Bernoulli distribution refers to an event that has two outcomes: failure or success

  • Sample set is {0, 1} and the probability of 1 occuring is p while the probability of 0 occuring is 1 - p (also written as q)
  • f(x) = (1 - p)1 - xpx
  • E(x) = p
  • VAR(x) = p(1 - p) = pq Binomial distributions refer to multiple Bernoulli distributions
  • Takes two parameters: n (number of trials) and p (chance of success)
    • p must be constant, n must be set, and each event must be independent from each other
  • f(x) = nCx (1 - p)n - xpx
    • Accessed in R with dbinom(x, n, p) where x is the desired number of successful trials
  • F(X) = pbinom(x, n, p)
  • E(X) = np
  • VAR(X) = np(1 - p) = npq

A variable can be expressed using tilde notation

  • For example, X ~ Binomial(n = 22, p = 1/4) indicates that X follows a binomial distribution where the number of trials is 22 and the chance of success is 0.25