Binomial Distribution - rFronteddu/general_wiki GitHub Wiki

The formula uses combinatorics to express the probability of getting x (first term) and the probability that the rest is not x (second term). We also define p(x) as the favored outcomes over the sample space.

• Binomial events are a sequence of identical Bernoulli events.
• X ~ B(n, p): "Variable X follows a Binomial distribution with n trials and a likelihood of p to succeed in each individual trial."
• Bern(p) = B(1, p)
• E(Bern) indicates which outcome we expect for a single trial
• E(B) indicates the number of times we expect to get a specific outcome.

The likelihood of getting an outcome a specific number of times where each trial is a Bern event. To get the desired outcome y times, we also need to get the alternative outcome n- y times. There is also more than one way to reach the same outcome but since the order doesn't matter, we use the formula for combination.

Example: Combination in which 4 out of 6 can be successful == pick 4 elements out of a sample space of 6.

• Since Y is a Bern event, the expected number of repetitions of the favored outcome will be the probability of the favored outcome multiplied by the number of trials, E(X) = p * n
• $σ^2 = E({(Y-μ)}^2=E(Y^2) - μ^2 = n * p ( 1 - p)$