Beta distribution

The beta distribution is a continuous probability distribution supported on the interval from zero to one. Its two positive shape parameters control its density, allowing forms that include uniform, U-shaped, and skewed distributions.

Probability density function: A probability density function describes the relative likelihood of values for a continuous random variable. The beta distribution’s shape is specified by its probability density.
Shape parameter: A shape parameter changes a probability distribution’s form without merely shifting or rescaling it. Both beta parameters alter how probability is arranged across the unit interval.
Bayesian statistics: Bayesian statistics updates probability distributions for unknown quantities using observed data and Bayes’ rule. The beta distribution is a standard prior and posterior for unknown Bernoulli probabilities.
Thomas Bayes: Thomas Bayes was an English statistician and Presbyterian minister whose work established an early form of Bayes’ theorem. Bayesian inference later made beta-distributed uncertainty about probabilities especially useful.
Beta-binomial distribution: The beta-binomial distribution is a discrete distribution obtained by mixing binomial probabilities over a beta-distributed success probability. Mixing binomial outcomes over this distribution produces a standard predictive model.
Expected value: The expected value is the probability-weighted average of a random variable’s possible values. For parameters α and β, the beta distribution’s mean is α/(α+β).
Bernoulli distribution: The Bernoulli distribution models a single binary outcome with a specified probability of success. A beta prior on its success probability yields a beta posterior after observations.
Pierre-Simon Laplace: Pierre-Simon Laplace was a French mathematician who developed probability theory and applied it extensively to inference. His work helped establish the mathematical framework in which beta integrals became useful.
Cumulative distribution function: A cumulative distribution function gives the probability that a random variable is at most a specified value. It converts the beta density into interval probabilities and quantiles.
Variance: Variance measures the expected squared deviation of a random variable from its mean. The two parameters jointly determine the distribution’s spread around its mean.

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