   | kurtosis = \frac{1-6p(1-p)}{np(1-p)} | entropy = \frac12 \log_2 \big( 2\pi e\, np(1-p) \big) + o \left( \frac{1}{n} \right) | mgf = (1-p + pe^t)^n \! | char = (1-p + pe^{it})^n \! | pgf = g(z) = \left[(1-p) + pz\right]^n. | fisher = g_n(p) = \frac{n}{p(1-p)} (for fixed n) }}  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 yes/no experiments, each of which yields success with probability p. a success/failure experiment is also called a bernoulli experiment or bernoulli trial; when 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 is a good approximation, and widely used. ==specification== ===probability mass function=== in general, if the random variable x follows the binomial distribution with parameters n and p, we write x&nbsp;~&nbsp;b(n,&nbsp;p). the probability of getting exactly k successes in n trials is given by the probability mass function: : f(k;n,p) = \pr(x = k) = {n\cho
Discrete_distributions Distributions_with_conjugate_priors Exponential_family_distributions Factorial_and_binomial_topics Probability_distributions