Bernoulli Distribution in Business Statistics: Mean and Variance

This document provides a comprehensive overview of the Bernoulli distribution, focusing on its fundamental concepts, formulas, and applications in business statistics, with a particular emphasis on its mean and variance.

11.1 Terminologies Associated with Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution that describes the outcome of a single Bernoulli trial. Key terminologies include:

• Bernoulli Trial: A random experiment with exactly two possible outcomes, typically labeled as "success" and "failure."
• Success: The outcome of interest in a Bernoulli trial.
• Failure: The outcome that is not of interest in a Bernoulli trial.
• Probability of Success ($p$): The probability of achieving the "success" outcome in a Bernoulli trial.
• Probability of Failure ($q$): The probability of achieving the "failure" outcome in a Bernoulli trial. By definition, $q = 1 - p$.

11.2 Formula of Bernoulli Distribution

The probability mass function (PMF) of a Bernoulli distribution is defined as follows:

$P(X=x) = \begin{cases} p & \text{if } x = 1 \text{ (success)} \ 1-p & \text{if } x = 0 \text{ (failure)} \end{cases}$

Where:

• $X$ is the random variable representing the outcome of a Bernoulli trial.
• $x$ is the value of the random variable (either 0 for failure or 1 for success).
• $p$ is the probability of success.

This can be more compactly written as:

$P(X=x) = p^x (1-p)^{1-x}$, for $x \in {0, 1}$

11.3 Mean and Variance of Bernoulli Distribution

The mean (expected value) and variance are key measures of a Bernoulli distribution's central tendency and spread, respectively.

11.3.1 Mean (Expected Value)

The mean of a Bernoulli distribution represents the average outcome over many trials. It is calculated as:

$E(X) = \sum_{x} x P(X=x)$ $E(X) = (1 \times p) + (0 \times (1-p))$ $E(X) = p$

Therefore, the mean of a Bernoulli distribution is equal to the probability of success ($p$).

11.3.2 Variance

The variance measures the dispersion of the distribution around its mean. It is calculated as:

$Var(X) = E[(X - E(X))^2]$ $Var(X) = \sum_{x} (x - E(X))^2 P(X=x)$ $Var(X) = (1 - p)^2 p + (0 - p)^2 (1-p)$ $Var(X) = (1 - 2p + p^2)p + p^2 (1-p)$ $Var(X) = p - 2p^2 + p^3 + p^2 - p^3$ $Var(X) = p - p^2$ $Var(X) = p(1-p)$

Alternatively, using the formula $Var(X) = E(X^2) - [E(X)]^2$:

$E(X^2) = (1^2 \times p) + (0^2 \times (1-p)) = p$ $Var(X) = p - p^2 = p(1-p)$

Therefore, the variance of a Bernoulli distribution is $p(1-p)$.

11.4 Properties of Bernoulli Distribution

• Discrete: The random variable can only take on a finite number of values (0 and 1).
• Two Outcomes: There are only two possible results for the trial.
• Independent Trials: Each Bernoulli trial is independent of previous trials.
• Constant Probability of Success: The probability of success ($p$) remains the same for every trial.
• Relationship to Binomial Distribution: A Bernoulli distribution is a special case of the binomial distribution where the number of trials ($n$) is 1.

11.5 Bernoulli Distribution Graph

The graph of a Bernoulli distribution typically consists of two bars, representing the probabilities of success (1) and failure (0).

• X-axis: Represents the outcome ($x=0$ for failure, $x=1$ for success).
• Y-axis: Represents the probability ($P(X=x)$).

The height of the bar at $x=1$ will be $p$, and the height of the bar at $x=0$ will be $1-p$.

  ^ Probability
  |
p +-------+
  |       |
  |       |
(1-p) +-------+
  |       |
  +-------+-------+------> Outcome
      0       1

11.6 Bernoulli Trial

A Bernoulli trial is the fundamental concept underlying the Bernoulli distribution. It is a single experiment or event that meets the following criteria:

1. Exactly two possible outcomes: These outcomes are mutually exclusive and exhaustive.
2. Probabilities are fixed: The probability of each outcome remains constant across all trials.
3. Independence: The outcome of one trial does not affect the outcome of any other trial.

Examples:

• Flipping a fair coin once (Heads/Tails).
• A single customer purchasing a product (Yes/No).
• A manufactured item passing a quality inspection (Pass/Fail).

11.7 Examples of Bernoulli Distribution

1. Coin Flip: Consider flipping a fair coin. Let "Heads" be a success (X=1) and "Tails" be a failure (X=0). The probability of success $p = 0.5$.

   • $P(X=1) = 0.5$
   • $P(X=0) = 1 - 0.5 = 0.5$
   • Mean: $E(X) = p = 0.5$
   • Variance: $Var(X) = p(1-p) = 0.5(0.5) = 0.25$

2. Customer Conversion: A marketing team runs an ad campaign. For a single user, the probability that they click the ad is $p = 0.1$. Let "click" be a success (X=1) and "no click" be a failure (X=0).

   • $P(X=1) = 0.1$
   • $P(X=0) = 1 - 0.1 = 0.9$
   • Mean: $E(X) = p = 0.1$
   • Variance: $Var(X) = p(1-p) = 0.1(0.9) = 0.09$

11.8 Applications of Bernoulli Distribution in Business Statistics

The Bernoulli distribution serves as a foundational building block for analyzing many business scenarios involving binary outcomes:

• Marketing and Sales:
  • Probability of a customer clicking an advertisement.
  • Probability of a sales lead converting into a sale.
  • Probability of a customer making a repeat purchase.
• Quality Control:
  • Probability of a product passing inspection.
  • Probability of a component being defective.
• Risk Management:
  • Probability of a loan defaulting.
  • Probability of an insurance claim being filed.
• Customer Service:
  • Probability of a customer being satisfied with a service.
  • Probability of a customer abandoning a support call.
• Operations:
  • Probability of a machine breaking down in a given period.
  • Probability of a specific event occurring in a process.

11.9 Difference between Bernoulli Distribution and Binomial Distribution

While related, the Bernoulli and Binomial distributions differ in the number of trials considered:

In essence, the Bernoulli distribution describes the outcome of a single event, while the Binomial distribution describes the number of successes in a series of identical and independent Bernoulli trials.