Probability Distributions
Select the distribution that best represents the uncertainty in your input data.
AI Distribution Suggestion
Not sure which distribution to use? Click the Suggest button in the variable block's detail panel to let AI recommend the most appropriate distribution with parameters.
| Feature | |
|---|---|
| How it works | AI analyzes the variable name and context to suggest a distribution type and realistic parameter values as a preview—your current distribution is never changed until you confirm |
| Compare & decide | The suggestion card shows both your current distribution (dimmed) and the AI suggestion (highlighted) with a preview curve so you can compare before applying |
| Apply or dismiss | Click ✓ to apply the suggestion to your block, or ✕ to dismiss without changes |
| Session memory | Suggestions persist per block during your session, so you can switch between blocks and return to review |
Normal (Gaussian)
meanstdThe classic bell curve, centered on the mean. Use when values cluster symmetrically around a central tendency.
| Property | |
|---|---|
| Applications | Measurement errors, human heights, test scores, short-term stock returns |
| Density | f(x) = (1/σ√2π) × e^(-(x-μ)²/2σ²) |
Uniform
minmaxEqual probability across all values in a range. Use when you know only the bounds, not the shape.
| Property | |
|---|---|
| Applications | Random number generation, arrival times within intervals, equally likely outcomes |
| Density | f(x) = 1/(max - min) for min ≤ x ≤ max |
Triangular
minmodemaxA three-point estimate: minimum, most likely, and maximum. Ideal for expert elicitation when data is limited.
| Property | |
|---|---|
| Applications | Project duration estimates, cost forecasting, subjective expert judgments |
| Density | Peaks at mode, linear slopes to min and max |
PERT
minmodemaxA modified Beta distribution that places 4× more weight on the mode than Triangular. Produces smoother, more realistic estimates for expert judgment.
| Property | |
|---|---|
| Applications | Project duration estimates, agile sprint planning, cost forecasting, risk analysis |
| Density | Beta(α, β) scaled to [min, max] where α = 1 + 4(mode-min)/(max-min) |
Log-Normal
meanstdAlways positive, right-skewed. In Carlo, the mean and std parameters represent the arithmetic mean and standard deviation of the output distribution (not log-space μ/σ). For example, mean=600, std=150 produces values centered around 600.
| Property | |
|---|---|
| Applications | Asset prices, income distributions, file sizes, multiplicative processes |
| Density | f(x) = (1/xσ√2π) × e^(-(ln(x)-μ)²/2σ²) |
Exponential
rate (λ)Models waiting times between independent events. Has the memoryless property—future wait time is independent of elapsed time.
| Property | |
|---|---|
| Applications | Time between arrivals, equipment failure intervals, radioactive decay |
| Density | f(x) = λe^(-λx) for x ≥ 0 |
Bernoulli
probability (p)A single binary trial: success (1) with probability p, failure (0) with probability 1−p.
| Property | |
|---|---|
| Applications | Yes/no decisions, pass/fail outcomes, event occurrence flags |
| Density | P(X=1) = p, P(X=0) = 1-p |
Binomial
n (trials)p (probability)Counts the number of successes in n independent Bernoulli trials. Each trial has the same probability p of success.
| Property | |
|---|---|
| Applications | Conversions out of visitors, defects per batch, coin flips, survey responses, A/B test success counts |
| Density | P(X=k) = C(n,k) × p^k × (1-p)^(n-k) |
Poisson
lambda (λ)Counts discrete events occurring independently at a constant average rate within a fixed interval.
| Property | |
|---|---|
| Applications | Customer arrivals per hour, defects per unit, website hits per minute |
| Density | P(X=k) = (λ^k × e^(-λ)) / k! |
Beta
alpha (α)beta (β)Flexible distribution bounded between 0 and 1. Shape adapts based on α and β parameters.
| Property | |
|---|---|
| Applications | Probabilities, conversion rates, proportions, Bayesian priors |
| Density | f(x) = x^(α-1)(1-x)^(β-1) / B(α,β) |
Gamma
shape (k)scale (θ)Positive-valued, right-skewed. Represents the sum of k independent exponential random variables.
| Property | |
|---|---|
| Applications | Wait times, rainfall amounts, insurance claims, aggregate demand |
| Density | f(x) = x^(k-1) × e^(-x/θ) / (θ^k × Γ(k)) |
Weibull
shape (k)scale (λ)Versatile for reliability modeling. Shape parameter controls whether failure rate increases, decreases, or remains constant over time.
| Property | |
|---|---|
| Applications | Equipment lifetime, material strength, wind speeds, survival analysis |
| Density | f(x) = (k/λ)(x/λ)^(k-1) × e^(-(x/λ)^k) |
Student's t
degrees of freedom (df)Similar to normal but with heavier tails, allowing more probability mass at extreme values. Approaches normal as df increases.
| Property | |
|---|---|
| Applications | Small sample inference, financial returns with fat tails, robust estimation |
| Density | Heavier tails than normal; converges to normal as df → ∞ |