plotG(data,char,size,clr) ... Σ(data)

26bcdacc-cb25-4e3a-ba89-a0de32f69893 41 4 5 decimal

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

data char size clr plotG 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 :

data Σ FD Cumulative Integration 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 :

Hypergeom ...

x Γ x Gamma :

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 39 :

n 12 :

p 20 :

x hypergeom N n p x hypergeom :

Continuous PDF Hypergeometric

0 x ≤ x hypergeom outside lower bound if

In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of 'k' successes in 'n' draws, without replacement, from a finite population of size 'N' that contains exactly 'K' successes, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of 'k' successes in 'n' draws with replacement.

scalar hypergeom x m n N hypergeom m 1 + Γ n 1 + Γ * m n + N - 1 + Γ * N 1 + Γ * x 1 + Γ x - n + 1 + Γ * m n + 1 + Γ * x m + N - 1 + Γ * x - N + 1 + Γ * / :

Discrete hypergeom X m n N f m! (n! * m n + N - (! * N! * X! (X - n + (! * m n + (! * X m + N - (! * X - N + (! * / :

x hypergeom x m n N hypergeom :

m 40 : n 20 : N 13 : X 1 N range : 14 mat

N 13 ≤

H X X m n N f vectorize augment : Hyper H o 6 red plotG : 21 line

option => numeric hyper x 0 N 0.05 range : j 1 x rows range h j el x j el hypergeom : for x h augment 31 line :

Guess Extrema range from 'hyper'

hyper Golden ratio

The hyperfast Golden-Ratio search [min/max]

x f x hypergeom : a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : 21 line

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema y1 y2 ≥ max in range ≡ y1 y2 ≤ min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 61 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

Ex f 35 y1 y2 ≥ 1012 -^Extrema 1 el :

Ex Ex :

Ex Ex f 12 mat 4.2389 0.2594 12 mat

hyper Hyper Ex Ex f + 15 black 1 5 mat 3 1 sys

hyper Hyper Ex Ex f + 15 black 15 mat 31 sys

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

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

N n m X h m X combin N m - n X - combin * N n combin / :

C X N n m X h vectorize augment :

COMBINE C o 5 red plotG :

COMBINE 1 1 sys

When we are interested in finding probabilities in the case where trials are not independent, as in the sampling without replacement, we obtain a Hypergeometric Distribution. A random variable 'X' has Hypergeometric Distribution if its probability distribution is given by h(N,n,m,X).

===============================================

'p' probability N n p x PDFhypergeom 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 30 :

p 10.5 :

x PDFhypergeom N n p x PDFhypergeom :

μ n p * N n + / : 3.5

σ N n * N n + p - (* p * 1 - N + n + (N n + (2^* / : 2.0843

PDF x 0 p 0.05 range : i 1 x rows range pdf i el x i el PDFhypergeom eval : for x pdf augment 31 line :

Normalized PDF/CDF Hypergeometric

PDF PDF Σ μ 0.65 μ 0.35 2 2 mat 3 1 sys

===== Section below is stand alone =====

Stem module

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

data char size clr plotG 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 :

x Γ x Gamma :

'p' probability N n p x PDFhypergeom 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 * / :

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 0.000948 10 0.000049 112 mat :

X XY 1 col : Y XY 2 col : 21 line

N n p SSE Y i el N n p X i el PDFhypergeom - (2^i 1 XY rows sum :

N 50 : n 30 : p 13 : 13 mat

N n p SSE 0.027

N 60 : n 40 : p 10 : 13 mat

<= best [N,n,p] from trial/error

N n p SSE 6107 -^*

PDF x 010 0.05 range : i 1 x rows range pdf i el N n p x i el PDFhypergeom eval : for x pdf augment 31 line :

Hist XY o 5 red plotG :

PDF Hist 2 1 sys

Fitting an histogram is a much different matter than fitting collected data set. When we fit a data set, we seek for the best model or at worst for the best approximation. When we attempt fitting an histogram, we are interested in the best underlying PDF. For this histogram/hypergeometric: 1. LM fails ... 2. CG poor ... 3. QN good Here, we fit purely: trial/error [a Monk job !] It could be automated in some ways [not offered].

Relative residuals Residuals i 1 XY rows range Δ i el 1 Y i el N n p X i el PDFhypergeom / - eval : for X Δ augment 21 line :

Relative residuals

Residuals Error Residuals 2 1 sys

Rework the program in convenient Smath code <= from unrecorded source N M Combin i 1 : k 1 : 12 mat i N M - 1 + (M^≤ j 1 M range tryV i j el i 1 - N M - 1 + (M j -^/ N M - 1 + mod trunc j + : for tryV i m el tryV i m 1 + el < (m 1 j 1 - product j 1 M range V k j el tryV i j el : for k k 1 + : 21 line computed if i i 1 + : 31 line while V 31 line :

Test 63 Combin :

Sanity check Test 123124125126134135136145146156234235236245246256345346356456203 mat