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. \(0, 0, 1, 0, 1, 1\) A Bernoulli Process does not mandate that the probability distributions of the \(X_i\) be identical. That is up to the model that we choose. For instance, the Binomial Random Variable assumes \(\mathbb{P}\{X_i = 1\} = p \ \ \forall i\) 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? $$ \sum_{k=0}^{n} \mathbb{P}\{X = k\} = \sum_{k=0}^{n} { n \choose k } p^k (1 - p)^{n-k} = \left( p + (1 - p) \right)^n = 1 $$ I totally didn’t understand how we got to \(\left( p + (1 - p) \right)^n\)....