Probability Theory

The Beta Distribution

The Beta Distribution is unique because it is bounded between [0, 1]. This makes it the perfect tool for modeling probabilities, proportions, and rates.

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

By adjusting the shape parameters Alpha (Successes) and Beta (Failures), we can mold this curve into almost any shape.

Key Behaviors

$\alpha = \beta$ : Symmetric. (Bell or U-shape).
$\alpha > \beta$ : Left Skew (High probability).
$\alpha < \beta$ : Right Skew (Low probability).
$\alpha = \beta = 1$ : Uniform (Total ignorance).

beta.control.alpha (

\alpha

beta.control.beta (

\beta

beta.stat.mean0.286α / (α+β)

beta.stat.var0.0255beta.stat.spread

beta.stat.skew0.596beta.skew.right

beta.stat.kurt-0.120beta.kurt.normal

beta.chart.pdf

2. The Link to Regression

In the real world, we often model Rates & Proportions (e.g., 'What % of users will click this ad?').

Linear Regression fails because it can predict values outside [0,1].
Logistic Regression fails because it assumes the outcome is binary (0 or 1), not a continuous percentage.

Beta Regression fills this gap. It assumes $Y \sim \text{Beta}(\mu, \phi)$ , allowing us to model the mean ( $\mu$ ) and precision ( $\phi$ ) of a rate directly.

The Link Function

g(\mu) = \ln\left(\frac{\mu}{1-\mu}\right) = X\beta

"Just like Logistic Regression, we map the (0,1) interval to the real line using the Logit link, but we model the distribution shape, not just the class probability."

3. The Project Simulation

This simulation replicates the Method of Moments (MOM) analysis from my research project.

By generating synthetic data from a known distribution, we can verify if our statistical estimators can accurately 'recover' the truth from the noise.

* Code translated from R to Python for web execution.

notebook.py

Console

Initializing Kernel...

Pyodide Kernel (Standard)○ Loading Libraries...

header.brand

Laboratory Index

Source Code Download Paper