api: YAML JSON TOON Database
version: 1.0.0
format: yaml
dataset:
  id: 1646
  slug: probability-distribution-reference
  title: Probability Distribution Selection Reference
  description: Common probability distributions with support, mean, and practical modeling boundaries.
  summary: Common probability distributions with support, mean, and practical modeling boundaries. Each row includes a practical application and a failure to check before using it.
  use_when:
    - Choose a distribution by support and mechanism.
    - Check parameter conventions before comparing implementations.
  caveats:
    - All parameters must satisfy the distribution’s domain constraints.
    - A good fit does not establish the mechanism that generated the data.
  related:
    - experimental-design-pitfalls
    - statistics-basics
  reviewed_at: "2026-10-12"
  sources:
    - title: NIST probability distributions handbook
      url: "https://www.itl.nist.gov/div898/handbook/eda/section3/eda36.htm"
  reading:
    - choose-a-probability-distribution-with-explicit-assumptions
  provenance: Original editorial reference notes by YJTOON, informed by the linked primary documentation. External specifications retain their own licenses.
  entry_count: 10
  fields:
    - section: data
      field: distribution
      type: string
      example: Bernoulli(p)
    - section: data
      field: support
      type: string
      example: 0 or 1
    - section: data
      field: mean
      type: string
      example: p
    - section: data
      field: use_and_boundary
      type: string
      example: One binary trial; define what counts as success.
  category: Mathematics
  category_slug: mathematics
  tags: mathematics,probability distribution reference,reference,engineering
  view_count: 1
  created_at: 1791798640
  updated_at: 1791798640
data:
  - distribution: Bernoulli(p)
    support: 0 or 1
    mean: p
    use_and_boundary: One binary trial; define what counts as success.
  - distribution: Binomial(n,p)
    support: Integers 0 through n
    mean: n*p
    use_and_boundary: Count successes in n independent equal-probability trials.
  - distribution: Geometric(p)
    support: Integers 1 and above
    mean: 1/p
    use_and_boundary: Count trials until first success; another convention counts failures from zero.
  - distribution: Poisson(lambda)
    support: Nonnegative integers
    mean: lambda
    use_and_boundary: Count events under a constant-rate independent-increment model.
  - distribution: Exponential(lambda)
    support: Nonnegative real values
    mean: 1/lambda
    use_and_boundary: Model memoryless waiting time; not every duration is memoryless.
  - distribution: Uniform(a,b)
    support: Real values from a to b
    mean: (a+b)/2
    use_and_boundary: Equal density on a bounded interval; endpoints have zero point probability.
  - distribution: Normal(mu,sigma)
    support: All real values
    mean: mu
    use_and_boundary: Model symmetric fluctuations; negative values may be inappropriate for some quantities.
  - distribution: Lognormal(mu,sigma)
    support: Positive real values
    mean: exp(mu+sigma^2/2)
    use_and_boundary: Model positive multiplicative variability; parameters describe the underlying normal.
  - distribution: Beta(alpha,beta)
    support: Real values from 0 to 1
    mean: alpha/(alpha+beta)
    use_and_boundary: Model a bounded proportion with positive shape parameters.
  - distribution: Gamma(shape,scale)
    support: Positive real values
    mean: shape*scale
    use_and_boundary: Model positive skewed quantities; rate and scale parameterizations differ.