Inspect beta boundary behavior before choosing shape parameters

Define the task

A beta distribution describes a continuous quantity between zero and one using two positive shape parameters. Decide whether exact zeros or ones represent genuine point masses in the data. An ordinary continuous beta model assigns no positive probability to those exact points, even when its density grows near a boundary.

Carry out the check

The mean is a/(a+b), so the same mean can arise from very different shapes. Shapes below one can concentrate density toward a boundary, while both shapes above one produce an interior mode. Use the variance a×b divided by ((a+b)²(a+b+1)) to check the spread implied by a chosen pair. Label chosen parameters as assumptions until a justified fitting or elicitation process has supplied them.

Worked hypothetical example

Hypothetical shapes (0.5,0.5), (1,1) and (2,2) all have mean 0.5. Their variances are 0.125, approximately 0.08333 and 0.05. The first is U-shaped, the second is uniform, and the third peaks at the center. A mean alone therefore cannot identify a useful simulation distribution. Keep the intended boundary behavior in the design record.

Method checked 11 November 2026 against R documentation: The Beta Distribution.