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.

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.

Shape (α): 2.0

Shape (β): 5.0

Mean (

\mu

): 0.29

Variance (

\sigma^2

): 0.03

Core Concepts

Probability Density Function (PDF)

The PDF is defined by two positive shape parameters, α and β.

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

$x$ is the variable (a probability, between 0 and 1).
$\alpha$ and $\beta$ can be thought of as counts of "successes" and "failures".
$B(\alpha, \beta)$ is the Beta function, which acts as a normalizing constant.

Cumulative Distribution Function (CDF)

The CDF is the regularized incomplete beta function.

F(x; \alpha, \beta) = I_x(\alpha, \beta)

This function is computationally complex and is almost always calculated using statistical software.

Expected Value & Variance

Expected Value (Mean)

E[X] = \frac{\alpha}{\alpha + \beta}

Variance

Var(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}

Key Derivations

Deriving the Expected Value (Mean)

We derive the mean by calculating the integral of

x \cdot f(x)

over its domain [0, 1].

Step 1: Set up the Integral for E[X]

The expected value is the integral of $x$ times the PDF.

E[X] = \int_{0}^{1} x \cdot \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)} dx

Step 2: Simplify the Integral

Combine the $x$ terms and move the constant Beta function outside.

E[X] = \frac{1}{B(\alpha, \beta)} \int_{0}^{1} x^{\alpha} (1-x)^{\beta-1} dx

Step 3: Recognize the New Integral

The new integral $\int_{0}^{1} x^{\alpha} (1-x)^{\beta-1} dx$ looks very similar to the definition of a Beta function itself. It is exactly equal to $B(\alpha+1, \beta)$ .

E[X] = \frac{B(\alpha+1, \beta)}{B(\alpha, \beta)}

Step 4: Expand Using the Gamma Function

Now, we use the definition of the Beta function in terms of the Gamma function: $B(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$ .

E[X] = \frac{\frac{\Gamma(\alpha+1)\Gamma(\beta)}{\Gamma(\alpha+1+\beta)}}{\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}}

This simplifies to:

E[X] = \frac{\Gamma(\alpha+1)}{\Gamma(\alpha)} \cdot \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+1)}

Step 5: Apply the Gamma Function Property

Using the property $\Gamma(z+1) = z\Gamma(z)$ , we have:

$\Gamma(\alpha+1) = \alpha\Gamma(\alpha)$
$\Gamma(\alpha+\beta+1) = (\alpha+\beta)\Gamma(\alpha+\beta)$

Substituting these in:

E[X] = \frac{\alpha\Gamma(\alpha)}{\Gamma(\alpha)} \cdot \frac{\Gamma(\alpha+\beta)}{(\alpha+\beta)\Gamma(\alpha+\beta)}

After canceling terms, we get our final result.

Final Mean Formula

E[X] = \frac{\alpha}{\alpha + \beta}