What is α0 (concentration)?

α0=Σα_i is the concentration (strength). Larger α0 makes samples cluster near the mean; smaller α0 creates more variability and corner-heavy samples.

What is mean + concentration input?

Instead of editing α directly, you can input a mean vector m (sums to 1) and a concentration α0, then α=m×α0 is computed automatically.

Dirichlet Distribution Generator & Visualizer (probability vectors)

What is a Dirichlet distribution?

A Dirichlet distribution is a distribution over probability vectors (x1,…,xK) where each component is non‑negative and the total sums to 1. This space is called a simplex.

α (alpha) can be interpreted like pseudo‑counts. The relative sizes of α determine the mean vector.
α0 = Σα_i is the concentration (strength): larger α0 ⇒ tighter around the mean; smaller α0 ⇒ more variability.
If some α_i < 1, samples tend to be sparse and stick to corners/edges; if all α_i > 1, mass is often inside the simplex.
K=2 is a special case: x1 ~ Beta(α1,α2) (this tool shows the Beta overlay and links to the Beta tool).

Common use cases: Bayesian priors for categorical probabilities, topic proportions, mixture weights, and probability‑like test data. You don’t need to enter personal information to use it.

Presets

Pick a practical preset (it regenerates instantly; you can tweak after applying).

Tip: For large K, use profile JSON for sharing instead of long URLs.

Generator

Choose a parameterization, generate samples, then inspect means, marginals, and diagnostics.

Parameterization

Dimension (K) Labels (comma-separated)

α (same for all components)

All components use α_i = α. Good starting point to see “corner vs center”.

Concentration (α0)

Enter a mean vector m (sums to 1). This tool derives α = m×α0.

Component	Mean (m_i)

All α values must be >0. Smaller values (<1) encourage sparse, corner-heavy vectors.

Component	α_i

Sample size (N) Bins (histograms)

RNG

Show components (marginals)

Up to 5 components are used for marginal histograms. (For large K, the checkbox list is hidden — use the index input.)

Show correlation heatmap (theory; small K only)

Preview rounding (decimals) Export rounding (optional) JSON mode

Copy format (preview)

Per-component stats

Component	Theory mean	Sample mean	Theory var	Sample var

Samples preview (first 20)

Profile JSON (save/restore settings)

Share URLs contain settings only. For large K, use profile JSON to save/restore without long URLs.

Import profile JSON

Tip: Don’t include confidential labels (customer names, etc.) in shared profiles.

FAQ

Why do components negatively correlate?

Because the components must sum to 1, increasing one component tends to decrease others. The theoretical covariance is negative for i≠j.

Why do samples stick to corners?

If any α_i<1 or α0 is small, the density can spike near simplex boundaries, creating sparse, corner-heavy vectors.

Does rounding affect Σ=1?

Yes. Rounding for export can make rows no longer sum to 1. Preview rounding is safe because it does not change the underlying samples.

Is seeded RNG secure?

No. Seeded mode is for reproducibility only. Use Secure (CSPRNG) for security-sensitive randomness.

Distributions hub
Browse distribution tools and randomness diagnostics.
Distribution sampler
A multi-distribution sampler (normal, gamma, beta, and more).
Beta distribution generator
K=2 Dirichlet is Beta — useful for single probability values.
Random JSON generator
Generate JSON test data (arrays & NDJSON).
Random CSV generator
Generate table-shaped test data in CSV.
Randomness tests
Quick sanity checks for randomness.
Probability & simulation guide
Learn and explore related topics.