Probability Distributions
Select the distribution that best represents the uncertainty in your input data.
Quick Selection Guide
Uncertain about shape? → Uniform or Triangular
Symmetric around a mean? → Normal
Positive only, right-skewed? → Log-Normal, Gamma, or Weibull
Binary outcome? → Bernoulli
Counting events? → Poisson
Modeling a probability? → Beta
Time until failure? → Exponential or Weibull
Normal (Gaussian)
The classic bell curve, centered on the mean. Use when values cluster symmetrically around a central tendency.
Common Applications
Measurement errors, human heights, test scores, short-term stock returns
Probability Density
f(x) = (1/σ√2π) × e^(-(x-μ)²/2σ²)
Uniform
Equal probability across all values in a range. Use when you know only the bounds, not the shape.
Common Applications
Random number generation, arrival times within intervals, equally likely outcomes
Probability Density
f(x) = 1/(max - min) for min ≤ x ≤ max
Triangular
A three-point estimate: minimum, most likely, and maximum. Ideal for expert elicitation when data is limited.
Common Applications
Project duration estimates, cost forecasting, subjective expert judgments
Probability Density
Peaks at mode, linear slopes to min and max
Log-Normal
Always positive, right-skewed. The natural logarithm of values follows a normal distribution.
Common Applications
Asset prices, income distributions, file sizes, multiplicative processes
Probability Density
f(x) = (1/xσ√2π) × e^(-(ln(x)-μ)²/2σ²)
Exponential
Models waiting times between independent events. Has the memoryless property—future wait time is independent of elapsed time.
Common Applications
Time between arrivals, equipment failure intervals, radioactive decay
Probability Density
f(x) = λe^(-λx) for x ≥ 0
Bernoulli
A single binary trial: success (1) with probability p, failure (0) with probability 1−p.
Common Applications
Yes/no decisions, pass/fail outcomes, event occurrence flags
Probability Density
P(X=1) = p, P(X=0) = 1-p
Poisson
Counts discrete events occurring independently at a constant average rate within a fixed interval.
Common Applications
Customer arrivals per hour, defects per unit, website hits per minute
Probability Density
P(X=k) = (λ^k × e^(-λ)) / k!
Beta
Flexible distribution bounded between 0 and 1. Shape adapts based on α and β parameters.
Common Applications
Probabilities, conversion rates, proportions, Bayesian priors
Probability Density
f(x) = x^(α-1)(1-x)^(β-1) / B(α,β)
Gamma
Positive-valued, right-skewed. Represents the sum of k independent exponential random variables.
Common Applications
Wait times, rainfall amounts, insurance claims, aggregate demand
Probability Density
f(x) = x^(k-1) × e^(-x/θ) / (θ^k × Γ(k))
Weibull
Versatile for reliability modeling. Shape parameter controls whether failure rate increases, decreases, or remains constant over time.
Common Applications
Equipment lifetime, material strength, wind speeds, survival analysis
Probability Density
f(x) = (k/λ)(x/λ)^(k-1) × e^(-(x/λ)^k)
Student's t
Similar to normal but with heavier tails, allowing more probability mass at extreme values. Approaches normal as df increases.
Common Applications
Small sample inference, financial returns with fat tails, robust estimation
Probability Density
Heavier tails than normal; converges to normal as df → ∞