The Beta Distribution Calculator helps you understand this versatile continuous probability distribution. Quickly compute the Probability Density Function (PDF), Mean, and Variance for specific shape parameters (alpha & beta) and a given probability (x), essential for Bayesian analysis and modeling proportions.
Formula:
Beta Distribution Formulas
The Beta distribution is defined by two positive shape parameters, α (alpha) and β (beta), and is defined on the interval [0, 1].
- Probability Density Function (PDF):
x: The probability value (between 0 and 1)α: Shape parameter alpha (a positive real number)β: Shape parameter beta (a positive real number)B(α, β): The Beta function, B(α, β) = Γ(α)Γ(β) / Γ(α+β)Γ: The Gamma function- Mean (Expected Value):
- Variance:
f(x; α, β) = (x(α-1) × (1-x)(β-1)) / B(α, β)
Where:
E[X] = α / (α + β)
Var[X] = (α × β) / ((α + β)2 × (α + β + 1))
Note: The Cumulative Distribution Function (CDF) involves the regularized incomplete beta function, which is computationally intensive and typically requires specialized statistical software or libraries for precise calculation. This calculator focuses on PDF, Mean, and Variance.