the beta distribution
/ BAY-tuh /
You want to model a quantity that is a fraction or a probability — a batting average, the chance a coin is biased, the proportion of defective parts — something that must lie between 0 and 1. The beta distribution is the flexible, all-purpose shape for exactly such bounded fractions. By turning two dials it can be flat, bell-shaped, U-shaped, or skewed to either end.
Write p ~ Beta(alpha, beta) with two shape parameters alpha > 0 and beta > 0. The density is f(p) = p^(alpha-1) (1 - p)^(beta-1) / B(alpha, beta) for 0 < p < 1, where B(alpha, beta) is the beta FUNCTION, just the normalizing constant. The mean is E[p] = alpha/(alpha + beta) — read alpha as 'successes' and beta as 'failures', so the mean is the success fraction. Large alpha + beta concentrates the mass tightly; alpha = beta = 1 gives the flat Uniform(0,1); alpha = beta > 1 gives a symmetric bell centred at 0.5; alpha < 1 and beta < 1 gives a U-shape piled at both ends.
Beta's star role is as the conjugate prior for a binomial proportion in Bayesian inference: if your belief about a probability is Beta(alpha, beta) and you then observe s successes and f failures, your updated belief is simply Beta(alpha + s, beta + f) — you literally add the counts. This makes learning about a proportion almost arithmetic. The natural caveat: beta lives strictly on (0, 1), so it is the wrong model for any quantity that can exceed 1 or go negative.
You start agnostic about a coin's bias, Beta(1, 1) = Uniform(0,1). After seeing 7 heads and 3 tails your belief becomes Beta(1 + 7, 1 + 3) = Beta(8, 4), with mean 8/12 which is about 0.67 — leaning toward heads but not certain.
Beta updates a proportion belief just by adding observed successes and failures.
The beta DISTRIBUTION and the beta FUNCTION B(alpha, beta) share a name but are different objects — the function is merely the constant that scales the density to area 1.