Bernoulli Processes

Bernoulli Process

A Bernoulli Process is a sequence of independent \({0, 1}\) - valued random variables \(X_1, X_2, X_3, …\), e.g. \(0, 0, 1, 0, 1, 1\)

Suppose you flip a coin repeatedly, and record \(0\) for tails and \(1\) for heads.

\(X\) is a binomial random variable if it takes the values \(0, 1, 2, …, n\) and

$$ \mathbb{P}\{X = k\} = { n \choose k } \cdot p^k \cdot (1 - p)^{n-k} $$

Sanity Check: Do the probabilities sum to 1?

$$ \sum_{k=0}^{n} \mathbb{P}\{X = k\} = \sum_{k=0}^{n} { n \choose k } p^k (1 - p)^{n-k} = \left( p + (1 - p) \right)^n = 1 $$