Distribution Formulas

Distribution Families Formula Reference

Comprehensive mathematical reference for probability distribution families with formulas, parameters, and statistical properties

Complete ReferenceMathematical FormulasProperties & Applications

Basic Distribution Families

Fundamental probability distributions with complete mathematical formulations

Binomial Distribution B(n,p)

Discrete

Exponential Family

Key Formulas:

pmf:P(X = k) = C(n,k) p^k (1-p)^(n-k)

cdf:F(k) = Σ(i=0 to k) C(n,i) p^i (1-p)^(n-i)

mean:E[X] = np

variance:Var(X) = np(1-p)

mgf:M(t) = (1-p+pe^t)^n

Parameters:

n ∈ {1,2,3,...}, p ∈ (0,1)

Support:

k ∈ {0,1,2,...,n}

Poisson Distribution P(λ)

Discrete

Exponential Family

Key Formulas:

pmf:P(X = k) = (λ^k e^(-λ)) / k!

cdf:F(k) = e^(-λ) Σ(i=0 to k) λ^i / i!

mean:E[X] = λ

variance:Var(X) = λ

mgf:M(t) = exp(λ(e^t - 1))

Parameters:

λ > 0 (rate parameter)

Support:

k ∈ {0,1,2,...}

Normal Distribution N(μ,σ²)

Continuous

Exponential Family

Key Formulas:

pdf:f(x) = (1/(σ√(2π))) exp(-(x-μ)²/(2σ²))

cdf:F(x) = Φ((x-μ)/σ)

mean:E[X] = μ

variance:Var(X) = σ²

mgf:M(t) = exp(μt + σ²t²/2)

Parameters:

μ ∈ ℝ (mean), σ > 0 (standard deviation)

Support:

x ∈ ℝ

Exponential Distribution E(λ)

Continuous

Exponential Family

Key Formulas:

pdf:f(x) = λe^(-λx)

cdf:F(x) = 1 - e^(-λx)

mean:E[X] = 1/λ

variance:Var(X) = 1/λ²

mgf:M(t) = λ/(λ-t) for t < λ

Parameters:

λ > 0 (rate parameter)

Support:

x ≥ 0

Advanced Distribution Families

Specialized distributions for statistical inference and advanced modeling

Gamma Distribution Γ(α,λ)

Continuous

Mathematical Formulas:

pdf:f(x) = (λ^α/Γ(α)) x^(α-1) e^(-λx)

cdf:F(x) = γ(α, λx) / Γ(α)

mean:E[X] = α/λ

variance:Var(X) = α/λ²

mgf:M(t) = (λ/(λ-t))^α for t < λ

Key Properties:

Additivity: Γ(α₁,λ) + Γ(α₂,λ) = Γ(α₁+α₂,λ)
Special cases: Γ(1,λ) = E(λ), Γ(n/2,1/2) = χ²(n)
Conjugate prior for Poisson rate parameter

Parameters:

α > 0 (shape), λ > 0 (rate)

Support:

x > 0

Chi-Square Distribution χ²(n)

Continuous

Mathematical Formulas:

pdf:f(x) = (1/(2^(n/2)Γ(n/2))) x^(n/2-1) e^(-x/2)

cdf:F(x) = γ(n/2, x/2) / Γ(n/2)

mean:E[X] = n

variance:Var(X) = 2n

mgf:M(t) = (1-2t)^(-n/2) for t < 1/2

Key Properties:

Definition: Σᵢ₌₁ⁿ Xᵢ² where Xᵢ ~ N(0,1)
Additivity: χ²(n₁) + χ²(n₂) = χ²(n₁+n₂)
Non-central χ²: when Xᵢ ~ N(aᵢ,1), parameter λ = Σaᵢ²

Parameters:

n ≥ 1 (degrees of freedom)

Support:

x > 0

t-Distribution t(n)

Continuous

Mathematical Formulas:

pdf:f(t) = (Γ((n+1)/2))/(√(nπ)Γ(n/2)) (1+t²/n)^(-(n+1)/2)

cdf:F(t) = ∫₋∞ᵗ f(u) du (no closed form)

mean:E[T] = 0 (n ≥ 2)

variance:Var(T) = n/(n-2) (n ≥ 3)

mgf:Does not exist

Key Properties:

Definition: T = X/√(K/n) where X ~ N(0,1), K ~ χ²(n)
Symmetric around 0
Heavier tails than normal distribution
Limit: t(n) → N(0,1) as n → ∞

Parameters:

n ≥ 1 (degrees of freedom)

Support:

t ∈ ℝ

F-Distribution F(m,n)

Continuous

Mathematical Formulas:

pdf:f(x) = (Γ((m+n)/2))/(Γ(m/2)Γ(n/2)) (m/n)^(m/2) x^(m/2-1) (1+(m/n)x)^(-(m+n)/2)

cdf:F(x) = I(mx/(n+mx); m/2, n/2)

mean:E[F] = n/(n-2) (n > 2)

variance:Var(F) = (2n²(m+n-2))/(m(n-2)²(n-4)) (n > 4)

mgf:Does not exist in general

Key Properties:

Definition: F = (K₁/m)/(K₂/n) where K₁ ~ χ²(m), K₂ ~ χ²(n)
Reciprocal property: 1/F ~ F(n,m)
Quantile relation: F₁₋α(m,n) = 1/Fα(n,m)
Connection: t²(n) ~ F(1,n)

Parameters:

m,n ≥ 1 (degrees of freedom)

Support:

x > 0

Beta Distribution B(a,b)

Continuous

Mathematical Formulas:

pdf:f(x) = (Γ(a+b))/(Γ(a)Γ(b)) x^(a-1) (1-x)^(b-1)

cdf:F(x) = I(x; a, b) (regularized incomplete beta function)

mean:E[X] = a/(a+b)

variance:Var(X) = ab/((a+b)²(a+b+1))

mgf:Complex form involving hypergeometric functions

Key Properties:

Special case: B(1,1) = U(0,1)
Relationship to F: if X ~ B(a,b), then X/(1-X) ~ Fisher Z(a,b)
Conjugate prior for binomial probability parameter
Flexible shapes: uniform, U-shaped, bell-shaped

Parameters:

a,b > 0 (shape parameters)

Support:

0 < x < 1

Exponential Family Theory

Unified mathematical framework for many important statistical distributions

General Form:

f(x;θ) = c(θ) exp{Σⱼ Qⱼ(θ)Tⱼ(x)} h(x)

Components Breakdown:

c(θ)

Normalizing constant

Ensures probability integrates to 1

c(θ) = 1/∫ exp{Σⱼ Qⱼ(θ)Tⱼ(x)} h(x) dx

Qⱼ(θ)

Natural parameters

Transform original parameters to canonical form

η = Q(θ) maps parameter space to natural parameter space

Tⱼ(x)

Sufficient statistics

Capture all information about θ from data

T(x) = (T₁(x), T₂(x), ..., Tₖ(x))

h(x)

Base measure

Reference measure independent of parameters

h(x) ≥ 0 for all x in support

Canonical Form:

f(x;η) = exp{η·T(x) - A(η)} h(x)

Log-partition function:

A(η) = log ∫ exp{η·T(x)} h(x) dx

Important Properties:

Cumulant generating function: K(t) = A(η + t) - A(η)

Mean: E[T(X)] = ∇A(η)

Covariance: Cov[T(X)] = ∇²A(η)

Maximum likelihood estimator: closed form solution

Sufficient statistics: finite dimensional

Conjugate priors exist

Distribution Relationships

Mathematical connections and transformations between distribution families

Gamma Family

Γ(1, λ) = E(λ)

Exponential as special case of Gamma

Γ(n/2, 1/2) = χ²(n)

Chi-square as special case of Gamma

Γ(α₁,λ) + Γ(α₂,λ) = Γ(α₁+α₂,λ)

Additivity property

Normal Derivatives

(X-μ)/σ ~ N(0,1)

Standardization

Σ(Xᵢ-μ)²/σ² ~ χ²(n)

Sum of squared standardized normals

X̄ ~ N(μ, σ²/n)

Sample mean distribution

t and F Connections

t(n) = N(0,1)/√(χ²(n)/n)

t-distribution definition

F(m,n) = (χ²(m)/m)/(χ²(n)/n)

F-distribution definition

t²(n) = F(1,n)

Squared t is F with 1 numerator df

Beta and Related

B(1,1) = U(0,1)

Uniform as special case of Beta

Y/(1-Y) ~ Z(a,b) if Y ~ B(a,b)

Beta to Fisher Z transformation

(n/m)F ~ Z(n/2,m/2) if F ~ F(n,m)

F to Fisher Z transformation

Moment Formulas

Mathematical formulas for computing distribution moments and shape measures

Raw Moments

μ'ₙ = E[Xⁿ]

μ'₁ = E[X] (mean)

μ'₂ = E[X²]

μ'₃ = E[X³]

μ'₄ = E[X⁴]

Central Moments

μₙ = E[(X-μ)ⁿ]

μ₁ = 0 (by definition)

μ₂ = Var(X) (variance)

μ₃ = E[(X-μ)³] (skewness measure)

μ₄ = E[(X-μ)⁴] (kurtosis measure)

Standardized Moments

Dimensionless measures of distribution shape

Skewness: γ₁ = μ₃/σ³

Kurtosis: γ₂ = μ₄/σ⁴

Excess kurtosis: γ₂ - 3

Coefficient of variation: CV = σ/μ

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