Discrete
| Name | Support | Mean | Variance | Used for |
|---|---|---|---|---|
| Bernoulli(p) | {0, 1} | p | p(1−p) | Single yes/no trial |
| Binomial(n, p) | 0..n | np | np(1−p) | Count of successes in n trials |
| Poisson(λ) | 0, 1, … | λ | λ | Rare-event counts (arrivals) |
| Geometric(p) | 1, 2, … | 1/p | (1−p)/p² | Trials until first success |
| Negative binomial | 0, 1, … | r(1−p)/p | r(1−p)/p² | Overdispersed counts |
| Uniform {1..n} | 1..n | (n+1)/2 | (n²−1)/12 | Discrete fair |
Continuous
| Name | Support | Mean | Variance | Used for |
|---|---|---|---|---|
| Uniform(a, b) | [a, b] | (a+b)/2 | (b−a)²/12 | Fair between bounds |
| Normal(μ, σ²) | ℝ | μ | σ² | CLT limit; measurement error |
| Log-normal | (0, ∞) | e^{μ+σ²/2} | varies | Positive-skewed quantities |
| Exponential(λ) | [0, ∞) | 1/λ | 1/λ² | Waiting time (memoryless) |
| Gamma(k, θ) | [0, ∞) | kθ | kθ² | Sum of exponentials |
| Beta(α, β) | [0, 1] | α/(α+β) | varies | Proportion / rate |
| Chi-square(k) | [0, ∞) | k | 2k | Sum of squared normals |
| Student t(ν) | ℝ | 0 | ν/(ν−2) | Sample mean with unknown σ |
| F(ν₁, ν₂) | [0, ∞) | — | — | ANOVA, variance ratios |
| Cauchy | ℝ | undefined | undefined | Heavy-tailed |
| Weibull(k, λ) | [0, ∞) | varies | varies | Reliability / life |
Rules of thumb
- Normal approximation to binomial: np > 5 and n(1−p) > 5.
- Poisson approximation to binomial: n large, p small, np moderate.
- Exponential is memoryless — the only continuous distribution with this property.
- Central Limit Theorem: sum of many IID finite-variance RVs → normal.
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