AOS4 Topic 4: Bernoulli Distribution
Bernoulli Distribution is a type of discrete probability distribution where every experiment conducted asks a question that can be answered only in yes or no. In other words, the random variable can be 1 with a probability p or it can be 0 with a probability (1 - p). Such an experiment is called a Bernoulli trial. A pass or fail exam can be modeled by a Bernoulli Distribution.
If we have a Binomial Distribution where n = 1 then it becomes a Bernoulli Distribution. As this distribution is very easy to understand, it is used as a basis for deriving more complex distributions. Bernoulli Distribution can be used to describe events that can only have two outcomes, that is, success or failure.
What is Bernoulli Distribution?
Bernoulli Distribution is a special kind of distribution that is used to model real-life examples and can be used in many different types of applications. A random experiment that can only have an outcome of either 1 or 0 is known as a Bernoulli trial. Such an experiment is used in a Bernoulli distribution.
Bernoulli Distribution Definition
A discrete probability distribution wherein the random variable can only have 2 possible outcomes is known as a Bernoulli Distribution. If in a Bernoulli trial the random variable takes on the value of 1, it means that this is a success. The probability of success is given by \(p\). Similarly, if the value of the random variable is 0, it indicates failure. The probability of failure is \(q = 1 - p\). Bernoulli distribution can be used to derive a binomial distribution, geometric distribution, and negative binomial distribution.
Bernoulli Distribution Example
Suppose there is an experiment where you flip a fair coin. If the outcome of the flip is heads, then you win. This means that the probability of getting heads is \(p = \frac{1}{2}\). If \(X\) is the random variable following a Bernoulli Distribution, we get \(P(X = 1) = p = \frac{1}{2}\).
Bernoulli Distribution Formula
A binomial random variable, \(X\), is also known as an indicator variable. This is because if an event results in success, then \(X = 1\), and if the outcome is a failure, then \(X = 0\). \(X\) can be written as \(X \sim \text{Bernoulli}(p)\), where \(p\) is the parameter. The formulas for Bernoulli distribution are given by the probability mass function (PMF) and the cumulative distribution function (CDF).
Probability Mass Function for Bernoulli Distribution
The probability mass function (PMF) for a Bernoulli distribution is:
\[ f(x, p) = \begin{cases} p & \text{if } x = 1 \\ 1 - p & \text{if } x = 0 \end{cases} \]We can also express this formula as:
\[ f(x, p) = p^x (1 - p)^{1 - x}, \quad x \in \{0, 1\} \]Cumulative Distribution Function for Bernoulli Distribution
The cumulative distribution function (CDF) of a Bernoulli random variable \(X\) is defined as:
\[ F(x, p) = \begin{cases} 0 & \text{if } x < 0 \\ 1 - p & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases} \]Mean and Variance of Bernoulli Distribution
The mean (expected value) and variance of a Bernoulli distribution are derived as follows:
Mean of Bernoulli Distribution
We know that:
\[ P(X = 1) = p \quad \text{and} \quad P(X = 0) = q = 1 - p \]The expected value (mean) is given by:
\[ E[X] = P(X = 1) \cdot 1 + P(X = 0) \cdot 0 = p \cdot 1 + (1 - p) \cdot 0 = p \]Variance of Bernoulli Distribution
The variance is calculated as:
\[ \text{Var}(X) = E[X^2] - (E[X])^2 \]Since \(E[X^2] = E[X]\), we get:
\[ \text{Var}(X) = p - p^2 = p(1 - p) = p \cdot q \]Thus, the variance of a Bernoulli distribution is \( \text{Var}(X) = p(1 - p) \).
Bernoulli Distribution Applications
Bernoulli distribution is a simple distribution and hence, is widely used in many industries. Given below are some applications of Bernoulli distribution:
- In medicine, Bernoulli distributions are used to model the events experienced by a single patient. These events could be disease, death, and so on.
- Logistic regressions use Bernoulli distribution to model the occurrence of certain events such as the specific outcome of a dice roll.
- Bernoulli distribution is also used as a basis to derive several other probability distributions that have applications in the engineering, aerospace, and medical industries.