Probability mass function | |||
Cumulative distribution function | |||
Notation | |||
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Parameters |
– number of trials – success probability for each trial | ||
Support | – number of successes | ||
PMF | |||
CDF | (the regularized incomplete beta function) | ||
Mean | |||
Median | or | ||
Mode | or | ||
Variance | |||
Skewness | |||
Excess kurtosis | |||
Entropy |
in shannons. For nats, use the natural log in the log. | ||
MGF | |||
CF | |||
PGF | |||
Fisher information |
(for fixed ) |
Part of a series on statistics |
Probability theory |
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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 q = 1 − p). 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 binomial test of statistical significance.[1]
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.