Beta Distribution Calculator

Beta distribution is a family of continuous probability distributions defined on the interval [0, 1]. It is parameterized by two positive shape parameters, α (alpha) and β (beta), which control the shape of the distribution. Depending on these parameters, it can be symmetric, skewed left, skewed right, U-shaped, or uniform.

Enter your desired shape parameters Alpha (α) and Beta (β), both of which must be positive numbers. Then enter a value of x between 0 and 1, and choose whether you want a left-tail P(X < x) or right-tail P(X > x) probability. The calculator will display the cumulative probability, PDF value, mean, variance, and standard deviation.

The PDF of the beta distribution is f(x) = [Γ(α+β) / (Γ(α)·Γ(β))] · x^(α−1) · (1−x)^(β−1) for x ∈ [0, 1]. The Gamma function Γ serves as a normalizing constant. The shape of this curve varies widely depending on the values of α and β.

The mean of a beta distribution is calculated as μ = α / (α + β). For example, if α = 2 and β = 5, the mean is 2 / (2 + 5) = 2/7 ≈ 0.2857.

A beta distribution is symmetric when α = β. In that case, the distribution is symmetric around x = 0.5. When α = β = 1, it becomes a uniform distribution. When α = β > 1, it is bell-shaped and symmetric.

Beta distribution is the conjugate prior for the Bernoulli and binomial likelihoods, meaning that if you start with a beta prior and observe binomial data, the posterior distribution is also a beta distribution. This makes it mathematically convenient for modeling probabilities, success rates, and proportions in Bayesian analysis.

Alpha (α) and beta (β) control the shape of the distribution. Higher α shifts probability mass toward 1, while higher β shifts it toward 0. When both are equal to 1, the distribution is uniform. When both are less than 1, the distribution is U-shaped. When both are greater than 1, it is unimodal and bell-shaped.

The variance of a beta distribution is σ² = (α · β) / [(α + β)² · (α + β + 1)]. This measures how spread out the distribution is. As α + β increases (with their ratio fixed), the variance decreases and the distribution becomes more concentrated.