽𝟐𝒎=[𝑣1(𝑥1)⋯ 𝑣1(𝑥𝑚)
⋮ ⋱ ⋮
𝑣𝑤𝑉(𝑥1)⋯ 𝑣𝑤𝑉(𝑥𝑚)]
whe e
𝑤𝑉2𝑚=2𝑚
[PRM4] Va ia ion ma ix 𝒘𝑽𝒏𝒎
𝑛 elemen s o class 𝑚.
𝒘𝑽𝒏𝒎=[𝑣1(𝑥1)⋯ 𝑣1(𝑥𝑚)
⋮ ⋱ ⋮
𝑣𝑤𝑉(𝑥1)⋯ 𝑣𝑤𝑉(𝑥𝑚)]
whe e
𝑤𝑉𝑛𝑚=𝑛𝑚;𝑛>𝑚
[PRM3] [nk] Pe mu a ion ma ix 𝒘𝑷𝒏(𝒌𝒎,𝒌𝒏−𝒎)
𝑛 elemen s o class 𝑚.
𝒘𝑷𝒏(𝒌𝒎,𝒌𝒎−𝒏)=[𝑝1(𝑥11)⋯ 𝑝1(𝑥𝑘1) 𝑝1(𝑥12)⋯ 𝑝1(𝑥𝑘2)
⋮ ⋱ ⋮ ⋮ ⋱ ⋮
𝑝𝑤𝑃(𝑥11)⋯ 𝑝𝑤𝑃(𝑥𝑘1) 𝑝𝑤𝑃(𝑥12)⋯ 𝑝𝑤𝑃(𝑥𝑘2)]
whe e
𝑤𝑃𝑛(𝑘𝑚,𝑘𝑛−𝑚)=𝑛!
∏𝑘𝑖
2𝑖=1 !;𝑛≥𝑚
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 and Ulam, 1949), o so wa e solu ions see
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
24
e.g. Solomon (1982), Dallal (1986, 1988), Peladeau (1993), Woo and Peladeau (1994),
Meh a e al. (2014), also Sch ausse (2024d).
[PV_] 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
𝒘𝑽𝟐𝒎 equi ed, s. Scambo (1997), Scambo and Sch ausse (2022, p. 7), espec i ely.
𝛩11=∑𝑥1𝑖
𝑛
𝑖=1 ,𝛩21=∑𝑥2𝑖
𝑛
𝑖=1 ,
𝛩2=(∑𝑥1𝑖
𝑛
𝑖=1 )2+(∑𝑥2𝑖
𝑛
𝑖=1 )2
wi h
𝑝𝑒𝑥𝑎𝑐𝑡=∑1
2𝑛
𝑖=1
2𝑛;𝜃𝑖≥𝛩
whe e
𝛩11,𝛩21 = one- ailed es alues
𝛩2 = wo- ailed es alue
[mPV_] Randomized pe mu a ion es mP o 2 pai ed samples (𝑥1|𝑥2)
Random sampling model, 𝑝- alue no andomized.
𝛩11=∑𝑥1𝑖
𝑛
𝑖=1 ,𝛩21=∑𝑥2𝑖
𝑛
𝑖=1 ,
𝛩2=(∑𝑥1𝑖
𝑛
𝑖=1 )2+(∑𝑥2𝑖
𝑛
𝑖=1 )2
wi h
𝑝=∑1
𝑀
𝑖=1
𝑀;𝜃𝑖≥𝛩
whe e
𝛩11,𝛩21 = one- ailed es alues
𝛩2 = wo- ailed es alue
𝑀 = simula ion cycles o e a ia ions 𝑤𝑉2𝑚=2𝑛
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
25
[PU_] 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).
𝛩11=∑𝑥𝑔1𝑖
𝑛1
𝑖=1 ,𝛩21=∑𝑥𝑔2𝑖
𝑛2
𝑖=1 ,
𝛩2=|𝑥𝑔1−𝑥𝑔2|
wi h
𝑝𝑒𝑥𝑎𝑐𝑡=∑1
𝑛!
𝑛1!⋅𝑛2!
𝑖=1
𝑛!
𝑛1!⋅𝑛2!;𝜃𝑖≥𝛩
whe e
𝛩11,𝛩21 = one- ailed es alues
𝛩2 = wo- ailed es alue
𝑛=𝑛1+𝑛2
[mPU_] Randomized pe mu a ion es mP o 2 independen samples (𝑥|𝑔)
Random sampling model, 𝑝- alue no andomized.
𝛩11=∑𝑥𝑔1𝑖
𝑛1
𝑖=1 ,𝛩21=∑𝑥𝑔2𝑖
𝑛2
𝑖=1 ,
𝛩2=|𝑥𝑔1−𝑥𝑔2|
wi h
𝑝=∑1
𝑀
𝑖=1
𝑀;𝜃𝑖≥𝛩
whe e
𝛩11,𝛩21 = one- ailed es alues
𝛩2 = wo- ailed es alue
𝑛=𝑛1+𝑛2
𝑀 = simula ion cycles o e pe mu a ions 𝑤𝑃𝑛(𝑘𝑚,𝑘𝑛−𝑚)=𝑛!
𝑛1!⋅𝑛2!
[B U_] Boo s ap es B o 2 independen samples (𝑥|𝑔)
Quenouille (1949), E on (1979, 1981, 1982).
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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L3(3) NTHROOT (p oduc ((L1(x)(1))^L1(x)(2),x,1,L3(1)))▸L3(5)
L3(3)/(Σ(L1(x)(2)/L1(x)(1),x,1,L3(1)))▸L3(6)
//n,AM,sumni,GAM,GGM,GHM
L3
END;
#end
//
B
BNMVTLG.pas
//BNMVTLG(p[e],a=k,n)/D.G.SCHRAUSSER/2025
//e.g.BNMVTLG(0.5,5,10)
#cas
BNMVTLG(P,K,N):=
BEGIN
B=0;
FOR I FROM 0 TO K DO
BINOMIAL(N,P,I)▸L4(I)
B=B+L4(I)
END;
D5=L4;L4={}
FOR I FROM 0 TO N DO
BINOMIAL(N,P,I)▸L5(I);
END;
D6=L5;L5={};
STARTAPP("S a is iken_1_Va ");
STARTVIEW(1);
"D5"▸H1(1);5▸H1(3);
"D6"▸H2(1);5▸H2(3);
//p
RETURN(B);
END;
#end
//
B U_.pas
//B U_(simula ion cycles B)/D.G.SCHRAUSSER/2025
//Boo s ap me hod, B
//2 independen samples (x|g)
//e.g.B U_(1000)
#cas
B U_(B):=
BEGIN
//L1L2 p o ided
{}▸L3;{}▸L4
{}▸L5;{}▸L6
0▸M11
0▸M12
0▸M2
SIZE(L1)▸N1
SIZE(L2)▸N2
N=N1+N2
ABS(mean(L1)-mean(L2))▸Q02
ΣLIST(L1)▸Q011
ΣLIST(L2)▸Q012
//
CONCAT(L1,L2)▸L9
MSGBOX("B U")
FOR J FROM 1 TO B
DO
//
FOR A FROM 1 TO N DO
L9(RANDINT(N))▸L0(A) END;
FOR A FROM 1 TO N1 DO

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
33
L0(A)▸L3(A) END;
FOR A FROM 1 TO N2 DO
L0(N1+A)▸L4(A) END;
ABS(mean(L3)-mean(L4))▸QJ2
ΣLIST(L3)▸QJ11
ΣLIST(L4)▸QJ12
IF QJ11≥Q011 THEN M11=M11+1 END;
IF QJ12≥Q012 THEN M12=M12+1 END;
IF QJ2≥Q02 THEN M2=M2+1 END;
QJ11▸L5(J)
QJ2▸L6(J)
END;
//
SORT(L5)▸L5
SORT(L6)▸L6
{}▸L9
{}▸L0
N1,N2,[Q011,Q012,Q02],M11/B,M12/B,M2/B
END;
#end
//
C
ch2VTLG.pas
//ch2VTLG(chi-squa ed,d )/D.G.SCHRAUSSER/2025
//e.g.ch2VTLG(2.65,1)[Ad ancedG aphing]
#cas
ch2VTLG(C2569,A7485):=
BEGIN
G=Gamma(A7485/2)
//P=∫((1/(2^(A7485/2)*G))*X^((A7485/2)-1)*e^(-X/2),X,0,C)
P=CHISQUARE_CDF(A7485,C2569)
A7485▸A
C2569▸C
"Y=(1/(2^(A/2)*G))*X^((A/2)-1)*e^(-X/2)"▸V1
"Y<(1/(2^(A/2)*G))*X^((A/2)-1)*e^(-X/2) AND Y>0 AND X<C AND X>0"▸V2
STARTAPP("E wei e e_G a iken");
STARTVIEW(1);
[1-P]
END;
#end
//
CIXY.pas
//CIXY(x,y'CI)/D.G.SCHRAUSSER/2022
//S anda d e o o p edic ion sy'x, CI
//e.g.CIXY(3,0.99[ZWERT,S a is ics_2_Va ,Sp eadshee ,Ad ancedG aphing]
#cas
CIXY(X,C):=
BEGIN
//C1C2 p o ided
STARTAPP("S a is iken_2_Va ");
STARTVIEW(−6)
A=Co
B=sY
D=MeanY
P edY(X)▸L3(2);
√(1-A^2)*B*NORMALD_ICDF(1-((1-C)/2))▸L3(4)
L3(4)+L3(2)▸L3(3)
L3(2)-L3(4)▸L3(1)
C▸L3(5)
ZWERT(L3(1),D,B)▸L4(1)
ZWERT(L3(2),D,B)▸L4(2)
ZWERT(L3(3),D,B)▸L4(3)
L4(1)▸U
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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L4(3)▸O
ZWERT(X,MeanX,sX)▸Q
STARTAPP("A bei sbla ");
"ŷ-"▸A1;L3(1)▸B1;L4(1)▸C1
"ŷ"▸A2;L3(2)▸B2;L4(2)▸C2
"ŷ+"▸A3;L3(3)▸B3;L4(3)▸C3
"±"▸A4;L3(4)▸B4;L3(4)/B▸C4
"CI"▸A5;L3(5)▸B5;
STARTAPP("E wei e e_G a iken");
STARTVIEW(1)
"Y=A*X"▸V3
"Y>0 AND (Y<A^(-1)*X AND Y>A*X) OR Y<0 AND (Y>A^(-1)*X AND Y<A*X)"▸V4
"Y=√(1-X^2)"▸V5
"Y=-1*√(1-X^2)"▸V6
"X<A AND X>0 AND Y<A AND Y>0"▸V7
CAS((X,Y)->((Y<O) AND (Y>U)) AND ((X==Q))▸V0)
//y'-,y',y'+,CI,sy'x,CIp
RETURN(L3);
END;
#end
//
CPLHX.pas
//CPLHX(complex numbe ,a+bi)/D.G.SCHRAUSSER/2022
//e.g.CPLHX(2+i/2),[Ad ancedG aphing]
#cas
CPLHX(C):=
BEGIN
C▸Z1
RE(Z1)▸R
IM(Z1)▸I
Z1▸L1(1)
ABS(Z1)▸L1(2)
ARG(Z1)▸L1(3)
"Y=R*X"▸V1
"Y=I"▸V2
"Y=√((R*X)^2+I^2)"▸V3
"Y=(I/ABS(I))*(π/2)-ATAN((R/I)*X)"▸V4
STARTAPP("E wei e e_G a iken")
STARTVIEW(1)
RETURN(L1);
END;
#end
//
CPLX.pas
//CPLX(complex numbe ,a+bi)/D.G.SCHRAUSSER/2022
//e.g.CPLX(2+i/2),[Ad ancedG aphing]
#cas
CPLX(C):=
BEGIN
C▸Z1
RE(Z1)▸R
IM(Z1)▸I
ABS(Z1)▸L1(1)
R▸X▸J
I▸K
"Y=√(1-X^2)"▸V5
"Y=-1*√(1-X^2)"▸V6
"Y=I"▸V7
"X=R"▸V8
"X<R AND X>0 AND Y>0 AND Y≤0.01"▸V0
"Y=(I/R)*X AND Y>0 AND X<R"▸V9
IF I<0 THEN
"Y=(I/R)*X AND Y<0 AND X<R"▸V9
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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END;
IF R<0 THEN
"Y=(I/R)*X AND Y>0 AND X>R"▸V9
END;
IF I<0 AND R<0 THEN
"Y=(I/R)*X AND Y<0 AND X>R"▸V9
END;
ARG(Z1)▸L1(2)
CONVERT(L1(2)_ ad,1_deg)▸L1(3)
RETURN(L1);
END;
#end
//
CPLX.pas
//CPLX2(complex numbe ,a+bi)/D.G.SCHRAUSSER/2022
//e.g.CPLX2(2+i/2),[CPLX,Sp eadshee ]//
#cas
CPLX2(C):=
BEGIN
CPLX(C)
STARTAPP("A bei sbla ");
"z"▸A1;Z1▸B1
"|z|"▸A2;L1(1)▸B2
"∡π"▸A3;L1(2)▸B3
"∡°"▸A4;L1(3)▸B4
END;
#end
//
D
DELTA2.pas
//DELTA2()/D.G.SCHRAUSSER/2025
//Some s' D o bina y alues [0,1]
#cas
DELTA2():=
BEGIN
SIZE(L1)▸N
{}▸L3
0▸X01
0▸X02
FOR I FROM 1 TO N DO
IF L1(I)=1 AND L2(I)=1 THEN
X01=X01+1
END;
IF L1(I)=1 AND L2(I)=0 THEN
X02=X02+1
END;
END;
X01/N▸L3(1)
X02/N▸L3(2)
L3(1)-L3(2)▸L3(3)
//pA,pB,D
app ox(L3)
END;
#end
//
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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E
EPSILON.pas
//EPSILON(x1,m1,m2,s12,d)/D.G.SCHRAUSSER/2022
//e.g.EPSILON(106,100,110,15,25)
#cas
EPSILON(X,M,N,S,D):=
BEGIN
G=Gamma((D+1)/2)/Gamma(D/2)
E=(N-M)/S
P=1-∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,E)
T=(((N+M)/2)-M)/S
H=1-∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,T)
Q=(X-M)/S
R=1-∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,Q)
U=∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,(X-N)/S)
B=1-U
"X>T AND X≤T"▸V1
"X>Q AND X≤Q"▸V2
//
D▸K
"X>0 AND X<E AND Y<0 AND Y>-0.01"▸V3
"Y<(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2)) AND Y>0 AND X>Q"▸V6
"Y<(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2)) AND Y>0 AND X<Q"▸V7
"Y=(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2))"▸V8
"Y=(G*(K*π)^(-1/2)*(1+((X)^2/K))^(-(K+1)/2))"▸V0
"Y<(G*(K*π)^(-1/2)*(1+((X)^2/K))^(-(K+1)/2)) AND Y>0 AND X>Q"▸V9
E▸L2(1);P▸L3(1);N▸L1(1)
T▸L2(2);H▸L3(2);T*S+M▸L1(2)
Q▸L2(3);R▸L3(3);Q*S+M▸L1(3)
U▸L4(3);B▸L5(3)
STARTAPP("A bei sbla ");
"ε"▸A1;L2(3)▸B1;L2(2)▸C1;L2(1)▸D1;
"x"▸A2;L1(3)▸B2;L1(2)▸C2;L1(1)▸D2;
"α"▸A3;L3(3)▸B3;L3(2)▸C3;L3(1)▸D3;
"β"▸A4;L4(3)▸B4;
"1-β"▸A5;L5(3)▸B5;
STARTAPP("E wei e e_G a iken")
STARTVIEW(1)
RETURN(L2(1),L3(3),L4(3));
END;
#end
//
EPSILON2.pas
//EPSILON2(epsilon,n,d ,pc i )/D.G.SCHRAUSSER/2022
//e.g.EPSILON2(0.38,100,99,0.95)
//op imal e ec size epsilon
//op imal alpha
//1-p(alpha op )
#cas
EPSILON2(E,N,D,K):=
BEGIN
# op ni
V=√(E^2*N)/2▸L6(2)
P=1-STUDENT_CDF(D,V)▸L6(3)
#e op e s ke
L=√((2*STUDENT_ICDF(D,K))^2/N)▸L6(1)
"X>0 AND X<L AND Y<0 AND Y>-0.02"▸V1
RETURN(L6);
END;
#end
//
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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EPSOLON3.pas
//EPSILON3(100,110,15,25,0.99)
#cas
EPSILON3(M,N,S,D,K):=
BEGIN
F=STUDENT_ICDF(D,K)
M+S*F▸L7(1)
N-S*F▸L7(2)
"X>E-F AND X≤E-F AND Y>0 AND Y<(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2))"
▸V4
"X>F AND X≤F AND Y>0 AND Y<(G*(K*π)^(-1/2)*(1+((X)^2/K))^(-(K+1)/2))"▸V5
RETURN(L7);
END;
#end
//
F
FVTLG.pas
//FVTLG(F,d 1,d 2)/D.G.SCHRAUSSER/2025
//e.g.FVTLG(2.8,10,5)
#cas
FVTLG(F,A,B):=
BEGIN
F▸X
CAS(Gamma((A+B)/2))▸H
CAS(Gamma(A/2))▸D
CAS(Gamma(B/2))▸E
CAS(H/(D*E))▸C
CAS((X,Y)->Y=C*((A/B)^(A/2)*X^((A/2)-1)*(1+(A/B)*X)^(−(((A+B)/2)))) AND X>0
▸V2)
CAS((X,Y)->Y<C*((A/B)^(A/2)*X^((A/2)-1)*(1+(A/B)*X)^(−(((A+B)/2)))) AND Y>0
AND X<F AND X>0▸V1)
FISHER_CDF(A,B,X)▸P
STARTAPP("E wei e e_G a iken")
STARTVIEW(1)
P,[1-P]
END;
#end
//
FX.pas
//FX_(cell coun a,b,c,d)/D.G.SCHRAUSSER/2025
//e.g.FX_(1,2,3,1)
//Exac hype geome ic 4- ield es acco ding o R. A. Fishe
//(Fishe Exac Tes ): Hype geome ic p obabili y p o cell a o he 4- ield
ini ial a angemen o all possible a angemen s a
//Exac signi icance le els p[exac 1], p[exac 2]
#cas
FX_(a,b,c,d):=
BEGIN
{}▸L1
1▸S
0▸X
0▸P20
0▸P21
0▸P3
a+b+c+d▸N
a+b▸z1
c+d▸z2
a+c▸s1
b+d▸s2
P0= (z1!*z2!*s1!*s2!)/(N!*a!*b!*c!*d!);

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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P0▸L2(1)
PRINT("P0-")
PRINT(P0)
PRINT("Pi-")
//
IF z1>s1 THEN max1=z1 ELSE max1=s1 END;
IF z1>s2 THEN max2=z1 ELSE max2=s2 END;
IF z2>s1 THEN max3=z2 ELSE max3=s1 END;
IF z2>s2 THEN max4=z2 ELSE max4=s2 END;
//
FOR I FROM 0 TO max1 DO
FOR J FROM 0 TO max2 DO
FOR K FROM 0 TO max3 DO
FOR L FROM 0 TO max4 DO
a1=I+J;
a2=K+L;
b1=I+K;
b2=J+L;
//
IF
a1=z1 AND
a2=z2 AND
b1=s1 AND
b2=s2
THEN
IF
I+J≠0 AND
K+L≠0 AND
I+K≠0 AND
J+L≠0
THEN
P10=(a1!*a2!*b1!*b2!)/(N!*I!*J!*K!*L!)
X+1▸X
P3=P10+P3
app ox(P10)▸L1(X)
IF app ox(P10)<app ox(P0) OR app ox(P10)=app ox(P0)
THEN
P20+P10▸P20 END;
PRINT(P10)
END;
END;
//
END;
END;
END;
END;
PRINT("p--")
PRINT(P3)
//
FOR I FROM 1 TO X DO
IF L1(I)=L2(1) THEN
0▸S
END;
IF S=1 THEN
L1(I)+P21▸P21
END;
END;
P21+P0▸P21
P22=1-(P21)
//sums, P0, C, p[exac 1], 1-p[exac 1], p[exac 2]
z1,z2,s1,s2,N,[P0],X,P21,P22,[P20]
END;
#end
//
FX_.pas
//FX_(cell coun a,b,c,d)/D.G.SCHRAUSSER/2025
//e.g.FX_(1,2,3,1)
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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//Exac hype geome ic 4- ield es acco ding o R. A. Fishe
//(Fishe Exac Tes ): Hype geome ic p obabili y p o cell a o he 4- ield
ini ial a angemen o all possible a angemen s a
//Exac signi icance le els p[exac 1], p[exac 2]
//(slow algo i hm)
#cas
FX_(a,b,c,d):=
BEGIN
{}▸L1
1▸S
0▸X
0▸P20
0▸P21
0▸P3
a+b+c+d▸N
a+b▸z1
c+d▸z2
a+c▸s1
b+d▸s2
P0= (z1!*z2!*s1!*s2!)/(N!*a!*b!*c!*d!);
PRINT("P0-")
PRINT(P0)
PRINT("Pi-")
//
FOR I FROM 0 TO N DO
FOR J FROM 0 TO N DO
FOR K FROM 0 TO N DO
FOR L FROM 0 TO N DO
a1=I+J;
a2=K+L;
b1=I+K;
b2=J+L;
//
IF
a1=z1 AND
a2=z2 AND
b1=s1 AND
b2=s2
THEN
IF
I+J≠0 AND
K+L≠0 AND
I+K≠0 AND
J+L≠0
THEN
P10=(a1!*a2!*b1!*b2!)/(N!*I!*J!*K!*L!)
X+1▸X
P3=P10+P3
app ox(P10)▸L1(X)
IF app ox(P10)<app ox(P0) OR
app ox(P10)=app ox(P0)
THEN
P20+P10▸P20 END;
PRINT(P10)
END;
END;
//
END;
END;
END;
END;
PRINT("p--")
PRINT(P3)
//
FOR I FROM 1 TO X DO
IF L1(I)=P0 THEN
0▸S
END;
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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IF S=1 THEN
L1(I)+P21▸P21
END;
END;
P21+P0▸P21
P22=1-(P21)
// sums, P0, C, p[exac 1], 1-p[exac 1], p[exac 2]
z1,z2,s1,s2,N,[P0],X,P21,P22,[P20]
END;
#end
//
G
GMVTLG.pas
//GMVTLG(pA, +1=n)/D.G.SCHRAUSSER/2025
//e.g.GMVTLG(1/6,10)
#cas
GMVTLG(P,N):=
BEGIN
MAKELIST(P*(1-P)^x,x,0,N-1)▸L1
MAKELIST(1-(1-P)^x,x,1,N)▸L2
STARTAPP("S a is iken_1_Va ");
STARTVIEW(1);
"L1"▸H1(1);6▸H1(3);
"L2"▸H2(1);5▸H2(3);
//P,p
L1(N),L2(N)
END;
#end
//
I
IC_M.pas
//IC_M(n o a iables k)/D.G.SCHRAUSSER/2025
//In e co ela ion ma ix, Pea son co ela ion
//e.g.IC_M(5)[pCo ]
//M1:
//M2:de %( ²×100)
//M3: - alue
//M4:2- ailed p
#cas
IC_M(K):=
BEGIN
//L1(n)(k) p o ided
{}▸L4
{}▸L5
{}▸L6
{}▸L7
{}▸L8
size(L1)▸N
FOR I FROM 1 TO K DO
FOR J FROM 1 TO K DO
MAKELIST({L1(X,I),L1(X,J)},X,1,N)▸L4
app ox(co ela ion(L4))▸R125▸L5(I,J) //
app ox(pCo (R125,N)(3))▸L0;L0(1)▸L6(I,J) //p2
app ox(pCo (R125,N)(1))▸L0;L0(1)▸L7(I,J) //
app ox(R125^2*100)▸L8(I,J) //de
END;
END;
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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L5▸M1
L8▸M2
L7▸M3
L6▸M4
//
END;
#end
//
K
KOR.pas
//KOR()/D.G.SCHRAUSSER/2025
//Pea son co /[pCo ]
#cas
KOR():=
BEGIN
//L1()(2) p o ided
SIZE(L1)(1)▸N
N-2▸d
co a iance_co ela ion(L1)▸L0
L0(2)▸
L0(1)▸c
pCo ( ,N)(2)▸p
pCo ( ,N)(3)▸p2
^2*100▸D
{}▸L0
//
d ,[c , ,D],p,[p2]
END;
#end
//
M
MCORR2.pas
//MCORR2()/D.G.SCHRAUSSER/2022
//Mul iple co ela ion R
//[S a is iken_2_Va ,A bei sbla ,G aph3D,FMCORR,MCORR]
#cas
MCORR2():=
BEGIN
//M1()(3) p o ided
//C1C2 o S1
STARTAPP("S a is iken_2_Va ");
STARTVIEW(−6)
M2=TRN(M1)
L7=M2(1);C1=L7
L7=M2(2);C2=L7
Do2VS a s(S1)
Co ▸L1(1)
MeanX▸L3(1);sX▸L4(1)
MeanY▸L3(2);sY▸L4(2)
//
L7=M2(1);C1=L7
L7=M2(3);C2=L7
Do2VS a s(S1)
Co ▸L1(2)
MeanY▸L3(3);sY▸L4(3)
//
L7=M2(2);C1=L7
L7=M2(3);C2=L7
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app ox(co ela ion(L4))▸ 1
pCo ( 1,N)(3)▸p 1
// yz
FOR I FROM 1 TO N DO
L2(I)▸L5(1)
L3(I)▸L5(2)
L5▸L4(I)
END;
app ox(co ela ion(L4))▸ 2
pCo ( 2,N)(3)▸p 2
// xy.z
p=( 0- 1* 2)
p= p/(sq (1- 1^2)*sq (1- 2^2))
p=app ox( p)
ZCo ( p,N)(1)▸L0
L0(1)*SQRT(N-2)▸z p
p p=NORMALD_CDF(z p)
IF p p>0.5 THEN p p=1-p p END;
p p2=2*p p
//pCo ( p,N)(3)▸p
//d , xy,p2, xz,p2, yz,p2, xy.z,p2
d ,[ 0,p 0],[ 1,p 1],[ 2,p 2],[ p,p p]
END;
#end
//
PRM1.pas
//PRM1(n pe m)/D.G.SCHRAUSSER/2025
//e.g.PRM1(5)/pe mu a ion ec o (p)n om L1
#cas
PRM1(N):=
BEGIN
//L1(N) p o ided
{}▸L0
{}▸L2
FOR A FROM 1 TO N DO
{RANDOM(),L1(A)}▸L0(A) END;
//
so (L0)▸L2
FOR A FROM 1 TO N DO
L2(A)▸L8;L8(2)▸L2(A) END;
//
L2
END;
#end
//
PRM2.pas
//PRM2(elemen s n)/D.G.SCHRAUSSER/2025
//Comple e pe mu a ion ma ix (P)n o elemen s n o 1 class,
//whe e P=n!
//e.g.PRM2(3)
#cas
PRM2(n):=
BEGIN
MAKELIST(1,P,1,n+1)▸L1
P=PERM(n,n)
0▸L1(1)
{}▸L2
0▸M1
1▸J
0▸I
0▸SW
//
WHILE I≠n AND L1(I)≤n DO
FOR I FROM 1 TO n DO

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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IF I=1 THEN L1(1)+1▸L1(1) END;
IF I=n AND L1(I)>n THEN BREAK END;
IF L1(I)>n THEN
1▸L1(I);L1(I+1)+1▸L1(I+1)
END;
END;//I
//
FOR K FROM 1 TO n DO
FOR L FROM K+1 TO n DO
IF L1(K)=L1(L) THEN
1▸SW
BREAK;
END;
END;
END;
IF SW=0 THEN
SUPPRESS(L1,n+1)▸L2(J);J+1▸J
END;
0▸SW
END;//while
//
L2▸M1
IF n=2 THEN
M1=[[1,2],[2,1]]
M1▸L2
END;
//
P,M1
END;
#end
//
PRM3.pas
//PRM3(elemen s n, class m)/D.G.SCHRAUSSER/2025
//Comple e pe mu a ion ma ix w(P)n(km,kn-m) o n elemen s o class m, whe e
P=n!/IIki!;n>=m
//equi alen o combina ion wi hou epe i ion Cn(m)
//e.g.PRM3(6,3)[PRM3a]
#cas
PRM3(n,m):=
BEGIN
MAKELIST(1,P,1,n+1)▸L1
0▸L1(1)
{}▸L2
0▸M1
1▸J
0▸I
0▸SW
//
WHILE I≠m AND L1(I)<n DO
FOR I FROM 1 TO m DO
IF I=1 THEN L1(1)+1▸L1(1) END;
IF I=m AND L1(I)>n THEN BREAK END;
IF L1(I)>n THEN
1▸L1(I)
L1(I+1)+1▸L1(I+1)
END;
END;//I
//
FOR K FROM 1 TO m DO
FOR L FROM K+1 TO m DO
IF L1(K)=L1(L) OR L1(K)>L1(L) THEN //<---
1▸SW
END;
END;
END;
IF SW=0 THEN
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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SUPPRESS(L1,n+1)▸L2(J)
J+1▸J
END;
0▸SW
END;//while
//
//L2▸M1
PRM3a(n,m)//
END;
#end
//
PRM3a.pas
//PRM3a(elemen s n,class m)/D.G.SCHRAUSSER/2025
//e.g.PRM3a(6,3)
#cas
PRM3a(N,M):=
BEGIN
{}▸L3
COMB(N,M)▸P
MAKELIST(x+1-1,x,1,N)▸L1
//
FOR J FROM 1 TO P DO
FOR I FROM 1 TO M DO
L2(J,I)▸L3(I)
END;
L3▸L4(J)▸M1
END;
FOR I FROM 1 TO P DO
L4(I)▸L5;DIFFERENCE(L5,L1)▸L7(I)
END;
FOR I FROM 1 TO P DO
CONCAT(L4(I),L7(I))▸L8(I)
END;
L4▸L2;L8▸L3
{}▸L4
{}▸L8
{}▸L5
{}▸L6
{}▸L7
L3▸M2
P,M2
END;
#end
//
PRM4.pas
//PRM4(elemen s n, class m)/D.G.SCHRAUSSER/2025
// a ia ion ma ix w(V)n(m), whe e V=n^m;n>=m
//e.g.PRM4(4,2)[PRM4a]
#cas
PRM4(n,m):=
BEGIN
MAKELIST(1,P,1,n+1)▸L1
0▸L1(1)
{}▸L2
0▸M1
1▸J
0▸I
0▸SW
//
WHILE I≠m AND L1(I)<n DO
FOR I FROM 1 TO m DO
IF I=1 THEN L1(1)+1▸L1(1) END;
IF I=m AND L1(I)>n THEN BREAK END;
IF L1(I)>n THEN
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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1▸L1(I)
L1(I+1)+1▸L1(I+1)
END;
END;//I
//
SUPPRESS(L1,n+1)▸L2(J);J+1▸J
END;//while
//
//L2▸M1
PRM4a(n,m)
END;
#end
//
PRM4a.pas
//PRM4a(elemen s n, class m)/D.G.SCHRAUSSER/2025
//e.g.PRM4a(4,3)
#cas
PRM4a(N,M):=
BEGIN
//L2 p o ided
V=N^M
X=M+1
FOR I FROM 1 TO V DO
L2(I)▸L3
SUPPRESS(L3,X,N)▸L4(I)
END;
L4▸M1
V,M1
END;
#end
//
PRM5.pas
//PRM5(cases m)/D.G.SCHRAUSSER/2025
// a ia ion ma ix w(V)2(m) o pai ed 2 sample design PV_,
//whe e V=2^m
//e.g.PRM5(3)
#cas
PRM5(N):=
BEGIN
M1=0
2^N▸P
X=−1
0▸Z
P/2▸A
//
FOR I FROM 1 TO N DO
FOR J FROM 1 TO P DO
X▸M1(J,I)
Z=Z+1
IF Z=A THEN X=X*−1;0▸Z; END;
END;
0▸Z
// A=A/2 //
A=A*0.5
END;
//
M1▸L3
P,M1
END;
#end
//
PRMDAT.pas
//PRMDAT( ows n, cols k)/D.G.SCHRAUSSER/2025
//e.g.PRMDAT(720,6)
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#cas
PRMDAT(N,K):=
BEGIN
//L3 p o ided
FOR J FROM 1 TO N DO
FOR I FROM 1 TO K DO
L3(M1(J,I))▸M2(J,I)
END;
END;
END;
#end
//
Q
Q01_.pas
//Q01_()/D.G.SCHRAUSSER/2025
//S a is ical pa ame e s 1.0
//[L1:Raw]
//L2:Dis ibu ion
//L3:z- alue
//L4:z´- alue
#cas
Q01_():=
BEGIN
//L1 p o ided
SORT(L1)▸L2 //dis
SIZE(L1)▸N
mean(L1)▸AM
s dde (L1)▸SD
s dde p(L1)▸SD1
a iance(L1)▸VA
VA1=VA*(N/(N-1)) //SD1^2
SEM=sq ((VA1/N))
VQ=SD/AM
QGM= N NTHROOT(p oduc (L1))
QHM=N/Σ(1/L1)
app ox(MAKELIST(((L2(X)-AM)/SD),X,1,N))▸L3 //z
app ox(MAKELIST(((L2(X)-AM)/SD1),X,1,N))▸L4 //z´
//
app ox(N,[AM,SEM],SD,SD1,VA,VA1,VQ,[QGM,QHM])
END;
#end
//
Q02_.pas
//Q02_()/D.G.SCHRAUSSER/2025
//S a is ical pa ame e s 2.0
//[L1:Raw]
//L2:Dis ibu ion
//L3:z- alue
#cas
Q02_():=
BEGIN
//L1 p o ided
SORT(L1)▸L2 //
SIZE(L1)▸N
mean(L1)▸AM
s dde (L1)▸SD
s dde p(L1)▸SD1
app ox(MAKELIST(((L2(X)-AM)/SD),X,1,N))▸L3 //
Σ(L3.^3)/N▸A3
sq (6/N)▸SA3
Σ(L3.^4)/N-3▸A4
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53
2*SA3▸SA4
Σ((L1 .- AM) .^ 3)*N/((N-1)*(N-2)*SD1^3)▸A31
A41=((N-1)*(N-2)*(N-3)*SD1^4)
EX1=Σ((L1 .- AM) .^ 4)*N*(N+1)
EX2=Σ((L1 .- AM) .^ 2)
EX2=3*EX2*EX2*(N-1)
A41=(EX1-EX2)/A41
NORMALD_CDF(A3/SA3)▸PA3
P2A3=2*PA3
IF PA3>0.5 THEN P2A3=2*(1-PA3) END;
NORMALD_CDF(A4/SA4)▸PA4
P2A4=2*PA4
IF PA4>0.5 THEN P2A4=2*(1-PA4) END;
//
app ox(N,[A3,A31],A3/SA3,[P2A3],[A4,A41],A4/SA4,[P2A4])
END;
#end
//
R
Di .pas
// Di ( 1,n1, 2,n2)/D.G.SCHRAUSSER/2025
//e.g. Di (0.78,12,0.34,8)[ZCo ]
#cas
Di (R1,N1,R2,N2):=
BEGIN
ZCo (R1,N1)(1)▸L2
L2(1)▸L1(1)
ZCo (R2,N2)(1)▸L2
L2(1)▸L1(2)
L1(1)-L1(2)▸L2(1)
sq ((1/(N1-3))+1/(N2-3))▸L2(2)
L2(1)/L2(2)▸L2(3)
NORMALD_CDF(L2(3))▸L2(4)
1-L2(4)▸L2(5)
2*L2(5)▸L2(6)
IF L2(5)>0.5 THEN
2*L2(4)▸L2(6)
END;
//Zd,sZd,z,p,1-p,p2
[L2(1),L2(2)],[L2(3)],L2(4),L2(5),[L2(6)]
END;
#end
//
RHO.pas
//RHO()/D.G.SCHRAUSSER/2025
//Spea man's ank co ela ion coe icien ho s/[pCo ]
#cas
RHO():=
BEGIN
//L1()(2) p o ided
size(L1)▸N
mean(L1)▸L3
MAKELIST((L1(I)(1)-L1(I)(2))^2,I,1,N)▸L2
Σ(L2)▸SUM
RHO=1-((6*SUM)/(N*(N^2-1)))
pCo (RHO,N)▸L4
//n, ho, ,p2 ho
app ox(N,[RHO,co ela ion(L1)],L4(3))
END;
#end
//

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54
RNK.pas
//RNK()/D.G.SCHRAUSSER/2025
//L1 o anking L3
#cas
RNK():=
BEGIN
//L1 p o ided
{}▸L2
{}▸L3
1▸R
0▸S1
0▸Q01
0▸V
SIZE(L1)▸N
SORT(L1)▸L2
FOR I FROM 2 TO N+1 DO
R▸L3(I-1)
IF L2(I-1)≠L2(I) THEN
IF S1=1 THEN
R=R+Q01
Q01=0;S1=0
1+V▸V
END;
R+1▸R
ELSE
Q01+1▸Q01;S1=1
END;
END;
Σ(L3)▸SR
//n, ies,Rsum,meanR
app ox(N,V,SR,[SR/N])
END;
#end
//
T
TAU.pas
//TAU()/D.G.SCHRAUSSER/2025
//Kendall's τ coe icien ( au-a)
#cas
TAU():=
BEGIN
//L1,L2 p o ided
SIZE(L1)▸N
FOR I FROM 1 TO N-1 DO
((SIGN(L1(I)-L1(I+1))))*((SIGN(L2(I)-L2(I+1))))▸L3(I)
IF L3(I)=1 THEN 0▸L3(I) END;
IF L3(I)=-1 THEN 1▸L3(I) END;
888▸M1(L1(I),L2(I))
END;
888▸M1(L1(N),L2(N))
//(N*(N-1)/2-2*ΣLIST(L3))/(N*(N-1)/2)▸ au_a
app ox(1-2*ΣLIST(L3)/(N*(N-1)/2))▸L4(1)
app ox((√(N*(N-1)))/(√(2*(2*N+5)))*3*L4(1))▸L4(2)
NORMALD_CDF(L4(2))▸L4(3)
1-L4(3)▸L4(4)
L4(4)*2▸L4(5)
IF L4(3)<0.5 THEN 2*L4(3)▸L4(5)
((COMB(N,2)-ΣLIST(L3))-ΣLIST(L3))/√((1/18)*N*(N-1)*(2*N+5))▸L4(6)
END;
// aua,z,p,1-p,p2
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[L4(1)],[L4(2)],L4(3),L4(4),[L4(5)]
END;
#end
//
TKV.pas
//TKV()/D.G.SCHRAUSSER/2025
//Va iance p/[C2V,pCo ]
#cas
TKV():=
BEGIN
//L1L2 p o ided
SIZE(L1)▸N
a iance(L1)▸s21
a iance(L2)▸s22
N-2▸d
C2V(2)
app ox(co ela ion(L6))▸
pCo ( ,N)(2)▸p
app ox(((s21-s22)*sq (N-2))/(2*sq (s21*s22*(1- ^2))))▸
STUDENT_CDF(d , )▸p
1-p▸p1
2*p▸p2
IF p2>1 THEN 2*(1-p)▸p2 END;
d ,[ ],[p ],[ ],p,p1,[p2]
END;
#end
//
TT_.pas
//TT_(y)/D.G.SCHRAUSSER/2025
//One-sample - es o es a iable y
//e.g TT_(5.3)
#cas
TT_(Y):=
BEGIN
//L1 p o ided
size(L1)▸N
N-1▸d
mean(L1)▸x
a iance(L1)▸s2
app ox((x-Y)/(sq (s2/(N-1))))▸
p=STUDENT_CDF(d , )
1-p▸p1
2*p▸p2
IF p2>1 THEN 2*(1-p)▸p2 END;
d ,[ ],p,p1,[p2]
END;
#end
//
TU_.pas
//TU_()/D.G.SCHRAUSSER/2025
// - es o unpai ed samples
#cas
TU_():=
BEGIN
//L1L2 p o ided
0▸Sx1
0▸Sx2
SIZE(L1)▸n1
SIZE(L2)▸n2
n1+n2-2▸d
mean(L1)▸x1
mean(L2)▸x2
FOR I FROM 1 TO n1 DO
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
56
Sx1=Sx1+(L1(I)-x1)^2 END;
FOR I FROM 1 TO n2 DO
Sx2=Sx2+(L2(I)-x2)^2 END;
=x1-x2
= /(√((Sx1+Sx2)/((n1-1)+(n2-1)))*√(1/n1+1/n2))
=app ox( )
p=STUDENT_CDF(d , )
p1=p
IF p>0.5 THEN p1=1-p END;
p2=2*p1
d ,[ ],p,p1,[p2]
END;
#end
//
TV_.pas
//TV_()/D.G.SCHRAUSSER/2025
// - es o pai ed samples
#cas
TV_():=
BEGIN
//L1()(2) p o ided
size(L1)▸n
n-1▸d
FOR I FROM 1 TO n DO
L1(I)(1)-L1(I)(2)▸L2(I)
L2(I)^2▸L3(I)
END;
ΣLIST(L2)▸Sxd
ΣLIST(L3)▸Sxd2
=Sxd/n
= ( /((√((Sxd2-Sxd^2/n)/(n-1)))/(√n)))
=app ox( )
p=STUDENT_CDF(d , )
p▸p1
IF p>0.5 THEN 1-p▸p1 END;
2*p1▸p2
//
d ,[ ],p,p1,[p2]
END;
#end
//
VTLG.pas
// VTLG( ,d )/D.G.SCHRAUSSER/2025
//e.g. VTLG(2.65,8)[Ad ancedG aphing]
#cas
VTLG(T857,D187):=
BEGIN
G25478=Gamma((D187+1)/2)/Gamma(D187/2)
P=∫(G25478*(D187*π)^(-1/2)*(1+(X^2/D187))^(-(D187+1)/2),X,−∞,T857)
T857▸C
D187▸D
G25478▸G
"Y=(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2))"▸V1
"Y<(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2)) AND Y>0 AND X<C"▸V2
"Y=(1/√(2*π))*e^((-1/2)*(X)^2)"▸V3
STARTAPP("E wei e e_G a iken");
STARTVIEW(1);
IF P>0.5 THEN P=1-P END;
P,[2*P]
END;
#end
//
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
57
V
VAR1.pas
//VAR1(n a )/D.G.SCHRAUSSER/2025
//e.g.VAR1(5)/ a ia ion ec o ( )n om L1
#cas
VAR1(N):=
BEGIN
//L1(N) p o ided
{}▸L2
FOR A FROM 1 TO N DO
L1(RANDINT(1,N))▸L2(A) END;
//
L2
END;
#end
//
VFC0.pas
//VFC0(cell coun a,b,c,d)/D.G.SCHRAUSSER/2025
//2×2 chi-squa ed es o independence
//Obse ed equencies abcd b
//Chi-squa ed, McNema wi h 2- ailed p
//e.g.VFC0(17,12,14,24)
#cas
VFC0(a,b,c,d):=
BEGIN
a+b+c+d▸N
a+b▸z1;c+d▸z2
a+c▸s1;b+d▸s2
z1/z2▸Z01;s1/s2▸S01
VFX=(N*(a*d-b*c)^2)/((a+b)*(c+d)*(a+c)*(b+d))
VFC=1-CHISQUARE_CDF(1,VFX)
MNX=(b-c)^2/(b+c)
//McNema Ya es co .
IF b+c<30 AND b+c>20 THEN
MNX=(ABS(b-c)-0.5)^2/(b+c)
END
pMNX=1-CHISQUARE_CDF(1,MNX)
//n,mnchi2,mnp2,chi2,p2
N,[MNX],[pMNX],[VFX],[VFC]
END;
#end
//
VFCH.pas
//VFCH(cell coun a,b,c,d)/D.G.SCHRAUSSER/2025
//2×2 chi-squa ed es o independence
//L0: Obse ed equencies abcd b
//L1: Expec ed equencies e
//L2,L3,L4: P obabili ies p(A^B), p(B|A), p(A|B)
//L5: Chi-squa ed,(w. Ya es co .)
//L6: 2- ailed sig. p2
//Chi-squa e McNema (w. Ya es co .) wi h p2
//e.g.VFCH(17,12,14,24)
//
#cas
VFCH(a,b,c,d):=
BEGIN
a▸L0(1)
b▸L0(2)
c▸L0(3)
d▸L0(4)
a+b+c+d▸N
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64
pTUX(n1,n2,x1,x2,s21,s22)
STUDENT_CDF(D+F-2,((A-B)/(√((C*D+E*F)/(D-1+F-
1))*(√(1/D)+√(1/F)))))
pTV_(L1)
s uden _cd (size(L1)-1,ΣLIST(MAKELIST(L1(A)-
(L2(A)),A,1,size(L1)))/(size(L1))/(√((ΣLIST(MAKELIST((L1(A)-
(L2(A)))^2,A,1,size(L1)))-ΣLIST(MAKELIST(L1(A)-
(L2(A)),A,1,size(L1)))^2/(size(L1)))/(size(L1)-
1))*1/(√(size(L1)-1))))
pz4F(a,b,c,d)[e.g.pz4F(11,20,80,58)]
NORMALD_CDF(((D-
(A+B+C+D)*((D+B)*(C+D)/(A+B+C+D)^2))/(√((A+B+C+D)*(1-
((D+B)*(C+D)/(A+B+C+D)^2))-(A+B+C+D)*(A+B+C+D-
1)*((D+B)*(C+D)/(A+B+C+D)^2)*(((D+B)*(C+D)/(A+B+C+D)^2)-((D+B-
1)*(C+D-1)/(A+B+C+D-1)^2))))))
pzBN(a,b)
NORMALD_CDF(((A-(A+B)/2)/(√((A+B)/4))))
R
R2D( ad)
X/π*180.000
bis(L1,L2)
((mean(L1)-
mean(L2))/s dde (CONCAT(L1,L2)))*SIZE(L1)*SIZE(L2)/((1/(√(2*π)
))*e^(-
(NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2)*SIZE(CONCA
T(L1,L2))^2)
bisR(L1,L2)
(2/(SIZE(L1)+SIZE(L2)))*(mean(L1)-mean(L2))
RED( )
A^2*100
RND1(n)
MAKELIST(RANDNORM,A,1,B)
RND2(n)
MAKELIST(RANDOM,A,1,B)

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65
pbis(L1,L2)
(mean(L1)-
mean(L2))/s dde (CONCAT(L1,L2))*√(SIZE(L1)*SIZE(L2)/(SIZE(CONC
AT(L1,L2)))^2)
xy(x1,x2)
app ox(co ela ion(L1,L2))
e (b,c,a,d), ad
COS((π/(1+√(B*C/(A*D)))))
xy_z( xy, xz, yz)
(A-B*C)/(√(1-B^2)*√(1-C^2))
y_xz( xy, xz, yz)
(A-B*C)/(sq (1-B^2))
Z(Z)
(e^(2*A)-1)/(e^(2*A)+1)
S
SCR(n,k,R)
1.00-((A-3.00)/(A-B-2.00))*((1.00-C^2.00)+((2.00/(A-
B)))*(1.00-C^2.00)^2.00)
SMG [SMG(A,B),SMG(sd,n)]
√((A^2)*(B/(B-1))/B)
SQR(x)
A^2
s bis(L1,L2)
√(SIZE(L1)*SIZE(L2))/((√SIZE(CONCAT(L1,L2))*SIZE(CONCAT(L1,L2)
)*1/(√(2*π)))*e^(-
(NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2))
s e (b,c,a,d)
√(((A+B)/(A+B+C+D))*((A+C)/(A+B+C+D))*((C+D)/(A+B+C+D))*((B+D)
/(A+B+C+D))/(A+B+C+D))*(1/(((1/(√(2*π)))*e^(-
(NORMALD_ICDF(((C+D)/(A+B+C+D)))^2)/2))*((1/(√(2*π)))*e^(-
(NORMALD_ICDF(((B+D)/(A+B+C+D)))^2)/2))))
sumd2(L1)
ΣLIST(MAKELIST((L1(A)-(L2(A)))^2,A,1,SIZE(L1)))
CAS inpu
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
66
L4:=ΣLIST(L3:=MAKELIST((L1(x)-(L2(x)))^2,x,1,size(L1)))
L4:=ΣLIST(L3:=MAKELIST(L1(x)-(L2(x)),x,1,size(L1)))
L4:=ΣLIST(L3:=MAKELIST(L1(x)-(L2(x)),x,1,size(L1)))^2
sumx2(L1,L2)
CAS inpu
ΣLIST(L3:=app ox(MAKELIST((L1(x)-mean(L1))^2,x,1,size(L1))))
T
TE (E ,A )
2^E/A^2
( ,n)
(R*√(N-2))/(√(1-R^2))
TRW(L1,L2)
(co ela ion(L1,L2)*√(SIZE(L1)-2))/(√(1-co ela ion(L1,L2)^2))
TKV(L1,L2)
(( a iance(L1)- a iance(L2))*sq (size(L1)-
2))/(2*sq ( a iance(L1)* a iance(L2)*(1-co ela ion(L1,L2))))
TT_(L1,y)
(mean(L1)-A)/(√(s dde (L1)^2/(SIZE(L1)-1)))
TU_(L1,L2)
(mean(L1)-mean(L2))/(sq ((ΣLIST(MAKELIST((L1(x)-
mean(L1))^2,x,1,size(L1)))+ ΣLIST(MAKELIST((L2(x)-
mean(L2))^2,x,1,size(L2))))/ (size(L1)-1+size(L2)-
1))*(sq (1/size(L1))+sq (1/size(L2))))
TUX(x1,x2,s21,n1,s22,n2)
(A-B)/(√((C*D+E*F)/(D-1+F-1))*(√(1/D)+√(1/F)))
TV_(L1)
ΣLIST(MAKELIST(L1(A)-
(L2(A)),A,1,size(L1)))/(size(L1))/(√((ΣLIST(MAKELIST((L1(A)-
(L2(A)))^2,A,1,size(L1)))-ΣLIST(MAKELIST(L1(A)-
(L2(A)),A,1,size(L1)))^2/(size(L1)))/(size(L1)-
1))*1/(√(size(L1)-1)))
U
U_1(L1,L2)
SIZE(L1)*SIZE(L2)+(((SIZE(L1))^2+SIZE(L1))/2)-ΣLIST(L1)
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
67
U_2(L1,L2)
SIZE(L1)*SIZE(L2)+(((SIZE(L2))^2+SIZE(L2))/2)-ΣLIST(L2)
X
x2F(a,b)
((A-((A+B)/2))^2/((A+B)/2))+(B-((A+B)/2))^2/((A+B)/2)
x4F(a,b,c,d)
(A+B+C+D)*(A*D-B*C)^2/((A+B)*(C+D)*(A+C)*(B+D))
x4FY(a,b,c,d) Ya es co 4< e<7
(A+B+C+D)*(ABS(A*D-B*C)-
((A+B+C+D)/2))^2/((A+B)*(C+D)*(A+C)*(B+D))
xMN(b,c)
(A-B)^2/(A+B)
xMNY(b,c) Ya es co 20<b+c<30
(ABS(A-B)-0.5)^2/(A+B)
xPHI(a,d,b,c)
(((A*D-B*C)/(√((A+C)*(B+D)*(A+B)*(C+D)))))^2*(A+B+C+D)
Z
z4F(a,b,c,d)
(D-(A+B+C+D)*((D+B)*(C+D)/(A+B+C+D)^2))/(√((A+B+C+D)*(1-
((D+B)*(C+D)/(A+B+C+D)^2))-(A+B+C+D)*(A+B+C+D-
1)*((D+B)*(C+D)/(A+B+C+D)^2)*(((D+B)*(C+D)/(A+B+C+D)^2)-((D+B-
1)*(C+D-1)/(A+B+C+D-1)^2))))
zBN(a,b)
(A-(A+B)/2)/(√((A+B)/4))
Z ( )
0.5*LN((1+A)/(1-A))
z bis(L1,L2)
(((mean(L1)-
mean(L2))/s dde (CONCAT(L1,L2)))*SIZE(L1)*SIZE(L2)/((1/(√(2*π)
))*e^(-
(NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2)*SIZE(CONCA
T(L1,L2))^2))/(√(SIZE(L1)*SIZE(L2))/((√SIZE(CONCAT(L1,L2))*SIZ
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68
E(CONCAT(L1,L2))*1/(√(2*π)))*e^(-
(NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2)))
z bisR(L1,L2)
(size(L1)*size(L2)+(((size(L1))^2+size(L1))/2)-ΣLIST(L1)-
size(L1)*size(L2)/2)/(sq (size(L1)*size(L2)*(size(L1)+size(L2
)+1)/12))
z ( 1, 2,n1,n2)
(0.5*LN((1+A)/(1-A))-0.5*LN((1+B)/(1-B)))/(√(1/(C-3)+1/(D-3)))
z xy_z( xy_z,n)
0.5*LN(((1+A)/(1-A)))*√(B-2)
ZWERT(x1,x,s)
(A-B)/C
zVAL(L1),e.g.L2:=zVAL(L1)
app ox(MAKELIST(((L1(X)-mean(L1))/s dde (L1)),X,1,SIZE(L1)))
zVALp(L1),e.g.L3:=zVAL(L1)
app ox(MAKELIST(((L1(X)-mean(L1))/s dde p(L1)),X,1,SIZE(L1)))
Applica ion unc ions
Func ion
To selec :
F01.pas
//F01()/D.G.SCHRAUSSER/2022
//Func ion: Equa ions 1.0
EXPORT F01()
BEGIN
"√(1-((X-W)/A)^2)*A+V"▸F1;
"−√(1-((X-W)/A)^2)*A+V"▸F2;
"√(1-((X-T)/B)^2)*B+U"▸F3;
"-√(1-((X-T)/B)^2)*B+U"▸F4;
"√(1-((X-R)/C)^2)*C+S"▸F5;
"-√(1-((X-R)/C)^2)*C+S"▸F6;
200▸A;
150▸B;
344▸C;
1▸W;
450▸T;
1000▸R;
"Func ion: Equa ions 1.0"
END;
//
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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F02.pas
//F02()/D.G.SCHRAUSSER/2022
//Func ion: Equa ions 2.0
EXPORT F02()
BEGIN
"NORMALD_CDF(0,1,X)"▸F1;
"NORMALD(0,1,X)"▸F2;
"STUDENT(50,X)"▸F3;
"STUDENT_CDF(50,X)"▸F4;
"CHISQUARE(1,X)"▸F5;
"CHISQUARE_CDF(1,X)"▸F6;
"FISHER_CDF(25,3,X)"▸F7;
"FISHER(25,3,X)"▸F8;
"0"▸F9;
"0"▸F0;
"Func ion: Equa ions 2.0"
END;
//
F03.pas
//F03()/D.G.SCHRAUSSER/2025
//Func ion: Equa ions 3.0
//F3-7:De i a i es o he s anda d no mal dis ibu ion unc ion, '(z)-
'''''(z)
EXPORT F03()
BEGIN
"NORMALD_CDF(0,1,X)"▸F1;
"NORMALD(0,1,X)"▸F2;
"∂((1/√(2*π))*e^((-1/2)*X^2),X=X)"▸F3;
"∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X)"▸F4;
"∂(∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X),X)"▸F5;
"∂(∂(∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X),X),X)"▸F6;
"∂(∂(∂(∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X),X),X),X)"▸F7;
"0"▸F8;
"0"▸F9;
"0"▸F0;
"Func ion: Equa ions 3.0"
END;
//
F04.pas
//F04()/D.G.SCHRAUSSER/2025
//Func ion: Equa ions 4.0
//F3:De i a i e o Gamma, '(x)
//F5-7:De i a i es o he exponen ial unc ion, (x)={ '(x)- '''(x)...}
EXPORT F04()
BEGIN
"CAS.Gamma(X)"▸F1;
"(X)!"▸F2;
"∂(Gamma(X),X=X)"▸F3;
"EXP(X)"▸F4;
"∂(e^X,X = X)"▸F5;
"∂(∂(e^X,X=X),X=X)"▸F6;
"∂(∂(∂(e^X,X=X),X=X),X=X)"▸F7;
"0"▸F8;
"0"▸F9;
"0"▸F0;
"Func ion: Equa ions 4.0"
END;
//

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F05.pas
//F05()/D.G.SCHRAUSSER/2025
//Func ion: Equa ions 5.0
//F2-5,F7-0:De i a i es o he ci cula unc ion, '( )- ''''( )
EXPORT F05()
BEGIN
"√(1-X^2)"▸F1;
"∂(√(1-X^2),X)"▸F2;
"∂(∂(√(1-X^2),X),X)"▸F3;
"∂(∂(∂(√(1-X^2),X),X),X)"▸F4;
"∂(∂(∂(∂(√(1-X^2),X),X),X),X)"▸F5;
"-√(1-X^2)"▸F6;
"∂(-√(1-X^2),X)"▸F7;
"∂(∂(-√(1-X^2),X),X)"▸F8;
"∂(∂(∂(-√(1-X^2),X),X),X)"▸F9;
"∂(∂(∂(∂(-√(1-X^2),X),X),X),X)"▸F0;
"Func ion: Equa ions 5.0"
END;
//
F06.pas
//F06()/D.G.SCHRAUSSER/2025
//Func ion: Equa ions 6.0
//F3-6:De i a i es o S uden 's- , '( )- ''''( ) EXPORT F06()
BEGIN
"STUDENT_CDF(D,X)"▸F1;
"(Gamma(((D+1)/2))/Gamma((D/2)))*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2)"▸F2;
"∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X)"▸F3;
"∂(∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X),X)"▸F4;
"∂(∂(∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X),X),X)"▸F5;
"∂(∂(∂(∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X),X),X),X)"▸F6;
""▸F7;
""▸F8;
""▸F9;
""▸F0;
"Func ion: Equa ions 6.0"
END;
//
F06_.pas
//F06_(d )/D.G.SCHRAUSSER/2025
//De i a i es o S uden ’s-
//e.g.F06_(5)[F06]
#cas
F06_(DF):=
BEGIN F06
DF▸D
G=Gamma(((DF+1)/2))/Gamma((DF/2))
D,G
END;
#end
//
F07.pas
//F07()/D.G.SCHRAUSSER/2025
//Func ion: Equa ions 7.0
//F3-6:De i a i es o chi², '( )- ''''( )
EXPORT F07()
BEGIN
"CHISQUARE_CDF(D,X)"▸F1;
"(1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2)"▸F2;
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"∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X)"▸F3;
"∂(∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X),X)"▸F4;
"∂(∂(∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X),X),X)"▸F5;
"∂(∂(∂(∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X),X),X),X)"▸F6;
""▸F7;
""▸F8;
""▸F9;
""▸F0;
"Func ion: Equa ions 7.0"
END;
//
F07_.pas
//F07_(d )/D.G.SCHRAUSSER/2025
//De i a i es o chi²
//e.g.F07_(1)[F07]
#cas
F07_(DF):=
BEGIN F07
DF▸D G=Gamma(DF/2)
D,G
END;
#end //
G aph 3D
To selec :
F01Z.pas
//F01Z()/D.G.SCHRAUSSER/2025
//G aph 3D: Equa ions 1.0
//FZ1-2:Gamma
//FZ4-9:Sphe ical unc ions
//FZ0:Sine
EXPORT F01Z()
BEGIN
"CAS.Gamma(Y)"▸FZ1;
"X^(Y-1)*e^(-X)"▸FZ2;
"(1/(2*π*√(1-R^2)))*e^((-1/(2*(1-R^2)))*(X^2-2*R*X*Y+Y^2))"▸FZ3;
"√((1-X^2)+(1-Y^2))"▸FZ4;
"−1*√((1-X^2)+(1-Y^2))"▸FZ5;
"√(1-X^2-Y^2)"▸FZ6;
"−1*√(1-X^2-Y^2)"▸FZ7;
"√((X-X^2)+(Y-Y^2))"▸FZ8;
"−1*√((X-X^2)+(Y-Y^2))"▸FZ9;
"SIN(X)*SIN(Y)*1.5"▸FZ0;
"G aph 3D: Equa ions 1.0"
END;
//
F02Z.pas
//F02Z()/D.G.SCHRAUSSER/2025
//G aph 3D: Equa ions 2.0
//Complex plane (z)=z,
//wi h z=|x+i|
//whe e (x,y=i)=√x²+y²
EXPORT F02Z()
BEGIN
"√(X^2+Y^2)"▸FZ1;
"(1/π)*e^(-(ABS(√(X^2+Y^2))^2))"▸FZ2;
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"X^((√(X^2+Y^2))-1)*e^(-X)"▸FZ3;
""▸FZ4;
""▸FZ5;
""▸FZ6;
""▸FZ7;
""▸FZ8;
""▸FZ9;
""▸FZ0;
"G aph 3D: Equa ions 2.0"
END;
//
F03Z.pas
//F03Z()/D.G.SCHRAUSSER/2025
//G aph 3D: Equa ions 3.0
//FZ1:S uden 's- su ace, ( ,d ),x[-4,4],y[1,10],z[0,0.5]
//FZ2:chi² su ace, (chi²,d ), x[0,5],y[1,10],z[0,0.5]
//FZ3-6:F space, (F,d 2);d 1={1,5,9,13},x[0,5],y[0,20],z[0,5]
EXPORT F03Z()
BEGIN
"(Gamma(((Y+1)/2))/Gamma((Y/2)))*(Y*π)^(-1/2)*(1+(X^2/Y))^(-(Y+1)/2)"▸FZ1;
"(1/(2^(Y/2)*Gamma((Y/2))))*X^((Y/2)-1)*e^(-X/2)"▸FZ2;
"(Gamma(((1+Y)/2))/(Gamma((1/2))*Gamma((1/2))))*(1/Y)^(1/2)*X^((1/2)-
1)*(1+(1/Y)*X)^(-(1+Y)/2)"▸FZ3;
"(Gamma(((5+Y)/2))/(Gamma((5/2))*Gamma((5/2))))*(5/Y)^(5/2)*X^((5/2)-
1)*(1+(5/Y)*X)^(-(5+Y)/2)"▸FZ4;
"(Gamma(((9+Y)/2))/(Gamma((9/2))*Gamma((9/2))))*(9/Y)^(9/2)*X^((9/2)-
1)*(1+(9/Y)*X)^(-(9+Y)/2)"▸FZ5;
"(Gamma(((13+Y)/2))/(Gamma((13/2))*Gamma((13/2))))*(13/Y)^(13/2)*X^((13/2)-
1)*(1+(13/Y)*X)^(-(13+Y)/2)"▸FZ6;
""▸FZ7;
""▸FZ8;
""▸FZ9;
""▸FZ0;
"G aph 3D: Equa ions 3.0"
END;
//
Sol e
To selec :
E01.pas
//E01()/D.G.SCHRAUSSER/2025
//Sol e: Equa ions 1.0
//E1: Addi i e p obabili y (P=p,X=pb,N=n; special addi ion heo em)
//E2: Nega i e binomial p obabili y (P=p,N=pnb,R=n,K=1,2..,I=0.0)
//E3: Binomial p obabili y (b,c)
//E4: S anda d no mal dis ibu ion z(0,1)
//E5: E ec size epsilon
//E6: Linea eg ession y'
//E7: S anda d e o o p edic ion y'+-C wi h zc i
//E8: p o co ela ion wi h n (2- ailed sig p2=2*(1-p);p>0.5)
EXPORT E01()
BEGIN
"X=1-(1-P)^N"▸E1;
"P=Σ(((A+B)!/(I!*(A+B-I)!))*2^(-I)*2^(-(A+B-I)),I,0,A)"▸E3;
"N=Σ(((K+I-1)!/(I!*(K-1)!))*P^K*(1-P)^I,I,0,R-K)"▸E2;
"P=NORMALD_CDF(0,1,Z)"▸E4;
"E=(A-X)/S"▸E5;
"Y=A*X+B"▸E6;
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
73
"C=(Z*√(1-R^2))*S"▸E7;
"P=STUDENT_CDF(N-2,(R*√(N-2))/√(1-R^2))"▸E8;
""▸E9;
""▸E0;
"Sol e: Equa ions 1.0"
END;
//
E02.pas
//E02()/D.G.SCHRAUSSER/2022
//Sol e: Equa ions 2.0
//E1: Ape u e alue A o exposu e alue E and shu e speed T
//E2: Exposu e alue E o illuminance lux L and ISO I
//E3: Magni ica ion M a ocal leng h F
//E4: Angle o iew W a ocal leng h F
EXPORT E02()
BEGIN
"A=1/√(2^(-E)*T)"▸E1;
"E=LN((L*I/250))*(LN(2))^(-1)"▸E2;
"M=F/50"▸E3;
"W=-0.95908335982/(1-1.00293098572*e^(0.000351450126836*F))"▸E4;
""▸E5;
""▸E6;
""▸E7;
""▸E8;
""▸E9;
""▸E0;
"Sol e: Equa ions 2.0"
END;
//
E03.pas
//E03()/D.G.SCHRAUSSER/2025
//Sol e: Equa ions 3.0
//E1: Resolu ion R om R0, 0, 1
//E2: Exposu e Value E om T , A
//E3: Ape u e B om A 0, ISO0, ISO1
EXPORT E03()
BEGIN
"(A*B^2/F^2)=R"▸E1;
"(LN(2))^(-1)*LN(T*A^2)=E"▸E2;
"A*e^(0.5*LN(S^(-1)*I))=B"▸E3;
""▸E4;
""▸E5;
""▸E6;
""▸E7;
""▸E8;
""▸E9;
""▸E0;
"Sol e: Equa ions 3.0"
END;
//
E04.pas
//E04()/D.G.SCHRAUSSER/2025
//Sol e: Equa ions 4.0
//E1: As onomical uni A om me e s M
//E2: Pa sec P om as onomical uni A
//E3: Pa sec P om pa allax X in millia cseconds mas
//E4: Ligh -yea L om pa sec P
//E5: Speed o ligh C om m/c M
//E6: Luminosi y dis ance P om dis ance modulus M
//E7: Radius R a a gi en dis ance D wi h angula diame e V°
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
80
.,5.,6.,9.,10.],[2.,3.,4.,7.,8.,1.,5.,6.,9.,10.],[1.,2.,5.,7.,8.,3.,4.,6.,9.,10.],[1.
,3.,5.,7.,8.,2.,4.,6.,9.,10.],[2.,3.,5.,7.,8.,1.,4.,6.,9.,10.],[1.,4.,5.,7.,8.,2.,3.,
6.,9.,10.],[2.,4.,5.,7.,8.,1.,3.,6.,9.,10.],[3.,4.,5.,7.,8.,1.,2.,6.,9.,10.],[1.,2.,6
.,7.,8.,3.,4.,5.,9.,10.],[1.,3.,6.,7.,8.,2.,4.,5.,9.,10.],[2.,3.,6.,7.,8.,1.,4.,5.,9.
,10.],[1.,4.,6.,7.,8.,2.,3.,5.,9.,10.],[2.,4.,6.,7.,8.,1.,3.,5.,9.,10.],[3.,4.,6.,7.,
8.,1.,2.,5.,9.,10.],[1.,5.,6.,7.,8.,2.,3.,4.,9.,10.],[2.,5.,6.,7.,8.,1.,3.,4.,9.,10.]
,[3.,5.,6.,7.,8.,1.,2.,4.,9.,10.],[4.,5.,6.,7.,8.,1.,2.,3.,9.,10.],[1.,2.,3.,4.,9.,5.
,6.,7.,8.,10.],[1.,2.,3.,5.,9.,4.,6.,7.,8.,10.],[1.,2.,4.,5.,9.,3.,6.,7.,8.,10.],[1.,
3.,4.,5.,9.,2.,6.,7.,8.,10.],[2.,3.,4.,5.,9.,1.,6.,7.,8.,10.],[1.,2.,3.,6.,9.,4.,5.,7
.,8.,10.],[1.,2.,4.,6.,9.,3.,5.,7.,8.,10.],[1.,3.,4.,6.,9.,2.,5.,7.,8.,10.],[2.,3.,4.
,6.,9.,1.,5.,7.,8.,10.],[1.,2.,5.,6.,9.,3.,4.,7.,8.,10.],[1.,3.,5.,6.,9.,2.,4.,7.,8.,
10.],[2.,3.,5.,6.,9.,1.,4.,7.,8.,10.],[1.,4.,5.,6.,9.,2.,3.,7.,8.,10.],[2.,4.,5.,6.,9
.,1.,3.,7.,8.,10.],[3.,4.,5.,6.,9.,1.,2.,7.,8.,10.],[1.,2.,3.,7.,9.,4.,5.,6.,8.,10.],
[1.,2.,4.,7.,9.,3.,5.,6.,8.,10.],[1.,3.,4.,7.,9.,2.,5.,6.,8.,10.],[2.,3.,4.,7.,9.,1.,
5.,6.,8.,10.],[1.,2.,5.,7.,9.,3.,4.,6.,8.,10.],[1.,3.,5.,7.,9.,2.,4.,6.,8.,10.],[2.,3
.,5.,7.,9.,1.,4.,6.,8.,10.],[1.,4.,5.,7.,9.,2.,3.,6.,8.,10.],[2.,4.,5.,7.,9.,1.,3.,6.
,8.,10.],[3.,4.,5.,7.,9.,1.,2.,6.,8.,10.],[1.,2.,6.,7.,9.,3.,4.,5.,8.,10.],[1.,3.,6.,
7.,9.,2.,4.,5.,8.,10.],[2.,3.,6.,7.,9.,1.,4.,5.,8.,10.],[1.,4.,6.,7.,9.,2.,3.,5.,8.,1
0.],[2.,4.,6.,7.,9.,1.,3.,5.,8.,10.],[3.,4.,6.,7.,9.,1.,2.,5.,8.,10.],[1.,5.,6.,7.,9.
,2.,3.,4.,8.,10.],[2.,5.,6.,7.,9.,1.,3.,4.,8.,10.],[3.,5.,6.,7.,9.,1.,2.,4.,8.,10.],[
4.,5.,6.,7.,9.,1.,2.,3.,8.,10.],[1.,2.,3.,8.,9.,4.,5.,6.,7.,10.],[1.,2.,4.,8.,9.,3.,5
.,6.,7.,10.],[1.,3.,4.,8.,9.,2.,5.,6.,7.,10.],[2.,3.,4.,8.,9.,1.,5.,6.,7.,10.],[1.,2.
,5.,8.,9.,3.,4.,6.,7.,10.],[1.,3.,5.,8.,9.,2.,4.,6.,7.,10.],[2.,3.,5.,8.,9.,1.,4.,6.,
7.,10.],[1.,4.,5.,8.,9.,2.,3.,6.,7.,10.],[2.,4.,5.,8.,9.,1.,3.,6.,7.,10.],[3.,4.,5.,8
.,9.,1.,2.,6.,7.,10.],[1.,2.,6.,8.,9.,3.,4.,5.,7.,10.],[1.,3.,6.,8.,9.,2.,4.,5.,7.,10
.],[2.,3.,6.,8.,9.,1.,4.,5.,7.,10.],[1.,4.,6.,8.,9.,2.,3.,5.,7.,10.],[2.,4.,6.,8.,9.,
1.,3.,5.,7.,10.],[3.,4.,6.,8.,9.,1.,2.,5.,7.,10.],[1.,5.,6.,8.,9.,2.,3.,4.,7.,10.],[2
.,5.,6.,8.,9.,1.,3.,4.,7.,10.],[3.,5.,6.,8.,9.,1.,2.,4.,7.,10.],[4.,5.,6.,8.,9.,1.,2.
,3.,7.,10.],[1.,2.,7.,8.,9.,3.,4.,5.,6.,10.],[1.,3.,7.,8.,9.,2.,4.,5.,6.,10.],[2.,3.,
7.,8.,9.,1.,4.,5.,6.,10.],[1.,4.,7.,8.,9.,2.,3.,5.,6.,10.],[2.,4.,7.,8.,9.,1.,3.,5.,6
.,10.],[3.,4.,7.,8.,9.,1.,2.,5.,6.,10.],[1.,5.,7.,8.,9.,2.,3.,4.,6.,10.],[2.,5.,7.,8.
,9.,1.,3.,4.,6.,10.],[3.,5.,7.,8.,9.,1.,2.,4.,6.,10.],[4.,5.,7.,8.,9.,1.,2.,3.,6.,10.
],[1.,6.,7.,8.,9.,2.,3.,4.,5.,10.],[2.,6.,7.,8.,9.,1.,3.,4.,5.,10.],[3.,6.,7.,8.,9.,1
.,2.,4.,5.,10.],[4.,6.,7.,8.,9.,1.,2.,3.,5.,10.],[5.,6.,7.,8.,9.,1.,2.,3.,4.,10.],[1.
,2.,3.,4.,10.,5.,6.,7.,8.,9.],[1.,2.,3.,5.,10.,4.,6.,7.,8.,9.],[1.,2.,4.,5.,10.,3.,6.
,7.,8.,9.],[1.,3.,4.,5.,10.,2.,6.,7.,8.,9.],[2.,3.,4.,5.,10.,1.,6.,7.,8.,9.],[1.,2.,3
.,6.,10.,4.,5.,7.,8.,9.],[1.,2.,4.,6.,10.,3.,5.,7.,8.,9.],[1.,3.,4.,6.,10.,2.,5.,7.,8
.,9.],[2.,3.,4.,6.,10.,1.,5.,7.,8.,9.],[1.,2.,5.,6.,10.,3.,4.,7.,8.,9.],[1.,3.,5.,6.,
10.,2.,4.,7.,8.,9.],[2.,3.,5.,6.,10.,1.,4.,7.,8.,9.],[1.,4.,5.,6.,10.,2.,3.,7.,8.,9.]
,[2.,4.,5.,6.,10.,1.,3.,7.,8.,9.],[3.,4.,5.,6.,10.,1.,2.,7.,8.,9.],[1.,2.,3.,7.,10.,4
.,5.,6.,8.,9.],[1.,2.,4.,7.,10.,3.,5.,6.,8.,9.],[1.,3.,4.,7.,10.,2.,5.,6.,8.,9.],[2.,
3.,4.,7.,10.,1.,5.,6.,8.,9.],[1.,2.,5.,7.,10.,3.,4.,6.,8.,9.],[1.,3.,5.,7.,10.,2.,4.,
6.,8.,9.],[2.,3.,5.,7.,10.,1.,4.,6.,8.,9.],[1.,4.,5.,7.,10.,2.,3.,6.,8.,9.],[2.,4.,5.
,7.,10.,1.,3.,6.,8.,9.],[3.,4.,5.,7.,10.,1.,2.,6.,8.,9.],[1.,2.,6.,7.,10.,3.,4.,5.,8.
,9.],[1.,3.,6.,7.,10.,2.,4.,5.,8.,9.],[2.,3.,6.,7.,10.,1.,4.,5.,8.,9.],[1.,4.,6.,7.,1
0.,2.,3.,5.,8.,9.],[2.,4.,6.,7.,10.,1.,3.,5.,8.,9.],[3.,4.,6.,7.,10.,1.,2.,5.,8.,9.],
[1.,5.,6.,7.,10.,2.,3.,4.,8.,9.],[2.,5.,6.,7.,10.,1.,3.,4.,8.,9.],[3.,5.,6.,7.,10.,1.
,2.,4.,8.,9.],[4.,5.,6.,7.,10.,1.,2.,3.,8.,9.],[1.,2.,3.,8.,10.,4.,5.,6.,7.,9.],[1.,2
.,4.,8.,10.,3.,5.,6.,7.,9.],[1.,3.,4.,8.,10.,2.,5.,6.,7.,9.],[2.,3.,4.,8.,10.,1.,5.,6
.,7.,9.],[1.,2.,5.,8.,10.,3.,4.,6.,7.,9.],[1.,3.,5.,8.,10.,2.,4.,6.,7.,9.],[2.,3.,5.,
8.,10.,1.,4.,6.,7.,9.],[1.,4.,5.,8.,10.,2.,3.,6.,7.,9.],[2.,4.,5.,8.,10.,1.,3.,6.,7.,
9.],[3.,4.,5.,8.,10.,1.,2.,6.,7.,9.],[1.,2.,6.,8.,10.,3.,4.,5.,7.,9.],[1.,3.,6.,8.,10
.,2.,4.,5.,7.,9.],[2.,3.,6.,8.,10.,1.,4.,5.,7.,9.],[1.,4.,6.,8.,10.,2.,3.,5.,7.,9.],[
2.,4.,6.,8.,10.,1.,3.,5.,7.,9.],[3.,4.,6.,8.,10.,1.,2.,5.,7.,9.],[1.,5.,6.,8.,10.,2.,
3.,4.,7.,9.],[2.,5.,6.,8.,10.,1.,3.,4.,7.,9.],[3.,5.,6.,8.,10.,1.,2.,4.,7.,9.],[4.,5.
,6.,8.,10.,1.,2.,3.,7.,9.],[1.,2.,7.,8.,10.,3.,4.,5.,6.,9.],[1.,3.,7.,8.,10.,2.,4.,5.
,6.,9.],[2.,3.,7.,8.,10.,1.,4.,5.,6.,9.],[1.,4.,7.,8.,10.,2.,3.,5.,6.,9.],[2.,4.,7.,8
.,10.,1.,3.,5.,6.,9.],[3.,4.,7.,8.,10.,1.,2.,5.,6.,9.],[1.,5.,7.,8.,10.,2.,3.,4.,6.,9
.],[2.,5.,7.,8.,10.,1.,3.,4.,6.,9.],[3.,5.,7.,8.,10.,1.,2.,4.,6.,9.],[4.,5.,7.,8.,10.
,1.,2.,3.,6.,9.],[1.,6.,7.,8.,10.,2.,3.,4.,5.,9.],[2.,6.,7.,8.,10.,1.,3.,4.,5.,9.],[3
.,6.,7.,8.,10.,1.,2.,4.,5.,9.],[4.,6.,7.,8.,10.,1.,2.,3.,5.,9.],[5.,6.,7.,8.,10.,1.,2
.,3.,4.,9.],[1.,2.,3.,9.,10.,4.,5.,6.,7.,8.],[1.,2.,4.,9.,10.,3.,5.,6.,7.,8.],[1.,3.,
4.,9.,10.,2.,5.,6.,7.,8.],[2.,3.,4.,9.,10.,1.,5.,6.,7.,8.],[1.,2.,5.,9.,10.,3.,4.,6.,
7.,8.],[1.,3.,5.,9.,10.,2.,4.,6.,7.,8.],[2.,3.,5.,9.,10.,1.,4.,6.,7.,8.],[1.,4.,5.,9.
,10.,2.,3.,6.,7.,8.],[2.,4.,5.,9.,10.,1.,3.,6.,7.,8.],[3.,4.,5.,9.,10.,1.,2.,6.,7.,8.
],[1.,2.,6.,9.,10.,3.,4.,5.,7.,8.],[1.,3.,6.,9.,10.,2.,4.,5.,7.,8.],[2.,3.,6.,9.,10.,
1.,4.,5.,7.,8.],[1.,4.,6.,9.,10.,2.,3.,5.,7.,8.],[2.,4.,6.,9.,10.,1.,3.,5.,7.,8.],[3.
,4.,6.,9.,10.,1.,2.,5.,7.,8.],[1.,5.,6.,9.,10.,2.,3.,4.,7.,8.],[2.,5.,6.,9.,10.,1.,3.
,4.,7.,8.],[3.,5.,6.,9.,10.,1.,2.,4.,7.,8.],[4.,5.,6.,9.,10.,1.,2.,3.,7.,8.],[1.,2.,7
.,9.,10.,3.,4.,5.,6.,8.],[1.,3.,7.,9.,10.,2.,4.,5.,6.,8.],[2.,3.,7.,9.,10.,1.,4.,5.,6
.,8.],[1.,4.,7.,9.,10.,2.,3.,5.,6.,8.],[2.,4.,7.,9.,10.,1.,3.,5.,6.,8.],[3.,4.,7.,9.,
10.,1.,2.,5.,6.,8.],[1.,5.,7.,9.,10.,2.,3.,4.,6.,8.],[2.,5.,7.,9.,10.,1.,3.,4.,6.,8.]
,[3.,5.,7.,9.,10.,1.,2.,4.,6.,8.],[4.,5.,7.,9.,10.,1.,2.,3.,6.,8.],[1.,6.,7.,9.,10.,2
.,3.,4.,5.,8.],[2.,6.,7.,9.,10.,1.,3.,4.,5.,8.],[3.,6.,7.,9.,10.,1.,2.,4.,5.,8.],[4.,
6.,7.,9.,10.,1.,2.,3.,5.,8.],[5.,6.,7.,9.,10.,1.,2.,3.,4.,8.],[1.,2.,8.,9.,10.,3.,4.,
5.,6.,7.],[1.,3.,8.,9.,10.,2.,4.,5.,6.,7.],[2.,3.,8.,9.,10.,1.,4.,5.,6.,7.],[1.,4.,8.
,9.,10.,2.,3.,5.,6.,7.],[2.,4.,8.,9.,10.,1.,3.,5.,6.,7.],[3.,4.,8.,9.,10.,1.,2.,5.,6.
,7.],[1.,5.,8.,9.,10.,2.,3.,4.,6.,7.],[2.,5.,8.,9.,10.,1.,3.,4.,6.,7.],[3.,5.,8.,9.,1

Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
81
0.,1.,2.,4.,6.,7.],[4.,5.,8.,9.,10.,1.,2.,3.,6.,7.],[1.,6.,8.,9.,10.,2.,3.,4.,5.,7.],
[2.,6.,8.,9.,10.,1.,3.,4.,5.,7.],[3.,6.,8.,9.,10.,1.,2.,4.,5.,7.],[4.,6.,8.,9.,10.,1.
,2.,3.,5.,7.],[5.,6.,8.,9.,10.,1.,2.,3.,4.,7.],[1.,7.,8.,9.,10.,2.,3.,4.,5.,6.],[2.,7
.,8.,9.,10.,1.,3.,4.,5.,6.],[3.,7.,8.,9.,10.,1.,2.,4.,5.,6.],[4.,7.,8.,9.,10.,1.,2.,3
.,5.,6.],[5.,7.,8.,9.,10.,1.,2.,3.,4.,6.],[6.,7.,8.,9.,10.,1.,2.,3.,4.,5.]]
L1={{1,2,3,4,5,6,7,8,9,10},{1,2,3,4,6,5,7,8,9,10},{1,2,3,5,6,4,7,8,9,10},{1,2,4,5,6,3
,7,8,9,10},{1,3,4,5,6,2,7,8,9,10},{2,3,4,5,6,1,7,8,9,10},{1,2,3,4,7,5,6,8,9,10},{1,2,
3,5,7,4,6,8,9,10},{1,2,4,5,7,3,6,8,9,10},{1,3,4,5,7,2,6,8,9,10},{2,3,4,5,7,1,6,8,9,10
},{1,2,3,6,7,4,5,8,9,10},{1,2,4,6,7,3,5,8,9,10},{1,3,4,6,7,2,5,8,9,10},{2,3,4,6,7,1,5
,8,9,10},{1,2,5,6,7,3,4,8,9,10},{1,3,5,6,7,2,4,8,9,10},{2,3,5,6,7,1,4,8,9,10},{1,4,5,
6,7,2,3,8,9,10},{2,4,5,6,7,1,3,8,9,10},{3,4,5,6,7,1,2,8,9,10},{1,2,3,4,8,5,6,7,9,10},
{1,2,3,5,8,4,6,7,9,10},{1,2,4,5,8,3,6,7,9,10},{1,3,4,5,8,2,6,7,9,10},{2,3,4,5,8,1,6,7
,9,10},{1,2,3,6,8,4,5,7,9,10},{1,2,4,6,8,3,5,7,9,10},{1,3,4,6,8,2,5,7,9,10},{2,3,4,6,
8,1,5,7,9,10},{1,2,5,6,8,3,4,7,9,10},{1,3,5,6,8,2,4,7,9,10},{2,3,5,6,8,1,4,7,9,10},{1
,4,5,6,8,2,3,7,9,10},{2,4,5,6,8,1,3,7,9,10},{3,4,5,6,8,1,2,7,9,10},{1,2,3,7,8,4,5,6,9
,10},{1,2,4,7,8,3,5,6,9,10},{1,3,4,7,8,2,5,6,9,10},{2,3,4,7,8,1,5,6,9,10},{1,2,5,7,8,
3,4,6,9,10},{1,3,5,7,8,2,4,6,9,10},{2,3,5,7,8,1,4,6,9,10},{1,4,5,7,8,2,3,6,9,10},{2,4
,5,7,8,1,3,6,9,10},{3,4,5,7,8,1,2,6,9,10},{1,2,6,7,8,3,4,5,9,10},{1,3,6,7,8,2,4,5,9,1
0},{2,3,6,7,8,1,4,5,9,10},{1,4,6,7,8,2,3,5,9,10},{2,4,6,7,8,1,3,5,9,10},{3,4,6,7,8,1,
2,5,9,10},{1,5,6,7,8,2,3,4,9,10},{2,5,6,7,8,1,3,4,9,10},{3,5,6,7,8,1,2,4,9,10},{4,5,6
,7,8,1,2,3,9,10},{1,2,3,4,9,5,6,7,8,10},{1,2,3,5,9,4,6,7,8,10},{1,2,4,5,9,3,6,7,8,10}
,{1,3,4,5,9,2,6,7,8,10},{2,3,4,5,9,1,6,7,8,10},{1,2,3,6,9,4,5,7,8,10},{1,2,4,6,9,3,5,
7,8,10},{1,3,4,6,9,2,5,7,8,10},{2,3,4,6,9,1,5,7,8,10},{1,2,5,6,9,3,4,7,8,10},{1,3,5,6
,9,2,4,7,8,10},{2,3,5,6,9,1,4,7,8,10},{1,4,5,6,9,2,3,7,8,10},{2,4,5,6,9,1,3,7,8,10},{
3,4,5,6,9,1,2,7,8,10},{1,2,3,7,9,4,5,6,8,10},{1,2,4,7,9,3,5,6,8,10},{1,3,4,7,9,2,5,6,
8,10},{2,3,4,7,9,1,5,6,8,10},{1,2,5,7,9,3,4,6,8,10},{1,3,5,7,9,2,4,6,8,10},{2,3,5,7,9
,1,4,6,8,10},{1,4,5,7,9,2,3,6,8,10},{2,4,5,7,9,1,3,6,8,10},{3,4,5,7,9,1,2,6,8,10},{1,
2,6,7,9,3,4,5,8,10},{1,3,6,7,9,2,4,5,8,10},{2,3,6,7,9,1,4,5,8,10},{1,4,6,7,9,2,3,5,8,
10},{2,4,6,7,9,1,3,5,8,10},{3,4,6,7,9,1,2,5,8,10},{1,5,6,7,9,2,3,4,8,10},{2,5,6,7,9,1
,3,4,8,10},{3,5,6,7,9,1,2,4,8,10},{4,5,6,7,9,1,2,3,8,10},{1,2,3,8,9,4,5,6,7,10},{1,2,
4,8,9,3,5,6,7,10},{1,3,4,8,9,2,5,6,7,10},{2,3,4,8,9,1,5,6,7,10},{1,2,5,8,9,3,4,6,7,10
},{1,3,5,8,9,2,4,6,7,10},{2,3,5,8,9,1,4,6,7,10},{1,4,5,8,9,2,3,6,7,10},{2,4,5,8,9,1,3
,6,7,10},{3,4,5,8,9,1,2,6,7,10},{1,2,6,8,9,3,4,5,7,10},{1,3,6,8,9,2,4,5,7,10},{2,3,6,
8,9,1,4,5,7,10},{1,4,6,8,9,2,3,5,7,10},{2,4,6,8,9,1,3,5,7,10},{3,4,6,8,9,1,2,5,7,10},
{1,5,6,8,9,2,3,4,7,10},{2,5,6,8,9,1,3,4,7,10},{3,5,6,8,9,1,2,4,7,10},{4,5,6,8,9,1,2,3
,7,10},{1,2,7,8,9,3,4,5,6,10},{1,3,7,8,9,2,4,5,6,10},{2,3,7,8,9,1,4,5,6,10},{1,4,7,8,
9,2,3,5,6,10},{2,4,7,8,9,1,3,5,6,10},{3,4,7,8,9,1,2,5,6,10},{1,5,7,8,9,2,3,4,6,10},{2
,5,7,8,9,1,3,4,6,10},{3,5,7,8,9,1,2,4,6,10},{4,5,7,8,9,1,2,3,6,10},{1,6,7,8,9,2,3,4,5
,10},{2,6,7,8,9,1,3,4,5,10},{3,6,7,8,9,1,2,4,5,10},{4,6,7,8,9,1,2,3,5,10},{5,6,7,8,9,
1,2,3,4,10},{1,2,3,4,10,5,6,7,8,9},{1,2,3,5,10,4,6,7,8,9},{1,2,4,5,10,3,6,7,8,9},{1,3
,4,5,10,2,6,7,8,9},{2,3,4,5,10,1,6,7,8,9},{1,2,3,6,10,4,5,7,8,9},{1,2,4,6,10,3,5,7,8,
9},{1,3,4,6,10,2,5,7,8,9},{2,3,4,6,10,1,5,7,8,9},{1,2,5,6,10,3,4,7,8,9},{1,3,5,6,10,2
,4,7,8,9},{2,3,5,6,10,1,4,7,8,9},{1,4,5,6,10,2,3,7,8,9},{2,4,5,6,10,1,3,7,8,9},{3,4,5
,6,10,1,2,7,8,9},{1,2,3,7,10,4,5,6,8,9},{1,2,4,7,10,3,5,6,8,9},{1,3,4,7,10,2,5,6,8,9}
,{2,3,4,7,10,1,5,6,8,9},{1,2,5,7,10,3,4,6,8,9},{1,3,5,7,10,2,4,6,8,9},{2,3,5,7,10,1,4
,6,8,9},{1,4,5,7,10,2,3,6,8,9},{2,4,5,7,10,1,3,6,8,9},{3,4,5,7,10,1,2,6,8,9},{1,2,6,7
,10,3,4,5,8,9},{1,3,6,7,10,2,4,5,8,9},{2,3,6,7,10,1,4,5,8,9},{1,4,6,7,10,2,3,5,8,9},{
2,4,6,7,10,1,3,5,8,9},{3,4,6,7,10,1,2,5,8,9},{1,5,6,7,10,2,3,4,8,9},{2,5,6,7,10,1,3,4
,8,9},{3,5,6,7,10,1,2,4,8,9},{4,5,6,7,10,1,2,3,8,9},{1,2,3,8,10,4,5,6,7,9},{1,2,4,8,1
0,3,5,6,7,9},{1,3,4,8,10,2,5,6,7,9},{2,3,4,8,10,1,5,6,7,9},{1,2,5,8,10,3,4,6,7,9},{1,
3,5,8,10,2,4,6,7,9},{2,3,5,8,10,1,4,6,7,9},{1,4,5,8,10,2,3,6,7,9},{2,4,5,8,10,1,3,6,7
,9},{3,4,5,8,10,1,2,6,7,9},{1,2,6,8,10,3,4,5,7,9},{1,3,6,8,10,2,4,5,7,9},{2,3,6,8,10,
1,4,5,7,9},{1,4,6,8,10,2,3,5,7,9},{2,4,6,8,10,1,3,5,7,9},{3,4,6,8,10,1,2,5,7,9},{1,5,
6,8,10,2,3,4,7,9},{2,5,6,8,10,1,3,4,7,9},{3,5,6,8,10,1,2,4,7,9},{4,5,6,8,10,1,2,3,7,9
},{1,2,7,8,10,3,4,5,6,9},{1,3,7,8,10,2,4,5,6,9},{2,3,7,8,10,1,4,5,6,9},{1,4,7,8,10,2,
3,5,6,9},{2,4,7,8,10,1,3,5,6,9},{3,4,7,8,10,1,2,5,6,9},{1,5,7,8,10,2,3,4,6,9},{2,5,7,
8,10,1,3,4,6,9},{3,5,7,8,10,1,2,4,6,9},{4,5,7,8,10,1,2,3,6,9},{1,6,7,8,10,2,3,4,5,9},
{2,6,7,8,10,1,3,4,5,9},{3,6,7,8,10,1,2,4,5,9},{4,6,7,8,10,1,2,3,5,9},{5,6,7,8,10,1,2,
3,4,9},{1,2,3,9,10,4,5,6,7,8},{1,2,4,9,10,3,5,6,7,8},{1,3,4,9,10,2,5,6,7,8},{2,3,4,9,
10,1,5,6,7,8},{1,2,5,9,10,3,4,6,7,8},{1,3,5,9,10,2,4,6,7,8},{2,3,5,9,10,1,4,6,7,8},{1
,4,5,9,10,2,3,6,7,8},{2,4,5,9,10,1,3,6,7,8},{3,4,5,9,10,1,2,6,7,8},{1,2,6,9,10,3,4,5,
7,8},{1,3,6,9,10,2,4,5,7,8},{2,3,6,9,10,1,4,5,7,8},{1,4,6,9,10,2,3,5,7,8},{2,4,6,9,10
,1,3,5,7,8},{3,4,6,9,10,1,2,5,7,8},{1,5,6,9,10,2,3,4,7,8},{2,5,6,9,10,1,3,4,7,8},{3,5
,6,9,10,1,2,4,7,8},{4,5,6,9,10,1,2,3,7,8},{1,2,7,9,10,3,4,5,6,8},{1,3,7,9,10,2,4,5,6,
8},{2,3,7,9,10,1,4,5,6,8},{1,4,7,9,10,2,3,5,6,8},{2,4,7,9,10,1,3,5,6,8},{3,4,7,9,10,1
,2,5,6,8},{1,5,7,9,10,2,3,4,6,8},{2,5,7,9,10,1,3,4,6,8},{3,5,7,9,10,1,2,4,6,8},{4,5,7
,9,10,1,2,3,6,8},{1,6,7,9,10,2,3,4,5,8},{2,6,7,9,10,1,3,4,5,8},{3,6,7,9,10,1,2,4,5,8}
,{4,6,7,9,10,1,2,3,5,8},{5,6,7,9,10,1,2,3,4,8},{1,2,8,9,10,3,4,5,6,7},{1,3,8,9,10,2,4
,5,6,7},{2,3,8,9,10,1,4,5,6,7},{1,4,8,9,10,2,3,5,6,7},{2,4,8,9,10,1,3,5,6,7},{3,4,8,9
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7,8,9,10,1,2,3,4,5}}
Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
82
wP12_2_10.da
M1=[[1,2,3,4,5,6,7,8,9,10,11,12],[1,3,2,4,5,6,7,8,9,10,11,12],[2,3,1,4,5,6,7,8,9,10,1
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4,5,6,7,9,10,11,12],[4,8,1,2,3,5,6,7,9,10,11,12],[5,8,1,2,3,4,6,7,9,10,11,12],[6,8,1,
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1,3,4,5,6,7,8,10,11,12],[3,9,1,2,4,5,6,7,8,10,11,12],[4,9,1,2,3,5,6,7,8,10,11,12],[5,
9,1,2,3,4,6,7,8,10,11,12],[6,9,1,2,3,4,5,7,8,10,11,12],[7,9,1,2,3,4,5,6,8,10,11,12],[
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7,8,9,10,11],[2,12,1,3,4,5,6,7,8,9,10,11],[3,12,1,2,4,5,6,7,8,9,10,11],[4,12,1,2,3,5,
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wV2_3.da
M1=[[-1,-1,-1],[-1,-1,1],[-1,1,-1],[-1,1,1],[1,-1,-1],[1,-1,1],[1,1,-1],[1,1,1]]
L1={{-1,-1,-1},{-1,-1,1},{-1,1,-1},{-1,1,1},{1,-1,-1},{1,-1,1},{1,1,-1},{1,1,1}}
wV2_9.da
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Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
83
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Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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Sch ausse , D. G. (2025). HP_P ime_MATH: Manual. h ps://www.academia.edu/130037807
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