Bernoulli Distribution

The Bernoulli distribution models a single trial with two possible outcomes — success or failure. It’s the building block for more complex distributions like the Binomial and is widely used in binary classification and yes/no decisions. ⚖️

\[ p_{X}(x) = \begin{cases} p & \text{if } x = 1, \\ q = 1-p & \text{if } x = 0. \end{cases} \]

• The discrete random variable can take value \(1\) with probability \(p\) or value \(0\) with probability \(q = 1 - p\)
• Not to be confused with the binomial distribution, since only one trial is being conducted.
• \(\mathbb{E}[X] = p\)
• \(Var[X] = pq\)
  

viewof p = Inputs.range([0, 1], {
  step: 0.01,
  value: 0.5,
  label: tex`p`,
  width: 200
})

data = {
  return [
    {outcome: "0", probability: 1 - p},
    {outcome: "1", probability: p}
  ];
}

Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 600,
  height: 400,
  y: {
    grid: true,
    label: "Probability",
    domain: [0, 1]
  },
  x: {
    label: "Outcome",
    padding: 0.2
  },
  marks: [
    Plot.barY(data, {
      x: "outcome",
      y: "probability",
      fill: "steelblue"
    }),
    Plot.ruleY([0])
  ]
})

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

Python Code: Bernoulli Distribution

To create Bernoulli distributed data using numpy:

import numpy as np

interval = [0,1]
size = (1000,1)
p = [1-0.5, 0.5]

data = np.random.choice(interval, size, p = p)