Bernoulli Distribution

Bernoulli distribution is a probability distribution that models the outcomes of a random experiment with two possible outcomes, success or failure. It is an essential topic in statistics, probability theory, and machine learning, having applications in economics, finance, and other domains. This article will discuss the fundamentals of the Bernoulli distribution, as well as its features and applications.

Table of Contents:

Bernoulli Distribution Introduction
Formulation of the Bernoulli Distribution
Characteristics of the Bernoulli Distribution
Mean and Dispersion
Asymmetry and Kurtosis
Probability Weighting Function
Uses for the Bernoulli Distribution
Economics
Finance
Engineering
Robotic Learning
Bernoulli vs. Binomial Distribution
Conclusion

Overview of the Bernoulli Distribution:

Bernoulli distribution was introduced by Swiss mathematician Jacob Bernoulli in his book Ars Conjectu (1713). It is used to model random experiments with only two possible outcomes, 1 (success) and 0 (failure) (failure). Examples of such tests include flipping a coin (heads or tails), shooting a free throw (success or failure), and testing a light bulb (functions or does not function).

In the Bernoulli distribution, p represents the likelihood of success, while q=1-p represents the probability of failure. The probability mass function (PMF) of Bernoulli distribution may therefore be expressed as follows:

f(x) = p^x*(1-p)^(1-x), where x=0 or 1., given that p is a positive real number.

Observe that f(0) equals q and f(1) equals p. Moreover, Bernoulli distribution can be represented as a binomial distribution with n=1, or a single trial.

Formula for the Bernoulli Distribution

Here is the formula for the Bernoulli distribution:

f(x) = p^x*(1-p)^(1-x), where x = 0 or 1, given that p is a positive real number.

where:

The probability mass function (PMF) of Bernoulli distribution is denoted by f(x).
x represents the potential outcome (0 or 1) of a random experiment.
p represents the likelihood of success (1) in the experiment.
q=1-p represents the experiment’s probability of failure (0).
Characteristics of the Bernoulli Distribution:

Several essential characteristics of the Bernoulli distribution make it a popular probabilistic model. These characteristics include:

Standard deviation:

The mean (expected value) of the Bernoulli distribution can be calculated by using the following formula:

E(X) = p

where X is the random variable representing the result of Bernoulli’s experiment.

The variance of the Bernoulli distribution can be calculated using the following formula:

Var(X) = E(X^2) – E(X)^2 = p (1-p)

Kurtosis and Skewness

The skewness and kurtosis of the Bernoulli distribution are 1-2p and -6p2+6p-1, respectively.

Probability Weighting Function:

Bernoulli distribution probability mass function (PMF) is given by:

f(x) = p^x*(1-p)^(1-x), where x = 0 or 1, given that p is a positive real number.

where f(x) is the chance of the Bernoulli experiment yielding x.

Uses of the Bernoulli Distribution

Bernoulli distribution has numerous applications in a variety of domains, such as:

Economics:

Bernoulli distribution is utilised in economics to represent the chance of success or failure of financial decisions, such as purchasing a stock or launching a new firm. In utility theory, it is also used to model the preferences of customers and decision-makers.

Finance:

Bernoulli distribution is utilised in finance to estimate the chance of a stock’s price rising or falling, as well as the probability of default in credit risk models. In addition, it is utilised for option pricing and risk management.

Engineering:

Bernoulli distribution is used to describe the likelihood of success or failure in engineering reliability analysis, quality control, and product testing. In signal processing, it is also used to simulate the distribution of random noise.

Automatic Learning:

Bernoulli distribution is used as a building component in machine learning for a number of models, including Naive Bayes, logistic regression, and Bernoulli restricted Boltzmann machines (RBM). Also, it is utilised in binary classification issues, such as spam identification and sentiment analysis.

Bernoulli versus Binomial Distributions:

Bernoulli distribution is closely associated with binomial distribution, which predicts the number of successes in a set number of separate Bernoulli trials. In other words, the binomial distribution is the sum of Bernoulli trials with identical success probabilities.

Below is the formula for binomial distribution:

f(k) = C(n,k)p^k(1-p)^(n-k), where k=0,1,2,…,n.

where:

PMF of binomial distribution is f(k).
k represents the number of successful outcomes from n trials.
The number of total trials is denoted by n.
The likelihood of success in each trial is denoted by p.

Conclusion:

Bernoulli distribution is a fundamental notion in probability theory and statistics, with applications in numerous disciplines, such as economics, finance, engineering, and machine learning. It is employed to simulate the results of a random experiment with only two possible outcomes, namely success or failure. It possesses a variety of essential characteristics, including mean, variance, skewness, kurtosis, and probability mass function. Bernoulli distribution is quite similar to the binomial distribution, which predicts the number of successes in a certain number of independent Bernoulli trials. Understanding Bernoulli distribution is essential for evaluating and simulating a vast array of actual phenomena.