Beta Distribution

Modeling probabilities, percentages, and proportions.

The "Probability of Probabilities" Distribution

The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. This makes it perfectly suited for modeling random variables that represent probabilities or proportions.

In quantitative finance, it's a powerful tool in Bayesian inference and risk modeling. For example, a credit analyst might use it to model the recovery rate on a defaulted loan (which must be between 0% and 100%). A trading strategist could use it to represent the probability of a particular signal being profitable.

The Formula
The probability density function (PDF) is given by:
f(x;α,β)=xα−1(1−x)β−1B(α,β)f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)}
  • xx is the variable (between 0 and 1).
  • α\alpha and β\beta are positive shape parameters.
  • B(α,β)B(\alpha, \beta) is the Beta function, which normalizes the total probability to 1.
Interactive Beta Distribution
Adjust the shape parameters (α and β) to see how they influence the distribution. Note the wide variety of shapes the Beta distribution can take.