plot(data,char,size,clr)

5e822bfd-9852-40b1-a07f-148531d08686 37 15 12 decimal

&[DATE] &[TIME] - &[FILENAME]

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Γ(x) [15 D] Alvaro

Γ.κ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh z 0.5 + : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ.κ 1 el Γ.κ κ 1 + el z κ + / κ 114 sum + (* zp 2^* e zg -^* eval 21 line :

Valid for Re(ζ)>0

Hypergeometric ...

N n p x hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 85 :

n 10 :

p 20 :

shift 1 :

x hypergeom N n p x hypergeom :

Continuous scalar PDF Hypergeometric. Use to range/initialise Golden ratio.

N n 5 x shift - hypergeom N n 10 x shift - hypergeom N n 15 x shift - hypergeom 3 1 sys

n k combin n! k! (n k - (! * / :

N n p k H p N * k combin N 1 p - (* n k - (combin * N n combin / eval :

N 100 : n 25 : p 0.55 : k 0 n range : 14 mat

μ n p * : 13.75

σ n p * 1 p - (* N n - N 1 - / * sqrt : 2.1651

HyperD k N n p k H vectorize augment :

Definition . . .

N . . . number of elements in population.

n . . . number of drawn samples.

p . . . probability of class 1

k . . . observations of class 1

The hypergeometric distribution is a discrete distribution.

It is used to describe the probability of finding k observations

of class 1 when n samples are drawn from a population of N

objects, and the probability of a single element of class 1

equals p.

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data HyperD o 5 black plot :

data HyperD μ 1 μ 1 - 2 2 mat μ σ - 0 + 200 blue 1 5 mat μ σ + 0 + 200 blue 1 5 mat 5 1 sys

N 60 : n 20 : p 23 / : k 0 n range : 14 mat

μ n p * : 13.3333

σ n p * 1 p - (* N n - N 1 - / * sqrt : 1.7359

STATED OTHERWISE: What is the probability of 25 successes in 50 draws from 100 elements that have 40 possible successes ?

HyperD k N n p k H vectorize augment :

data HyperD o 5 black plot :

sanity Mathematica 1005040100 / 25 H 0.0207

data HyperD μ 1 μ 1 - 2 2 mat μ σ - 0 + 200 blue 1 5 mat μ σ + 0 + 200 blue 1 5 mat 5 1 sys

From a bag of mixed 20 red/blue balls [8 blue, 12 red] Draw 15 balls ... what is the probability of drawing 5 blue, 10 red ?

2015820 / 5 H 0.2384

... add 12 balls in the bag, draw 15 ... What is the new probability of drawing 5 blue balls ? Now, we don't know how many blue balls to expect, then define an expectation 'E'= Expected probability of the outcome

N 32 : n 15 : E 13 : p E N / : 14 mat

N n p 5 H 0.2102

EXPECT 10 :

N 157 : n 50 : p EXPECT N / : k 0 EXPECT range : 14 mat

μ n p * : 3.1847

σ n p * 1 p - (* N n - N 1 - / * sqrt : 1.4301

HyperD k N n p k H vectorize augment :

data HyperD o 5 black plot :

seek 5/EXPECT S 5 :

data HyperD μ 1 μ 1 - 2 2 mat μ σ - 0 + 200 blue 1 5 mat μ σ + 0 + 200 blue 1 5 mat S N n p S H + 20 black 1 5 mat 6 1 sys

probability of 'S'/EXPECT N n p S H 0.1204

The Hypergeometric Distribution:

Whwn we are interested in finding probabilities in the case where the trials are not independent,

as in sampling without replacement, we obtain a Hypergeometric Distribution. A random variable

X has a hypergeometric distribution if its probability is given by

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N n m X H m X combin N m - n X - (combin * N n combin / eval :

N 100 : n 15 : m 50 : X 0 n range : 14 mat

HyperD X N n m X H vectorize augment :

data HyperD o 5 black plot :

n 2 / 1 n 2 / 1 - 2 2 mat HyperD data 3 1 sys n 2 / 1 n 2 / 1 - 2 2 mat HyperD HyperD 3 1 sys

Normalized Hypergeometric ... PDF, CDF

N n p x PDF N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 60 :

n 40 :

p 10 :

shift 0 :

x pdf N n p x PDF eval :

Continuous scalar PDF Hypergeometric.

N n 5 x shift - PDF N n 10 x shift - PDF N n 15 x shift - PDF 3 1 sys

option: Int. acc u CDF x pdf x 0 u int :

2 CDF 0.0861

Carlos adaptive integration

]]> ∫Adaptive(f,a,b,N,ε)

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

Program algo style is a bit slower than open style, more convenient. Adapted from Carlos code [20180110]

end 20 :

normalized PDF x f 6040 end x PDF :

'N' iterations
precision N 1000 : ε 1012 -^: 21 sys

User inputs

lower limit [any]
upper limit ≤ end a 0 : b end : 21 sys

t0 1 time :

Technical acccuracy ... 1 for a well footed RHS
which depends upon 'end' &  'ε' f a b N ε ∫Adaptive 0.999998

1 time t0 - 77.3 s *

match the codes [U, p] end 10 :

data Σ Finite differences Cumul Integral vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

data U 0 end 0.1 range : i 1 U rows range vy i el 6040 end U i el PDF eval : for U vy augment 31 line :

Finite differences Cumulative Integral

data Σ data 2 1 sys

Fitting an histogram as PDF

N 60 : n 40 : p 10 : 13 mat

XY 0 0.004355 1 0.034 2 0.115 3 0.22 4 0.264 5 0.208 6 0.108 7 0.037 8 0.007864 9 0.0009478 10 0.00004897 112 mat :

pts XY o 5 black plot :

0 0.0044 o 5 black 1 0.034 o 5 black 2 0.115 o 5 black 3 0.22 o 5 black 4 0.264 o 5 black 5 0.208 o 5 black 6 0.108 o 5 black 7 0.037 o 5 black 8 0.0079 o 5 black 9 0.0009 o 5 black 10 4.897 10 5 - ^ * o 5 black 11 5 mat N n p x PDF 2 1 sys