The binomial distribution is an experiment of $n$ independent, identical Bernoulli trials where $p$ is the proportion of successes. - The number of trials $n$ is fixed in advance - Trials are identical and result in success or failure (i.e., Bernoulli trials) - Trials are independent ## Notation $X \sim Bin(n, p)$ ## PMF $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ ## Expected Value $E(X) = np$ ## Variance $V(X) = np(1-p)$ S = {(x1, x2, x3, ..., xn)} | xi = 1 if success or 0 if failure Cardinality = 2^n P(X=0) = P({0, 0, 0, ..., 0}) = (1-p)^n P(X=1) = P({1, 0, 0, ..., 0}, {P(0, 1, 0, ..., 0)}) = np(1-p)^(n-1) P(X=2) = P({1, 1, 0, ..., 0}, {1, 0, 1, ..., 0}) = choose(n, 2) p^2(1-p)^(n-2) P(X=k) = choose(n, k) p^k (1-p)^(n-k) Observe that, from the [[binomial theorem]], the sum of all k's equals 1, confirming that we do have a PMF. Sigma choose(n,k) p^k (1-p)(n-k) #expand