Beta Distribution Calculator

What is the Beta Distribution?

The Beta distribution is a continuous probability distribution defined on the interval $[0, 1]$, parameterized by two positive shape parameters, $\alpha$ (alpha) and $\beta$ (beta). It is widely used in Bayesian statistics as a conjugate prior for binomial and Bernoulli distributions.

Probability Density Function (PDF)

The probability density function of the Beta distribution is given by:

$$f(x; \alpha, \beta) = \frac{x^{\alpha - 1}(1 - x)^{\beta - 1}}{B(\alpha, \beta)}$$

Where $B(\alpha, \beta)$ is the Beta function:

$$B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$$

Key Statistical Properties

• Mean ($\mu$): $\frac{\alpha}{\alpha + \beta}$
• Variance ($\sigma^2$): $\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$
• Mode: $\frac{\alpha - 1}{\alpha + \beta - 2}$ (for $\alpha, \beta > 1$)
• Skewness: $\frac{2(\beta - \alpha)\sqrt{\alpha + \beta + 1}}{(\alpha + \beta + 2)\sqrt{\alpha \beta}}$

Applications of Beta Distribution

The Beta distribution models random variables constrained to finite intervals, such as proportions, percentages, conversion rates, and task completion times in project management (PERT/CPM).

Related Tools

Explore related tools to assist with your calculations:

• Beta Function Calculator
• Binomial Probability Distribution Calculator