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Concept
Binomial distribution
Summary
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.
In general, if the random variable X follows the binomial distribution with parameters n ∈ and p ∈ [0,1], we write X ~ B(n, p). The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:
for k = 0, 1, 2, ..., n, where
is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows: k successes occur with probability pk and n − k failures occur with probability . However, the k successes can occur anywhere among the n trials, and there are different ways of distributing k successes in a sequence of n trials.
In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as
Looking at the expression f(k, n, p) as a function of k, there is a k value that maximizes it.
Official source
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