Bernoullirandom

Bernoullirandom refers to a concept in probability theory and statistics that describes a random variable following a Bernoulli distribution. Named after the Swiss mathematician Jacob Bernoulli, this distribution models experiments with exactly two possible outcomes: success or failure. Each outcome is associated with a probability, typically denoted as *p* for success and *1 − p* for failure.

A Bernoulli random variable is a discrete random variable that takes the value 1 with probability *p* (indicating success) and 0 with probability *1 − p* (indicating failure). It is the simplest case of a binary random variable and serves as the foundation for more complex distributions, such as the binomial distribution, which generalizes repeated Bernoulli trials.

In practical applications, Bernoulli random variables are used to model scenarios like coin flips, where success might be defined as landing heads, or medical trials, where success could represent recovery from a treatment. The expected value (mean) of a Bernoulli random variable is *p*, and its variance is *p(1 − p)*, reflecting the uncertainty inherent in probabilistic outcomes.

The probability mass function (PMF) of a Bernoulli random variable *X* is given by:

- *P(X = 1) = p*

- *P(X = 0) = 1 − p*

This distribution is fundamental in statistical modeling, hypothesis testing, and machine learning, particularly in algorithms that rely on binary classification. Its simplicity makes it a useful building block for more advanced probabilistic frameworks.