Binomial distribution

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 ${\displaystyle q=1-p}$). 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 popular binomial test of statistical significance.

Binomial distribution

Probability mass function
Cumulative distribution function
Notation	${\displaystyle B(n,p)}$
Parameters	${\displaystyle n\in \{0,1,2,\ldots \}}$ – number of trials ${\displaystyle p\in [0,1]}$ – success probability for each trial ${\displaystyle q=1-p}$
Support	${\displaystyle k\in \{0,1,\ldots ,n\}}$ – number of successes
PMF	${\displaystyle {\binom {n}{k}}p^{k}q^{n-k}}$
CDF	${\displaystyle I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )}$ (the regularized incomplete beta function)
Mean	${\displaystyle np}$
Median	${\displaystyle \lfloor np\rfloor }$ or ${\displaystyle \lceil np\rceil }$
Mode	${\displaystyle \lfloor (n+1)p\rfloor }$ or ${\displaystyle \lceil (n+1)p\rceil -1}$
Variance	${\displaystyle npq=np(1-p)}$
Skewness	${\displaystyle {\frac {q-p}{\sqrt {npq}}}}$
Ex. kurtosis	${\displaystyle {\frac {1-6pq}{npq}}}$
Entropy	${\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)}$ in shannons. For nats, use the natural log in the log.
MGF	${\displaystyle (q+pe^{t})^{n}}$
CF	${\displaystyle (q+pe^{it})^{n}}$
PGF	${\displaystyle G(z)=[q+pz]^{n}}$
Fisher information	${\displaystyle g_{n}(p)={\frac {n}{pq}}}$ (for fixed ${\displaystyle n}$)

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.