Beta Distribution Calculator
Compute beta distribution probabilities, PDF, CDF, mean, variance, mode, and skewness with shape parameters α and β. Free online calculator with interactive PDF curve chart.
About This Calculator
The Beta Distribution Calculator computes probabilities and statistics for the beta distribution — a flexible family of continuous probability distributions defined on the interval [0, 1]. Given two positive shape parameters α (alpha) and β (beta), the calculator computes the probability density function (PDF), cumulative distribution function (CDF), mean, variance, skewness, and mode. The interactive PDF curve chart shows the full distribution shape across the [0, 1] domain.
The beta distribution's PDF formula is f(x; α, β) = x^(α-1) × (1-x)^(β-1) / B(α, β), where B(α, β) is the beta function. The CDF is computed using the regularized incomplete beta function I_x(α, β) via continued fraction expansion. The mean equals α/(α+β), variance equals αβ/[(α+β)²(α+β+1)], and the mode is (α-1)/(α+β-2) when both α, β > 1.
The beta distribution is widely used in Bayesian statistics, project management (PERT), A/B testing, reliability engineering, and modeling random proportions. Its shape varies dramatically with parameters — from uniform (α=β=1) to U-shaped (α=β<1) to bell-shaped (α,β>1) to J-shaped (one parameter much smaller than the other). This calculator supports the standard beta distribution (also called the beta distribution of the first kind).
Regional Notes
The beta distribution is a universal mathematical concept used across all regions — in the United States it is commonly taught in AP Statistics and used in Bayesian A/B testing; in India it features in engineering and statistics curricula; and in the United Kingdom it appears in A-level Further Statistics and university-level Bayesian analysis courses. The mathematical formulas and interpretation are identical worldwide.
Frequently Asked Questions
What is the beta distribution?
The beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is parameterized by two positive shape parameters, alpha (α) and beta (β), which control the shape of the distribution. Depending on the values of α and β, the beta distribution can be symmetric, skewed, U-shaped, bell-shaped, or J-shaped, making it one of the most flexible families of probability distributions.
How do you calculate the PDF of a beta distribution?
The probability density function (PDF) of the beta distribution is f(x; α, β) = x^(α-1) × (1-x)^(β-1) / B(α, β), where B(α, β) is the beta function (normalizing constant). For x exactly 0 or 1, the PDF is defined as 0 when the shape parameter is less than 1, and is infinite in certain cases.
What is the mean and variance of a beta distribution?
The mean of a beta distribution is α/(α+β). The variance is (αβ)/((α+β)²(α+β+1)). The mode is (α-1)/(α+β-2) when both α and β are greater than 1. The skewness is 2(β-α)√(α+β+1)/((α+β+2)√(αβ)), where positive skewness occurs when α < β.
When is beta distribution used in real-world applications?
Beta distribution is widely used in Bayesian statistics as a conjugate prior for binomial, negative binomial, and geometric distributions. It is applied in project management (PERT), A/B testing, Bayesian A/B analysis, reliability engineering, modeling random proportions, and representing prior beliefs about probabilities in machine learning.
What does the beta distribution look like for different parameters?
When α = β, the beta distribution is symmetric about 0.5. If α = β = 1, it becomes a uniform distribution. If α = β < 1, the distribution is U-shaped. If α, β > 1, it is bell-shaped. When α < β, the distribution is right-skewed, and when α > β, it is left-skewed. The flexibility of shapes makes beta distribution valuable for modeling various types of data on the [0, 1] interval.
How do you calculate the cumulative probability (CDF) for beta distribution?
The cumulative distribution function (CDF) for the beta distribution is given by the regularized incomplete beta function I_x(α, β). This function does not have a simple closed form and is typically computed numerically using continued fraction expansion or series approximation. Our calculator uses continued fraction methods for accurate CDF computation.
Can x be outside the [0, 1] interval for beta distribution?
No, the beta distribution is only defined for values of x in the closed interval [0, 1]. The distribution has support only on this interval, meaning probability is zero outside [0, 1]. When x = 0 or x = 1, the PDF may be zero or infinite depending on the shape parameters.
What is the relationship between beta distribution and gamma distribution?
Beta distribution is related to the gamma distribution: if X ~ Gamma(α, θ) and Y ~ Gamma(β, θ) are independent, then X/(X+Y) follows a Beta(α, β) distribution. This relationship explains why the beta distribution is naturally defined on [0, 1] and why it serves as the conjugate prior for binomial proportions.