How this calculator works
Both shapes must be positive, with a numerical limit of 10,000 each. Equal shapes produce symmetry around 0.5. Shapes below one may produce an unbounded endpoint density, which is reported in words while cumulative probabilities remain finite. The support is fixed at [0, 1].
Formula
f(x) = x^(α-1)(1-x)^(β-1)/B(α,β); E[X] = α/(α+β); Var[X] = αβ/[(α+β)²(α+β+1)].
Worked example
For α = β = 2, the density at 0.5 is 1.5, the cumulative probability is 0.5 and the variance is 0.05.
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Frequently asked questions
Does α = β = 1 have a special meaning?
It gives the uniform distribution on [0, 1].
Can I enter a percentage as x?
Enter the proportion itself: use 0.25 for 25%. The cumulative-probability field has a separate percent selector.
Are zero shapes allowed?
No. This tool handles ordinary continuous beta distributions, rather than limiting point masses.