Related Distributions
NIST/SEMATECH Section 1.3.6.7 Related Distributions
Probability distributions are connected through special cases, transformations, and limiting forms. Understanding these relationships helps select the right model and simplify computations. For critical value tables, see Distribution Tables.
Distribution Function Definitions
| Function | Symbol | Definition |
|---|---|---|
| PDF / PMF | f(x) | Probability density (continuous) or mass (discrete) |
| CDF | F(x) | P(X ≤ x), cumulative probability |
| Quantile (PPF) | Q(p) | Inverse CDF, F inverse of p |
| Survival | S(x) | 1 - F(x), exceedance probability |
| Hazard | h(x) | f(x) / S(x), instantaneous failure rate |
Normal Distribution Family
The Normal distribution is the central reference for much of classical statistics.
| Relationship | Description |
|---|---|
| Normal -> Lognormal | If X ~ Normal, then exp(X) ~ Lognormal |
| Normal -> Chi-Square | Sum of k squared standard normals ~ Chi-Square(k) |
| Normal -> t-Distribution | Z / sqrt(Chi-Square(k)/k) ~ t(k) |
| Chi-Square -> F-Distribution | (Chi-Square(d1)/d1) / (Chi-Square(d2)/d2) ~ F(d1,d2) |
| Normal -> Power Normal | Box-Cox transform of Normal |
| Lognormal -> Power Lognormal | Box-Cox transform of Lognormal |
Exponential Distribution Family
The Exponential distribution connects to several lifetime and reliability models.
| Relationship | Description |
|---|---|
| Exponential -> Gamma | Sum of k Exponentials ~ Gamma(k, beta) |
| Exponential -> Weibull | Weibull(shape=1) = Exponential |
| Exponential -> Double Exponential | Difference of two Exponentials |
| Gamma -> Chi-Square | Chi-Square(k) = Gamma(k/2, 2) |
| Gamma -> Exponential | Gamma(1, beta) = Exponential(beta) |
| Exponential -> Fatigue Life | Derived via cycle-to-failure model |
Limiting Distributions
| From | To | Condition |
|---|---|---|
| t-Distribution | Normal | df -> infinity |
| Chi-Square | Normal | df -> infinity (approx.) |
| F-Distribution | Chi-Square | df2 -> infinity, multiply by df1 |
| Binomial | Normal | n -> infinity (CLT) |
| Binomial | Poisson | n -> infinity, p -> 0, np = lambda |
| Poisson | Normal | lambda -> infinity |
| Gamma | Normal | shape -> infinity (CLT) |
Flexible Distribution Families
| Distribution | Role |
|---|---|
| Beta | Flexible shape on [0,1]; models proportions. Uniform is Beta(1,1). |
| Tukey-Lambda | Shape parameter sweeps from Cauchy-like (lambda=-1) through logistic (lambda=0) to uniform-like (lambda=1). Used with PPCC Plot for distribution selection. |
| Extreme Value | Models maxima of large samples (Gumbel). Related to Weibull via transformation. |
Discrete Distributions
| Distribution | Support | Key Use |
|---|---|---|
| Binomial | 0, 1, …, n | Fixed trials, constant probability |
| Poisson | 0, 1, 2, … | Rare events per unit time/space |
Discrete distributions use PMF (probability mass function) rather than PDF. Both converge to Normal under appropriate limits (see table above).
Choosing a Distribution
Start with the PPCC Plot or Probability Plot to identify candidate families, then validate with Anderson-Darling or K-S Test. Use the relationships above to simplify analysis when transformations connect your data to a better-known distribution.