Beta Distribution

Definition

Core Statement

The Beta Distribution is a continuous distribution defined on the interval [0, 1]. It is parametrized by two shape parameters, $α$ (Alpha) and $β$ (Beta). It is widely used to model probabilities or proportions, and serves as the conjugate prior for the Bernoulli, Binomial, and Geometric distributions.

Purpose

Bayesian Inference: Representing belief about a probability (e.g., "I think the conversion rate is between 2% and 5%").
Modeling Rates: Click-through rates, batting averages, defect rates.
Project Management: PERT distribution (Optimistic vs Pessimistic estimates).

Intuition: "Successes and Failures"

You can think of $α$ and $β$ as pseudo-counts of prior history:

$α - 1$ : Number of Successes.
$β - 1$ : Number of Failures.

Parameters	Shape	Interpretation
Beta(1, 1)	Flat (Uniform)	"I have no idea." (All probs equally likely).
Beta(2, 2)	Mound at 0.5	"Weakly believe it's fair."
Beta(100, 100)	Sharp Spike at 0.5	"Strongly believe it's fair."
Beta(10, 1)	Skewed Right (near 1)	"Almost certain to succeed."
Beta(0.5, 0.5)	U-Shape (bathtub)	"Either 0 or 1, but not middle."

Worked Example: Batting Average

Problem

A new baseball player appears.
Estimate his batting average.

Prior: League average is 0.260. We use a prior of Beta(81, 219) (Mean = $81 / 300 = 0.27$ , effectively 300 "prior at-bats").
Data: In his first game, he hits 1 out of 1. (100% average!).
Naive Mean: $1.000$ (Way too high).

Bayesian Update:

New $α = 81 + 1 = 82$ .
New $β = 219 + 0 = 219$ .
Posterior Mean: $82 / (82 + 219) = 82 / 301 \approx 0.272$ .

Conclusion: The massive weight of the prior ("He's a rookie") keeps the estimate grounded. One hit doesn't make him a god. This is regularization.

Key Properties

Domain: $x \in [0, 1]$ .
Mean: $μ = \frac{α}{α + β}$ .
Mode: $\frac{α - 1}{α + β - 2}$ (for $α, β > 1$ ).

Python Implementation

import numpy as np
from scipy.stats import beta
import matplotlib.pyplot as plt

x = np.linspace(0, 1, 100)

# 1. Uninformed Prior
y1 = beta.pdf(x, 1, 1)

# 2. Strong Belief in Fairness
y2 = beta.pdf(x, 50, 50)

# 3. Skewed Belief (Low prob)
y3 = beta.pdf(x, 2, 8)

plt.plot(x, y1, label='Beta(1,1) [Uniform]')
plt.plot(x, y2, label='Beta(50,50) [Peaked]')
plt.plot(x, y3, label='Beta(2,8) [Low Rate]')
plt.legend()
plt.title("Beta Distribution Shapes")
plt.show()

Bernoulli Distribution - Beta is derived from it.
Bayesian Statistics - Heavy user of Beta.
Conjugate Prior - Mathematical property making Beta useful.
Dirichlet Distribution - Multivariate generalization (Beta for >2 categories).