A **Bernoulli process** is a [[Sequence|sequence]] of [[Probabilistic Independence|independent]] Bernoulli trials: [[Experiment|experiments]] that have two possible outcomes: success and failure, meaning that if the [[Probability|probability]] of success is $p$, then the probability of failure is $q = 1-p$. When drawing [[Random Sample|random samples]] from a [[Population|population]] without replacement, the sample size $n$ must be less than or equal to a portion of the population size $5\%N$ for it to be considered a Bernoulli process. ### Distributions Different [[Random Variable|random variables]] may be extracted from the Bernoulli process by measuring the outcome differently: | Source | Distribution | [[Expectation]] | [[Variance]] | | ------------------------------------- | ---------------------------------- | ------------------ | ------------ | | Success in a single trial | [[Bernoulli Random Variable]] | $p$ | $pq$ | | Number of successes | [[Binomial Random Variable]] | $np$ | $npq$ | | Number of trials until the first success | [[Geometric Distribution]] | $\frac{1}{p}$ | $\frac{q}{p^2}$ | | The trial on which the $k$-th success | [[Negative Binomial Distribution]] | $\frac{k}{p}$ | $\frac{kq}{p^2}$ |