Introduction

The Bernoulli distribution is a fundamental concept in probability theory, widely used to model binary events with only two possible outcomes: success (usually denoted as 1) or failure (usually denoted as 0). Named after Swiss mathematician Jacob Bernoulli, this distribution holds significant importance in various fields, including statistics, economics, and engineering. In this chapter, we will explore the key characteristics, probability mass function, mean, variance, and real-world examples of the Bernoulli distribution.

Bernoulli Distribution: Characteristics and Probability Mass Function

The Bernoulli distribution is characterized by its simplicity and applicability to situations with dichotomous outcomes. Let’s denote the probability of success as $p$ and the probability of failure as $q=1−p$. The probability mass function (PMF) of the Bernoulli distribution is given by:

$$p, \text{ if } x = 1$$ $$q, \text{ if } x = 0$$

Here, $X$ represents the random variable, and $x$ takes the values 0 or 1.

Mean and Variance of Bernoulli Distribution

• Mean: The mean $mu$ of a Bernoulli distribution is calculated as:

$$\mu = p $$

• Variance: The variance $sigma^2$ of a Bernoulli distribution is calculated as: $$ \sigma^2 = p \cdot q $$

Real-World Examples

The Bernoulli distribution finds practical application in various scenarios. Consider the following examples:

• Coin Tossing: When flipping a fair coin, the outcome can be either heads (success) or tails (failure). Here, $p = q = 0.5$, making it a Bernoulli distribution.

• Online Purchase: In e-commerce, a customer either makes a purchase (success) or does not (failure). The probability of a successful purchase is $p$, while the probability of not making a purchase is $q$.