Beta Distribution

The Beta distribution models probabilities themselves — perfect for random variables bounded between 0 and 1. It’s commonly used in Bayesian inference as a prior for binomial proportions due to its flexibility in shape. 🔧

\[ f_{X}(x) = {\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}\,x^{\alpha -1}(1-x)^{\beta -1} \]

where \(\Gamma\) is the gamma function defined as:

\[ \Gamma (z)=\int _{0}^{\infty}t^{z-1}e^{-t}\,dt \]

• Gamma functions are used to model factorial functions of complex numbers \(z\).
• Beta functions are used to model behavior of random variables in intervals of finite length.
• \(E[X] = \frac{\alpha}{\alpha+\beta}\)
• \(Var[X] = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)

viewof alpha = Inputs.range([0.1, 10], {
  step: 0.1,
  value: 1,
  label: tex`\alpha`,
  width: 200
})

viewof beta = Inputs.range([0.1, 10], {
  step: 0.1,
  value: 1,
  label: tex`\beta`,
  width: 200
})

// Gamma function approximation using Lanczos approximation
function gamma(z) {
    const p = [676.5203681218851, -1259.1392167224028, 771.32342877765313,
        -176.61502916214059, 12.507343278686905, -0.13857109526572012,
        9.9843695780195716e-6, 1.5056327351493116e-7];
    
    if (z < 0.5) {
        return Math.PI / (Math.sin(Math.PI * z) * gamma(1 - z));
    }
    
    z -= 1;
    let x = 0.99999999999980993;
    for (let i = 0; i < p.length; i++) {
        x += p[i] / (z + i + 1);
    }
    
    const t = z + p.length - 0.5;
    return Math.sqrt(2 * Math.PI) * Math.pow(t, z + 0.5) * Math.exp(-t) * x;
}

// Beta function using gamma function
function betaFunc(x, y) {
    return (gamma(x) * gamma(y)) / gamma(x + y);
}

// Beta probability density function
function betaPDF(x, a, b) {
    if (x <= 0 || x >= 1) return 0;
    return Math.pow(x, a - 1) * Math.pow(1 - x, b - 1) / betaFunc(a, b);
}

// Generate points for the beta distribution curve
points = Array.from({length: 100}, (_, i) => {
  let x = 0.001 + i * 0.01;
  return { x, y: betaPDF(x, alpha, beta) };
});

Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 600,
  height: 400,
  y: {
    grid: true,
    label: "Density"
  },
  x: {
    label: "x",
    domain: [0, 1]
  },
  marks: [
    Plot.line(points, {x: "x", y: "y", stroke: "steelblue"}),
    Plot.ruleY([0])
  ]
})

html`<div style="text-align: center; margin-top: 1em;">
  <p>${tex`\mathbb{E}[X] =`} ${(alpha/(alpha + beta)).toFixed(3)}</p>
  <p>${tex`\text{Var}(X) =`} ${((alpha * beta)/((alpha + beta)**2 * (alpha + beta + 1))).toFixed(3)}</p>
</div>`

Python Code: Beta Distribution

To create Beta distributed data using numpy:

import numpy as np 

alpha = 0.5
beta = 0.5
size = (1000,1)

data = np.random.beta(alpha, beta, size)