Bernoulli Distribution

🐾 Bernoulli Distribution

The Bernoulli distribution is a discrete distribution modeling a single trial with only two possible outcomes: success (1) or failure (0).

📌 Definition

Let $X \sim \text{Bernoulli}(p)$, where $0 \leq p \leq 1$. Then:

\[P(X = x) = \begin{cases} p & \text{if } x = 1 \\ 1 - p & \text{if } x = 0 \\ 0 & \text{otherwise} \end{cases}\]

📋 Key Properties

Property	Value
Parameter(s)	$p \in [0, 1]$
Support	$x \in {0, 1}$
Mean	$\mathbb{E}[X] = p$
Variance	$\text{Var}(X) = p(1 - p)$
Mode	$1$ if $p > 0.5$, else $0$
Entropy	$-p \log p - (1 - p) \log(1 - p)$
MGF	$M_X(t) = 1 - p + pe^t$

🌍 Real-World Examples

1. Email Spam Filter
   Whether an email is spam ($1$) or not spam ($0$) can be modeled using a Bernoulli distribution.

2. Product Defect Detection
   A quality control test for a manufactured item can be modeled as success (passes inspection = $1$) or failure (defective = $0$).

Notes

1. A Bernoulli trial is the foundation for many more complex distributions (e.g., Binomial, Geometric).
2. Models binary outcomes: yes/no, success/failure, on/off, etc.
3. Very useful in binary classification, A/B testing, digital communication, and Bayesian inference.
4. The Bernoulli is a special case of the Binomial distribution with $n = 1$.
5. If you repeat Bernoulli trials and count the number of successes, you get a Binomial distribution.

🐍 Python Example

import numpy as np
from scipy.stats import bernoulli

# Define the probability of success
p = 0.7

# Create a Bernoulli distribution
rv = bernoulli(p)

# Sample 10 values
samples = rv.rvs(size=10)
print("Samples:", samples)

# Probability of success and failure
print("P(X=1):", rv.pmf(1))
print("P(X=0):", rv.pmf(0))

# Mean and variance
print("Mean:", rv.mean())
print("Variance:", rv.var())