Binomial distribution

Binomial distribution
Probability mass function
Probability mass function for the binomial distribution
Cumulative distribution function
Cumulative distribution function for the binomial distribution
Notation
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 )
Binomial distribution for p = 0.5
with n and k as in Pascal's triangle

The probability that a ball in a Galton box with 8 layers (n = 8 ends up in the central bin (k = 4 is 70/256.

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.

  1. ^ Westland, J. Christopher (2020). Audit Analytics: Data Science for the Accounting Profession. Chicago, IL, USA: Springer. p. 53. ISBN 978-3-030-49091-1.

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