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Choose a probability distribution by matching the variable’s type and support first, then verifying how the data were generated and which parameter convention your reference uses. Counts, measurements, proportions, waiting times and inferential statistics require different families; a familiar curve is not enough.
A defensible selection sequence
- Classify the outcome. Decide whether observations are discrete (individual values have probability mass) or continuous (probability is assigned to intervals through a density). NIST’s distribution gallery separates families this way.
- Check the support. Confirm whether values can be any real number, only nonnegative, restricted to [0,1], or integers from zero to a fixed maximum. A model whose support permits impossible values is not appropriate without an explicit transformation or censoring model.
- State the generating assumptions. For example, the basic binomial model requires a fixed number of trials, two mutually exclusive outcomes per trial and the same success probability on every trial (NIST’s binomial definition).
- Write parameter conventions beside symbols. “Rate” and “scale” can be reciprocals. NIST notes that references may use different, mathematically equivalent parameterizations (gallery).
- Name the purpose. A distribution for describing or generating observed data is not automatically the right reference distribution for a test statistic. The t distribution, for instance, is chiefly used for confidence intervals and hypothesis tests rather than as a general data-generating model (NIST’s t entry).
Common discrete distributions
| Family | Outcome and support | Parameters and assumptions | Typical use and cautions |
|---|---|---|---|
| Bernoulli | One binary outcome: 0 or 1. | Success probability p. | One trial. A binomial distribution with n = 1 is the corresponding repeated-trial special case. |
| Binomial | Success count X ∈ {0,…,n}. | Fixed n trials and fixed success probability p. | Number of successes under those assumptions. Its probability is P(X=x)=C(n,x)px(1−p)n−x; mean is np and standard deviation is √(np(1−p)) (NIST). |
| Poisson | Nonnegative integer event count. | Usually a rate/mean λ tied to a stated exposure. | Candidate for event counts, but support alone is insufficient: define exposure and assess independence, stationarity and heterogeneity rather than assuming a Poisson process. |
| Discrete uniform | Values in a finite, explicitly stated set. | Every listed value has equal probability. | Use only when equal probabilities are substantively justified; it is not the same as a continuous uniform distribution. |
Common continuous distributions
| Family | Support and shape | Parameters | When it is useful |
|---|---|---|---|
| Normal (Gaussian) | All real values; symmetric bell-shaped density. | Location μ and scale σ (often reported through variance σ²). | Symmetric measurement variation and many modeling residuals when diagnostics support it. NIST defines the location and scale form at its glossary. |
| Student t | All real values; symmetric with heavier tails at lower degrees of freedom. | Degrees of freedom ν. | Critical values, confidence intervals and tests. NIST says its shape approaches normality as ν grows and calls the approximation quite good for ν > 30; that reference threshold is not a universal modeling rule (NIST). |
| Continuous uniform | Bounded interval [a, b] with constant density. | Lower and upper bounds a, b. | A bounded baseline when every value in the interval is equally plausible. For continuous variables, a density at one point is not a point probability; probabilities are areas over intervals. |
| Exponential | Nonnegative waiting time or lifetime. | Scale β > 0, or rate λ=1/β depending on convention. | Constant-hazard reliability or waiting-time settings. In the scale form, hazard is 1/β and survival is exp(−x/β) for x ≥ 0 (NIST). |
| Gamma | Positive values, often right-skewed. | Shape plus a second parameter specified as either scale or rate. | Flexible positive quantities and waiting-time models. Always label the second parameter’s convention. |
| Beta | Bounded continuous value in [0,1]. | Two shape parameters. | Proportions or probabilities whose mass near 0, 1 or the center is represented by suitable shapes. |
| Chi-square | Nonnegative continuous value. | Degrees of freedom. | Inferential procedures and variance-related statistics; specify the test and degrees of freedom. |
| F | Nonnegative continuous ratio family. | Numerator and denominator degrees of freedom. | Inferential procedures such as variance-ratio and ANOVA statistics; it is a reference distribution tied to that procedure. |
| Lognormal | Positive, often strongly right-skewed; log-transformed values are normal. | Location and scale on the log scale. | Positive multiplicative measurements when a log scale is scientifically plausible. |
| Weibull | Nonnegative lifetime or duration with flexible hazard. | Shape and scale (convention varies). | Reliability and survival settings where a constant hazard is implausible. |
| Cauchy | All real values with very heavy tails. | Location and scale. | Specialized robust or theoretical applications; its undefined mean and variance make ordinary moment-based summaries inappropriate. |
NIST’s gallery provides the broader list and standard forms, while noting that location/scale transformations and parameter conventions differ among references: Gallery of Distributions.
Normal, binomial and Poisson: the frequent comparison
| Question | Normal | Binomial | Poisson |
|---|---|---|---|
| Discrete or continuous? | Continuous. | Discrete. | Discrete. |
| Support | All real numbers. | Integers 0 through fixed n. | All nonnegative integers. |
| Core parameters | μ, σ. | n, p. | λ linked to exposure. |
| Process assumption | Symmetric continuous variation is a reasonable approximation. | Fixed number of comparable independent trials with common p. | Event-count process assumptions and explicit exposure. |
| Typical question | How does a measurement vary? | How many of n trials succeed? | How many events occur in an exposure? |
A normal approximation to a count may be useful in some inferential settings, but it does not change the count’s discrete support or prove that the underlying process is normal. Check the approximation’s purpose and error before using it.
Modeling distribution versus inferential reference distribution
Use a modeling distribution to represent how observations could arise or to generate plausible data. Use a reference distribution to calibrate a statistic, critical region or confidence interval. Student t, chi-square and F commonly serve the second role. Their appearance in a procedure does not mean the raw business measurement itself follows that family.
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Parameter and data-quality pitfalls
- Unlabeled λ: for exponential models, state whether λ is a rate or whether β is the scale; they are reciprocals in the cited one-parameter forms.
- Support violations: do not fit an unrestricted normal model to a quantity that cannot be negative or exceed 1 without explaining a transformation or approximation.
- Dependence and heterogeneity: repeated observations, changing event rates, clusters or mixtures can invalidate binomial or Poisson assumptions even when the histogram looks familiar.
- Censoring and exposure: survival times with censoring and event counts observed for unequal exposure require models that include those mechanisms.
- Visual overconfidence: “roughly normal” describes appearance, not proof of an underlying process or valid inferential assumptions.
- Formula mismatch: align parameter names, units and conventions before comparing formulas from two references; distinct expressions may be equivalent under different definitions.
A compact decision checklist
- Write the measurement and its units.
- Mark it as binary, bounded count, unbounded count, proportion, positive duration, bounded continuous value or unrestricted continuous value.
- List impossible values and physical bounds.
- Describe trial count, exposure, independence, hazard, censoring and heterogeneity as applicable.
- Choose candidate families whose support matches those facts.
- Declare every parameter convention (especially rate versus scale) and degrees of freedom.
- Use plots and diagnostics to check shape, tails and dependence, then assess whether the model’s purpose is description, prediction, simulation or inference.
Further reference
NIST maintains a broad survey of distribution families in its Engineering Statistics Handbook gallery. For a historical survey of distribution tables, see Raghu N. Kacker and I. Olkin, A Survey of Tables of Probability Distributions, published in 2005 in the Journal of Research of NIST: NIST publication page.
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