List of All Probability-functions Formulas

All Probability-functions Formulas List

Modified Bessel Function

Formula:



First Kind Bessel Function

Second Kind Bessel Function

First Kind Modified Bessel Function

Second Kind Modified Bessel Function

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Beta Function

Formula Used:


Where,
Β - Beta Function
x - X Value
y - Y Value

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Incomplete Beta Function

Formula:


Incomplete Beta Value


Where,
Β - Beta Function
a - a Value
b - b Value
x - Upper Limit

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Gamma Function

Formula:




Where,
Γ - Gamma Function
z - Z Value

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Incomplete Gamma Function


Formula Used:


Where,
Γ - gamma Function
u - Complex variable
v - Real value

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Gaussian Error (ERF) and Complementary Error (ERFC) Function

Formula:


Gaussian Error Value (erf)

Gaussian Error Value

Gaussian Complementary Error Value (erfc)

Gaussian Complementary Error Value

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Digamma Function

Formula:


Digamma Value


Where,
Γ - Gamma Function
x - X Value

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Hankel Function

Formula:


First Kind
Second Kind


Where,
Jα = First Kind Bessel Function
Yα = Second Kind Bessel Function

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Riccati Bessel Function

Formula:


First Kind of Riccati Bessel Function

Second Kind of Riccati Bessel Function



Where,
Jn = First Kind Bessel Functions
Yn = Second Kind Bessel Functions

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Spherical Bessel Function

Formula:

First Kind of Bessel Function

Second Kind of Bessel Function


Where,
Jn - First Kind of Bessel Function
Yn - Second Kind of Bessel Function

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Spherical Hankel Function

Formula:

First Kind of Hankel Function
First Kind of Hankel Function


Where,
jn - First Kind of Hankel Function
yn - Second Kind of Hankel Function

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Ellipticgamma Function

Formula:

Elliptic Gamma Value

Where,
Γ - Gamma Function
z, p, and q - Complex Variables

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Log Gamma Distribution

Formula Used:


Where,
Γ - gamma Function
z - Complex Variable

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PSI Polygamma Function

Formula Used:


Where,
Γ - gamma Function
z - Complex Variable
m - Order of Ploygamma function

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Reciprocal Gamma Function

Formula Used:


Where,
Γ - gamma Function
z - Complex Variable

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Trigamma Function

Formula Used:
Second Kind Polygamma =

Where,
z - Complex Variable

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Normal CDF

Formula:

Probability Function f(x, μ, σ) = 1 / (√ (2 x Π) x a x e(-1 / 2 x ((b - c) / a)2)
Lower cumulative distribution P(x, μ, σ) = -∞∫x f(x, μ, σ) dt
Upper cumulative distribution Q(x, μ, σ) = x∫∞ f(x, μ, σ) dt


Where,
a = Standard Deviation σ
b = Percentile x
c = Mean μ

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Significance Level

Formula:

For One Tailed
l = 100 - c

For Two Tailed
l = (100 - c) / 2


Where,
l = Significance Level
c = Confidence Level

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