Gaussian Distribution

The Gaussian (or Normal) distribution is the most fundamental continuous distribution in statistics. Characterized by its bell-shaped curve, it models natural phenomena and underpins many inferential techniques due to the Central Limit Theorem. 🌍

\[ f_{X}(x)=\frac{1}{\sqrt{2\pi\sigma^{2} }}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \]

• Used frequently to represent real-valued random variables whose distributions are not known.
• Its importance is derived from the central limit theorem that states, under some conditions, the average of many samples of a random variable is itself a random variable that converges to a Gaussian distribution as it increases.
• \(E[X] = \mu\)
• \(Var[X] = \sigma^{2}\)

viewof mu = Inputs.range([-1, 1], {
  step: 0.1,
  value: 0,
  label: tex`\mu`,
  width: 200
})

viewof sigma = Inputs.range([0.1, 2], {
  step: 0.1,
  value: 1,
  label: tex`\sigma`,
  width: 200
})

// Generate points for the normal distribution curve
pointsGaussian = {
  const x = d3.range(-5, 5, 0.1);
  return x.map(x => ({
    x,
    y: (1 / (sigma * Math.sqrt(2 * Math.PI))) * 
       Math.exp(-0.5 * Math.pow((x - mu) / sigma, 2))
  }));
}

Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 600,
  height: 400,
  y: {
    grid: true,
    label: "Density"
  },
  x: {
    label: "x",
    domain: [-5, 5]
  },
  marks: [
    Plot.line(pointsGaussian, {x: "x", y: "y", stroke: "steelblue"}),
    Plot.ruleY([0])
  ]
})

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

Python Code: Gaussian Distribution

To create Gaussian distributed data using numpy:

import numpy as np

mu = 0
sigma = 1
size = (1000,1)

data = np.random.normal(mu, sigma, size)