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flowchart TD
Multi["Multinomial"] -->|"K = 2"| Bin["Binomial"]
Bin -->|"n = 1"| Ber["Bernoulli"]
Bin -.->|"CLT"| Norm["Normal"]
Bin -.->|"n large, p small"| Poi["Poisson"]
Poi -.->|"large λ"| Norm
Norm -->|"sum of squares"| ChiSq["Chi-squared"]
Norm -->|"N / √(χ²/n)"| Tdist["Student t"]
ChiSq -->|"ratio"| Fdist["F"]
Tdist -->|"t²"| Fdist
Tdist -.->|"large df"| Norm
Gam["Gamma"] -->|"k = 1"| Exp["Exponential"]
Gam -->|"k = n/2"| ChiSq
Wei["Weibull"] -->|"k = 1"| Exp
Dir["Dirichlet"] -->|"K = 2"| Beta["Beta"]
GEV["GEV"] -->|"ξ = 0"| Gum["Gumbel"]
Beta -.->|"conjugate"| Bin
Gam -.->|"conjugate"| Poi
Dir -.->|"conjugate"| Multi
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classDef extended fill:#F3E5F5,stroke:#7B1FA2,stroke-width:2px,color:#4A148C
class Multi,Bin,Ber,Poi discrete
class Norm,ChiSq,Tdist,Fdist classical
class Gam,Exp,Beta,Wei applied
class Dir,GEV,Gum extended
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51 Distribution Relationship Map
Probability distributions do not exist in isolation. They form an interconnected web where one distribution emerges as a special case of another, transforms into a third under algebraic operations, or converges to a fourth as a parameter tends to infinity. Understanding these relationships deepens intuition, simplifies computation (a Normal approximation often replaces an intractable exact distribution), and reveals the internal coherence of statistical theory.
The diagram below shows the central skeleton of this network for 15 hub distributions. Solid arrows denote special-case or transformation links; dashed arrows mark limiting approximations and Bayesian conjugate-prior relationships. The full interactive map — with 38 distributions and 54 edges — is available in the Shiny app that follows.
51.1 Interactive Relationship Map
The static diagram above shows only the main skeleton. The interactive map below includes all 38 distributions covered in this handbook and 54 typed relationships. Use the sidebar controls to filter by edge type and distribution group. Click any node to see its PDF/PMF formula, support, and links to the relevant chapter and Shiny app.
51.2 Relationship Types
The map encodes four types of relationships, each represented by a distinct visual style.
51.2.1 Special Case / Generalization
A distribution \(A\) is a special case of distribution \(B\) when fixing one or more parameters of \(B\) recovers \(A\) exactly. The arrow points from the more general distribution to the special case. For example, the Bernoulli distribution is a Binomial with \(n = 1\), the Exponential is a Gamma with shape \(k = 1\), and the Beta distribution is a Dirichlet with \(K = 2\). Recognising special-case links avoids redundant derivations: properties of the Bernoulli follow immediately from the Binomial, and marginal distributions of the Dirichlet are Beta by construction.
51.2.2 Transformation
Two distributions are linked by a transformation when an algebraic operation on a random variable from one distribution produces a random variable from the other. The sum of squared standard Normal variates follows a Chi-squared distribution; the ratio of two independent Chi-squared variates (each divided by their degrees of freedom) follows an F distribution; and squaring a Student \(t\) variate yields an \(F(1, n)\) variate. These relationships underpin the derivations of classical test statistics.
51.2.3 Limiting Approximation
A limiting link indicates that one distribution converges to another as a parameter tends to a boundary value. The Binomial converges to the Normal (Central Limit Theorem) and to the Poisson (rare-event limit). The Student \(t\) converges to the Normal as degrees of freedom grow. These approximations justify practical shortcuts: for large samples, a \(z\)-test replaces the \(t\)-test, and a Normal approximation replaces an exact Binomial calculation.
51.2.4 Bayesian Conjugate Prior
A conjugate prior link connects a prior distribution to the likelihood it is conjugate to. The Beta distribution is the conjugate prior for Bernoulli and Binomial likelihoods, the Gamma is conjugate for the Poisson rate, and the Dirichlet is conjugate for the Multinomial. Conjugacy guarantees that the posterior belongs to the same family as the prior, enabling closed-form Bayesian updating without numerical integration.
51.3 Legend
The colour and shape conventions used in the interactive map are summarised below.
Node groups:
| Group | Colour | Shape | Distributions |
|---|---|---|---|
| Discrete | Light blue | Square | Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Multinomial, Poisson |
| Classical inference | Light green | Circle | Normal, Chi, Chi-squared (1p & 2p), Student t, F |
| Applied modelling | Light orange | Circle | Exponential, Lognormal, Gamma, Beta, Weibull, Pareto, Inv. Gamma, Rayleigh, Erlang, Logistic, Laplace, Gumbel, Cauchy, Triangular, Power, Beta Prime, Corr. r |
| Extended toolkit | Light purple | Diamond | Dirichlet, GEV, Fréchet, Noncentral t, Noncentral F, Inv. Chi-squared, Maxwell-Boltzmann |
Edge types:
| Relationship | Colour | Line style |
|---|---|---|
| Special case / generalization | Blue | Solid |
| Transformation | Orange | Solid |
| Limiting / approximation | Green | Dashed |
| Conjugate prior | Purple | Dotted |