Map of Probability Distributions

How distributions relate to each other — special cases, limiting cases, transformations, mixtures, and conjugate priors — split into 8 focused panels instead of one dense poster. Every solid-border box links to that distribution's full reference page.

How to read the arrows:Special caseLimiting caseMixture / compoundBayesian / conjugateSum / aggregationTransformationSolid-border boxes link to a full reference page · dashed-border boxes aren't published yet

Discrete Distributions

Panel 1 of 8

Distributions over counts: successes in trials, events per interval, or categories per draw.

Building blocks

Bernoulli (p)

0/1 outcome

n=1

Binomial (n, p)

successes in n trials

n→∞, p=λ/n

Poisson (λ)

events in an interval; scales to Poisson(λt) over time t

Categorical (p_{1}, \dots, p_{k})

one trial, k categories — reduces to Bernoulli when k=2, and to Multinomial's n=1 case

Waiting-time & sampling variants

Geometric (p)

trials to 1st success

r=1

NegBinomial (r, p)

trials to r-th success

Hypergeometric (N, K, n)

successes, sampling w/o replacement

N large

→ Binomial(n,p) (see above)

Hypergeometric (N, K, n)

(same box as above)

ψ=1

Fisher’s / Wallenius’ Noncentral HG

biased (weighted) sampling

Multi-category

Multinomial (n, p_{1}, \dots, p_{k})

counts in k categories

conjugate prior

Dirichlet (α_{1}, \dots, α_{k})

on the simplex

Overdispersed & compound variants

Beta - Binomial (n, α, β)

Binomial with p ~ Beta

Dirichlet-Multinomial (n, α_{1}, \dots, α_{k})

Multinomial with p ~ Dirichlet

NegBinomial (r, p)

also arises as Poisson with λ ~ Gamma

Zero-Inflated Poisson

Hurdle Poisson

Zero-Inflated NegBinomial

Hurdle NegBinomial

Each box above compounds or truncates the count distribution it's built from: Binomial → Beta-Binomial, Multinomial → Dirichlet-Multinomial, Poisson → Negative Binomial / its Zero-Inflated & Hurdle variants, Negative Binomial → its own Zero-Inflated & Hurdle variants.

Not shown: Discrete Uniform, Skellam, Zipf, Zipf–Mandelbrot, Logarithmic, Yule–Simon, Conway–Maxwell–Poisson, Negative-/Multivariate-Hypergeometric — independent or specialized enough to skip the family tree below. Browse all 65 in the full Distributions library.

Continuous: The Normal Family

Panel 2 of 8

The bell curve and everything built directly from it: absolute-value and squaring transforms, and the t/F/Chi-Square trio used throughout hypothesis testing.

Normal & its absolute-value transforms

Normal (μ, σ^{2})

the bell curve

Y=|X|

Folded Normal (μ, σ)

|X| for general μ

μ=0

Half-Normal (σ)

|X| when X is zero-mean

t / F / Chi-Square family

χ_{ν}^{2}

= Gamma(ν/2, ½)

√

χ_{ν}

Chi distribution

ν=2

Rayleigh (σ)

= σ·√(Chi₂)

t_{ν}

Student's t

ν→∞

Normal (0, 1)

standard normal

t_{ν}

(same box as above)

T²

F (1, ν)

F with 1 numerator df

F (d_{1}, d_{2})

ratio of two scaled χ²'s

d₂→∞ (× d₁)

χ_{d_{1}}^{2}

(same box as above)

Not shown: Cauchy (= t with 1 degree of freedom, grouped instead with the heavy-tailed families on the Bounded (0,1) & Other Families panel), and the noncentral t / F / Chi-Square / Beta distributions (same families with an added noncentrality parameter, one reference page each). Browse all 65 in the full Distributions library.

Continuous: Positive (0, ∞) Distributions

Panel 3 of 8

Waiting times, hazards, and other quantities that can't go negative.

Exponential family & waiting times

Exponential (λ)

memoryless

k=1

Weibull (k, λ)

flexible hazard

Exponential (λ)

(same box as above)

α=1

Gamma (α, λ)

waiting time for the α-th event; sum of α iid Exponentials (Erlang) when α is an integer

Gamma (α, λ)

(same box as above)

Y=1/X

Inverse Gamma (α, λ)

Log-transform & flexible-tail families

Normal (μ, σ^{2})

see Normal-family panel

Y=eˣ

Log-Normal (μ, σ^{2})

log of normal

Log-Logistic (α, β)

heavy tail, sigmoidal survival curve

Generalized Gamma (δ, μ, σ)

superfamily — contains Gamma, Weibull, Exponential, Chi-Square as special cases and Log-Normal as a limiting case

Inverse Gaussian (λ, μ)

first-passage time of Brownian motion with drift

Not shown: Lévy (a special case of Inverse Gamma at shape ½). Browse all 65 in the full Distributions library.

Continuous: Bounded (0,1) & Other Families

Panel 4 of 8

Distributions confined to a fixed interval, plus the common heavy-tailed and flexible-shape families that don't fit neatly elsewhere — closing with a quick-reference card of transformation identities.

Bounded (0,1)

Uniform (0, 1)

α=β=1

Beta (α, β)

very flexible; see Beta-Binomial & Dirichlet on other panels

Kumaraswamy (α, β)

Beta-like, closed-form CDF (not literally nested)

Other important families

Skew Normal (ξ, ω, α)

reduces to Normal at α=0

Cauchy (x_{0}, γ)

heavy tails; = Student's t with 1 df

Laplace (μ, b)

double exponential

Logistic (μ, s)

sigmoid CDF

Generalized Pareto (ξ, σ, μ)

peaks-over-threshold

Johnson S_{U} /S_{B}

transformation family

Asymmetric Laplace (μ, b, κ)

quantile regression

Tweedie (p, ϕ, μ)

1<p<2: compound Poisson-Gamma

Common transformations (quick reference)

X \sim N (μ, σ^{2}) ⟹ a X + b \sim N (a μ + b, a^{2} σ^{2})

X \sim N (μ, σ^{2}) ⟹ e^{X} \sim LogNormal (μ, σ^{2})

X \sim Gamma (α, λ) ⟹ c X \sim Gamma (α, λ / c)

X \sim Beta (α, β) ⟹ 1 - X \sim Beta (β, α)

X_{1} \sim χ_{ν_{1}}^{2}, X_{2} \sim χ_{ν_{2}}^{2} indep. ⟹ X_{1} + X_{2} \sim χ_{ν_{1} + ν_{2}}^{2}

Not shown: the Generalized Pareto distribution is covered on the Extreme Value Theory panel, and Tweedie's p=1/p=2 boundary cases (scaled Poisson and Gamma) are covered on the Discrete and Positive-distributions panels respectively. Browse all 65 in the full Distributions library.

Multivariate & Matrix Distributions

Panel 5 of 8

Vectors and matrices of random variables, and the distributions used for their covariance structure and Bayesian updating.

Multivariate Normal & scatter matrices

MVN (μ, Σ)

vector of normals

scatter matrix of n draws

Wishart_{ν} (Σ)

Wishart_{ν} (Σ)

(same box as above)

W⁻¹

Inverse Wishart_{ν} (Ψ)

conjugate prior for Σ

MV- t_{ν} (μ, Σ)

heavier-tailed multivariate analog

ν→∞

MVN (μ, Σ)

(same box as above)

Matrix-valued & compositional

Matrix Normal (M, U, V)

normal in rows & columns

1 dim collapses

MVN (μ, Σ)

(same box as above)

Dirichlet (α_{1}, \dots, α_{k})

on the simplex — see Discrete panel

Logistic-Normal (μ, Σ)

alternative for compositional data

Joint conjugate priors

Normal (μ, σ^{2})

μ, σ² both unknown

joint prior

Normal-Inverse-Gamma

MVN (μ, Σ)

μ, Σ both unknown (same box as above)

joint prior

Normal-Inverse-Wishart

Browse all 65 in the full Distributions library.

Survival / Reliability (Time to Event)

Panel 6 of 8

The same waiting-time families from the Positive-distributions panel, viewed through their hazard function — how the instantaneous risk of the event changes over time.

Hazard-shape family

Exponential (λ)

constant hazard

k=1

Weibull (k, λ)

increasing/decreasing hazard

Weibull (k, λ)

(same box as above)

Log-Logistic (α, β)

sigmoidal survival curve

Log-Normal (μ, σ^{2})

common lifetime model

Accelerating & flexible-tail hazards

Gompertz (η, λ)

accelerating hazard

γ=0

Gompertz-Makeham (η, λ, γ)

adds age-independent mortality

Generalized Gamma (δ, μ, σ)

superfamily — see Positive-distributions panel

Not shown: Burr XII (very flexible tail, no reference page yet). Competing Risks & Mixture Models are modeling frameworks that combine several event-type distributions, not single named distributions, so they don't get their own card here.