Binomial random variable

binomial

Probability mass function
Cumulative distribution function
Notation	B(n, p)
Parameters	n ∈ N0 — number of trials p ∈ [0,1] — success probability in each trial
Support	k ∈ { 0, …, n } — number of successes
pmf	${\displaystyle \textstyle {n \choose k}\,p^{k}(1-p)^{n-k}}$
CDF	${\displaystyle \textstyle I_{1-p}(n-k,1+k)}$
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	${\displaystyle np(1-p)}$
Skewness	${\displaystyle {\frac {1-2p}{\sqrt {np(1-p)}}}}$
Ex. kurtosis	${\displaystyle {\frac {1-6p(1-p)}{np(1-p)}}}$
Entropy	${\displaystyle {\frac {1}{2}}\log _{2}{\big (}2\pi e\,np(1-p){\big )}+O\left({\frac {1}{n}}\right)}$ in shannons. For nats, use the natural log in the log.
MGF	${\displaystyle (1-p+pe^{t})^{n}\!}$
CF	${\displaystyle (1-p+pe^{it})^{n}\!}$
PGF	${\displaystyle G(z)=\left[(1-p)+pz\right]^{n}.}$
Fisher information	${\displaystyle g_{n}(p)={\frac {n}{p(1-p)}}}$ (for fixed ${\displaystyle n}$)

${\displaystyle g_{n}(p)={\frac {n}{p(1-p)}}}$

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 remains a good approximation, and is widely used.