<p> plot(data,char,size,clr)</p>

5e822bfd-9852-40b1-a07f-148531d08686 20 4 5 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 :

fails wrt factorial N n m X hyperD N! (n! (* N n + m - (! (* m! (* X! (X - n + (! (* N n + (! (* X N + m - (! (* X - m + (! (* / eval :

N 45 : n 10 : m 20 : X 0 m range : 14 mat

X N n m X hyperD vectorize augment

Hypergeometric ...

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 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

Hypergeometric ... PDF, CDF

x Γ x Gamma :

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 85 :

n 10 :

p 20 :

shift 1 :

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

Simpson integrator does NOT
recognize pseudo-scalar pdf(x) x CDF x pdf x 0 x int :

2 CDF #

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 05 0.1 range : i 1 U rows range vy i el N n 15 U i el 1 + PDF eval : for U vy augment 31 line :

Finite differences Cumulative Integral

data Σ data 2 1 sys

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 :

support 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