Beta distribution

The Beta distribution lives on the interval (0, 1), making it the natural choice to model probabilities, proportions, and rates. Its two shape parameters, $α$ and $β$ , make it extraordinarily flexible.

f (x; α, β) = \frac{x ^{α - 1} ( 1 - x ) ^{β - 1}}{B ( α , β )}

mean = 0.286 mode = 0.200 Think of it as: α−1 = 1.0 heads, β−1 = 4.0 tails observed

α (alpha) 2.0 β (beta) 5.0

Beta (α, β)

PDF. When used as a coin-flip prior, α−1 observed heads and β−1 tails give this posterior shape.

Shape guide

α, β	Shape
α = β = 1	Uniform — no information
α = β > 1	Symmetric bell — certainty near the centre
α > β	Right-skewed — favours higher values
α < 1 or β < 1	U-shaped or J-shaped — mass at the extremes

The Bayesian story

If you start with a flat prior (Beta(1, 1)) and observe h heads and t tails from a coin, the posterior probability of the coin’s bias is Beta(1 + h, 1 + t). The annotation below the chart translates: α − 1 = heads observed, β − 1 = tails observed.

This is Bayesian updating in closed form. Each new flip shifts the distribution, and with enough evidence the posterior narrows to a spike around the true bias — regardless of what you started with.