Beta Distribution Calculator
Calculate probability density (PDF), cumulative probability (CDF), mean, variance, and quantiles for the Beta distribution.
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).
Frequently Asked Questions
What happens when $\alpha = 1$ and $\beta = 1$?
When $\alpha = 1$ and $\beta = 1$, the Beta distribution reduces to the continuous Uniform distribution $U(0, 1)$.
What is the relationship with the Binomial distribution?
The Beta distribution serves as the conjugate prior distribution for the parameter $p$ of a Binomial distribution in Bayesian inference.
How are $\alpha$ and $\beta$ interpreted in A/B testing?
In Bayesian A/B testing, $\alpha - 1$ represents observed successes (conversions) and $\beta - 1$ represents observed failures.
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