1. What is the Beta Distribution?

The Beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, α and β. It is often used to model random variables limited to intervals of finite length.

2. How Does the Calculator Work?

The calculator plots the probability density function (PDF) of the Beta distribution:

Where:

• \( x \) — Random variable (0 ≤ x ≤ 1)
• \( \alpha \) — First shape parameter (α > 0)
• \( \beta \) — Second shape parameter (β > 0)
• \( B(\alpha,\beta) \) — Beta function

Explanation: The shape of the distribution depends on the values of α and β. When both are 1, it becomes the uniform distribution.

3. Importance of Beta Distribution

Details: The Beta distribution is widely used in Bayesian statistics, order statistics, and modeling random variables that represent proportions or probabilities.

4. Using the Calculator

Tips: Enter positive values for α and β parameters. The graph will show how changing these parameters affects the shape of the distribution.

5. Frequently Asked Questions (FAQ)

Q1: What are typical uses of the Beta distribution?
A: It's commonly used in Bayesian analysis, modeling percentages, A/B testing, and describing uncertainty about probabilities.

Q2: What's the relationship between α and β parameters?
A: α controls the shape near 0, β controls the shape near 1. When α = β, the distribution is symmetric.

Q3: What's the mean of the Beta distribution?
A: The mean is α/(α+β). The variance is αβ/[(α+β)²(α+β+1)].

Q4: How does this relate to the Dirichlet distribution?
A: The Dirichlet is a multivariate generalization of the Beta distribution.

Q5: What special cases does the Beta distribution include?
A: It includes the uniform distribution (α=β=1), arcsine distribution (α=β=0.5), and more.