A Bernoulli random variable refers to a discrete random variable that can take only two possible outcomes, typically labeled as "success" and "failure." It is named after Swiss mathematician Jacob Bernoulli, who extensively studied probability theory during the 18th century.
In a Bernoulli random variable, the probability of success, denoted as 'p', remains constant throughout the entire distribution. Conversely, the probability of failure, denoted as 'q' (where q = 1 - p), also remains constant. These probabilities are independent of each other, meaning that the outcome of one trial has no influence on the outcome of subsequent trials.
The distribution associated with a Bernoulli random variable can be represented as a probability mass function (PMF). This PMF provides the probability of observing each possible outcome, often denoted as X. For a Bernoulli random variable, the PMF takes the form P(X = x) = p^x * q^(1-x), where x is either 0 or 1.
Some classic examples of Bernoulli random variables include flipping a fair coin (where heads could be considered a success and tails a failure), conducting an experiment where an event occurs or does not occur (e.g., a light bulb turning on or off), or determining the success or failure of a medical treatment.
The Bernoulli random variable is fundamental to numerous probability models, statistical inference techniques, and applications in various fields, such as medicine, finance, and engineering.
