"Binomial model" redirects here. For the binomial model in options pricing, see Binomial options pricing model.
Binomial distribution
Probability mass function
Cumulative distribution function
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 )
Part of a series on statistics
Probability theory
Probability
Axioms
Determinism
System
Indeterminism
Randomness
Probability space
Sample space
Event
Collectively exhaustive events
Elementary event
Mutual exclusivity
Outcome
Singleton
Experiment
Bernoulli trial
Probability distribution
Bernoulli distribution
Binomial distribution
Exponential distribution
Normal distribution
Pareto distribution
Poisson distribution
Probability measure
Random variable
Bernoulli process
Continuous or discrete
Expected value
Variance
Markov chain
Observed value
Random walk
Stochastic process
Complementary event
Joint probability
Marginal probability
Conditional probability
Independence
Conditional independence
Law of total probability
Law of large numbers
Bayes' theorem
Boole's inequality
Venn diagram
Tree diagram
v
t
e
Binomial distribution for 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 .
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.[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.
^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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Global Information
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