Γ(x) [15 D] Alvaro

b550e24d-319a-4ece-a4b1-3663e5996f4c 42 3 5 decimal

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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

Part: 1 . . . native Smath Γ(x) ... 4 D

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Normalized Hypergeometric PDF

rejects integrator x Γ x Gamma :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

X Discrete probability ≡ x scalar probability ≡ 21 line

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 pdf x m n N hypergeom :

m 60 : n 30 : N 16 : X 0 N range : 14 mat H X X m n N f vectorize augment : Hyper H o 6 red plot : 31 line

option => numeric hyper x 0 N 0.1 range : j 1 x rows range h j el x j el pdf : for x h augment 31 line :

μ n N * m n + / : σ m n * N * m n + N - (* m n + 1 - (m n + (2^* / : 21 mat 5.333 2.956 21 mat

Infinitesimal h 0.01 :

x f x pdf eval :

1rst derivative x D1 x 2 h * - f x 2 h * + f - 4 h * / - :

2nd derivative x D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

Cumul

data Σ Finite differences Cumul 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 :

Hypergeometric PDF over data

hyper Hyper μ 0.5 μ 0.125 - 2 2 mat 3 1 sys

1rst & 2nd infinitesimal Deriv

x D1 x D2 2 1 sys

Σ cumulative Hypergeometric

hyper Σ 12 1 5 1 2 2 mat 2 1 sys

The Hypergeometric Distribution . . .

When 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 hypergeometric distribution if its probability distribution is given by => h(X,n,N,M)

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X n N M h M X combin N M - n X - combin * N n combin / ≡

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

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 * : σ n p * 1 p - (* N n - N 1 - / * sqrt : 21 mat 13.75 2.165 21 mat

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

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

μ n p * : σ n p * 1 p - (* N n - N 1 - / * sqrt : 21 mat 13.333 1.736 21 mat

sanity Mathematica 1005040100 / 25 H 0.0207

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 ?

HyperD k N n p k H vectorize augment :

data HyperD o 5 black plot :

2015820 / 5 H 0.2384

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

... 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

Part: 2 . . . advanced Γ(x) ... 15 D

]]> 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

Normalized Hypergeometric PDF

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

scalar hypergeom x m n N PDF 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 pdf x m n N PDF eval :

m 60 : n 20 : N 7 : X 0 N range : 14 mat H X X m n N f vectorize augment : Hyper H o 6 red plot : 31 line

option => numeric hyper x 0 N 0.1 range : j 1 x rows range h j el x j el pdf : for x h augment 31 line :

μ n N * m n + / : σ m n * N * m n + N - (* m n + 1 - (m n + (2^* / : 21 mat 1.75 1.213 21 mat

Infinitesimal h 0.01 :

x f x pdf :

1rst derivative x D1 x 2 h * - f x 2 h * + f - 4 h * / - :

2nd derivative x D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

Cumul

data Σ Finite differences Cumul 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 :

Hypergeometric PDF over data

hyper Hyper 2 1 sys

1rst & 2nd infinitesimal Deriv

x D1 x D2 2 1 sys

Σ cumulative Hypergeometric

hyper Σ

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

2 CDF 0.5652

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 x 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

Discrete cumulative algo style

# f Σ ε 1012 -^: U 010 0.1 range : i 1 U rows range Cumul i el x f x U 1 el U i el int eval : for U Cumul augment 41 line :

Assign user as function x f x pdf :

x f Σ 12 1 5 1 2 2 mat 2 1 sys

Resume Hypergeometric . . .

1. Native Gamma(x) is only 4 decimals, much quicker than the 15 D Γ(x)

2. You can integrate the vectorized Gamma(x), but CAN NOT the depending PDF.

3. The 15 D Γ(x) is more demanding in calculation load, but supports the integrator

[non-vectorized] as well as the cumulative algo.

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x Gamma vectorize x 1 π int 2.611