What is beta distribution?
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. See also our Inverse Normal Distribution Calculator.
How do I use this beta distribution calculator?
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.
What is the probability density function (PDF) of the beta distribution?
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.
How do I calculate the expected value (mean) of a beta distribution?
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. You might also find our Probability Density f(x) — Cauchy Distribution useful.
How do I check if a beta distribution is symmetric?
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.
Why is beta distribution popular in Bayesian inference?
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.
What do alpha and beta control in beta distribution?
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.
What is the variance of a beta distribution?
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.