# All Probability-and-distributions Formulas List

## Statistical Power

#### Formula:

p = 1 - β

Where,

p = Statistical Power of Test
β = Beta

## Permutation and Combination

#### Formula:

Permutation
nPr = n! / ( n - r )!

Combination
nCr = nPr / r!

## Normal Distribution

#### Normal Distribution Formula :

X < mean = M - Z
X > mean = M + Z
Z = (X-M) / Ï

Where,

M = Mean.
Ï = Standard Deviation.
X = Normal Random Variable

## Probability Density Function of Normal Distribution, Standard Normal Distribution

Formula :
PDF of Normal Distribution = P(x) = (1/(σsqrt(2π)))e-(x-m)2 / (2σ2)
Standard Normal Distribution = P(x) = (1/sqrt(2π))e-(x2 / 2)

## Binomial Distribution

#### Formula:

P (X = r) = nCr pr (1-p)n-r

Where,

Combination nCr = ( n! / (n- r)! ) / r!
n = Number of Events
r = Number of Success
p = Probability of Success

## Negative Binomial Distribution

Negative Binomial Distribution Formula :
Negative Binomial Distribution P(X = r) = n-1Cr-1 pr (1-p)n-r
where,
Combination n-1Cr-1 = ( (n-1)! / ((n-1)-(r-1))! ) / (r-1)!

## Poisson Distribution

#### Poisson-distribution Formula:

Poisson Distribution f(x) = eλx / x!

## Hypergeometric Distribution

#### Formula:

Hypergeometric Distribution h(x, N, n, k) = [kCx] [N - kCn - x] / [NCn]

Where,

k = Number of Selected Items from the Population Size
x = Random Variable
N = Total Population Size
n = Total Sample Size

## Z Score to P Value

#### Formula: ## Exponential Distribution

Formula:
P(x) = ae-ax,
where,
a is the parameter of the distribution,
x is the random variable,
P(x) is the probability density function.

## Expected Value E(x)

#### Formula:

E[x] = n x p

Where,

n is the number of trials,
p is the probability of a successful outcome.

## Weibull Probability Distribution Function

Formula Used : ## Beta Distribution Function

Formula Used:
Probability Density Function (pdf) = Cumulative Distribution Function (cdf) = where,
Β(α,β) - Beta Function
Βx(α,β)- Incomplete Beta Function

## Cauchy Distribution

Formula Used: ## Gamma Distribution Function

Formula Used: ## Laplace Distribution

Formula Used: ## Continuous Uniform Distribution

Formula Used:
Probability Density = Lower Cumulative Distribution = Upper Cumulative Distribution = ## Weibull Cumulative Distribution, Probability Density

Formula Used: ## Inverse / Reciprocal Gamma Distribution

Formula Used:
Probability Density Function (pdf) = Cumulative Distribution Function (cdf) = ## Diagnostic Post Test Probability

Formula:
O = p1 / ( 1 - p1 ),
p2 = O * L,
p = p2 / ( 1 + p2 ),

Where,
p1 is the pretest probability,
O is the pretest odds,
p2 is the posttest odds,
L is the likelihood ratio,
p is the posttest probability.

## Multinomial Distribution

Formula: ## Rayleigh Distribution

Formula:  ## Gumbel Distribution for PDF, CDF

Formula:  ## Geometric Distribution

#### Formula:

P(x) = qxp

Where ,

p = Probability of success for a single trial
q = Probability of failure for a single trial ( = 1-p )
x = Total Occurrence - 1

## Dirichlet Multinomial Distribution

Formula: Where,

Pr(Z | α ) is Dirichlets Multinominal Distribution
A is ∑k αk (Sigma) represents summation of Parameter vector (α) values
N is ∑k nk represents summation of Independent trial (n) values
k is number of trials
Γ (gamma) represents factorial function of (x-1)
Π (pi) represents product function

## Standard Distribution

Formula Used:
Mean (M)= Sum of random values / n
Standard Sample Deviation where,
X - sample values
M - mean value
n - number of samples values

## Kolmogorov Smirnov Test

#### Formula: Where,

D = Maximum Value of Normal Distribution,
N = Numbeformr of Statistic Data,
F = Kolmogorov Smirnov (KS) Index.

## Student T Test

#### Formula:  Where

X1 - Group one data,
X2 - Group two data,
t - test statistic
n1,n2 - Group values count

## Degrees of Freedom

d = n1 - 1

#### Two Sample T-Test Formula:

df = (n1 + n2) - 2

Where,

df = Degree of Freedom
n1 = Total Number in Sequence 1
n2 = Total Number in Sequence 2

## Combination nCr

#### Formula:

C(n,r) = n! / ( r! (n - r)! )

Where,

C(n,r) = Combinations nCr
n = Number of Sample Points in Set
r = Number of sample Points in Each Combination

## Fisher Z Transformation

#### Formula:

Zr = ( 1 / 2 ) [ ln ( 1 + r ) - ln ( 1 - r ) ]
SEzr = ( 1 / √( n - 3 ) )

Where,

Zr = Z Score
SEzr = Standard Error
r = Correlation Coefficient
n = Size

## Standard Normal Distribution

#### Formula:

Z = (x - μ) / σ

Where,

Z = Standardized Random Variable
x = The Value that is being Standardized
μ = Mean of the Distribution
σ = Standard Deviation of the Distribution

## Geometric Probability

#### Formula:

r = (1 - p)
u = rx
f = u x p
l = 1 - r(x + 1)
m = r / p

Where,

u = Upper CDF
l = Lower CDF
f = Probability of Mass
m = Mean
p = Probability of Success
x = Percentile x

## Index Of Dispersion

#### Formula:

Index of Dispersion = σ2 / μ

Where,

μ = Mean
σ2 = Variance

## Permutation

#### Formula:

nPr = n! / ( n - r )!

Where,

nPr = Permutations
n = Number of sample points in set n
r = Number of sample points in each permutation

## T Score

#### Formula :

t-Observed :
t - Observed = (Sample Mean - Population Mean) / Standard Deviation

Mean :
Mean = Sum of X values / N(Number of Values)

Sample Standard Deviation : ## Log Odds and Odds

#### Formula:

When Observed Proportion is Given
o = p / (1 - p)
l = log (p / (1 - p))

When Odds is Given
p = o / (1 + o)
l = log(o)

When Log Odds is Given
o = Exp (l)
p = o / (1 - o)

Where,

o = Odds
l = Log Odds
p = Observed Proportion

## Bivariate Distribution

#### Formula:

Probability Density (f) = ( 1 / ( 2 π √ ( 1 - p2))) × ( e ( - ( x2 - 2pxy + y2)) / ( 2 ( 1 - p2)))

Where,

f = Probability Density
x = Percentile x
y = Percentile y
p = Correlation Coefficient

## Logistic Distribution

#### Formula:

E = e- ((x - a) / b)
p = E / (b x (1 + E)2)
lcdf = 1 / (1 + E)
ucdf = 1- lcdf

Where,

E = Exponent Value
x = Percentile
a = Location Parameter
b = Scale Parameter
p = Probability Density Function (pdf)
lcdf = Lower Cumulative Density Function (lcdf)
ucdf = Upper Cumulative Density Function (ucdf)

## Number Lock Permutations

#### Formula:

Combination Without Repitition = a! / b! x (a - b)!
Combination With Repitition = (a + b - 1)! / b! x ((a + b - 1) - b)!
Variation Without Repitition = a! / (a - b)!
Variation With Repitition = ab

Where,

a = Element to Choose from
b = Elements Chosen

## T Function and V Function

#### Formula:

t = 1 / (2π) tan-1a

Where,

t = T-Function and V-Function
a = Coefficient (y = ax)

## Arcsine Distribution

#### Formula:

Cumulative Distribution Function = (2 / π) * (arcsine (√ (x)))
Probability Distribution Function = 1 / (π √ (x (1 - x)))

Where,

x = Parameter

## Folded Normal Distribution

#### Formula: Where,

x = Upper Limit
μ = Mean
σ2 = Variance
erf=Error function

## Percentile Mean Standard Deviation

#### Formula:

50th Percentile = Mean
84th Percentile = Mean + Standard Deviation
97.5th Percentile = Mean + (2 x Standard Deviation)