# Binomial distribution

## Probability distribution / From Wikipedia, the free encyclopedia

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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 popular binomial test of statistical significance.^{[1]}

**Quick Facts**Notation, Parameters ...

Probability mass function | |||

Cumulative distribution function | |||

Notation | $B(n,p)$ | ||
---|---|---|---|

Parameters |
$n\in \{0,1,2,\ldots \}$ – number of trials $p\in [0,1]$ – success probability for each trial $q=1-p$ | ||

Support | $k\in \{0,1,\ldots ,n\}$ – number of successes | ||

PMF | ${\binom {n}{k}}p^{k}q^{n-k}$ | ||

CDF | $I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )$ (the regularized incomplete beta function) | ||

Mean | $np$ | ||

Median | $\lfloor np\rfloor$ or $\lceil np\rceil$ | ||

Mode | $\lfloor (n+1)p\rfloor$ or $\lceil (n+1)p\rceil -1$ | ||

Variance | $npq=np(1-p)$ | ||

Skewness | ${\frac {q-p}{\sqrt {npq}}}$ | ||

Excess kurtosis | ${\frac {1-6pq}{npq}}$ | ||

Entropy |
${\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)$ in shannons. For nats, use the natural log in the log. | ||

MGF | $(q+pe^{t})^{n}$ | ||

CF | $(q+pe^{it})^{n}$ | ||

PGF | $G(z)=[q+pz]^{n}$ | ||

Fisher information |
$g_{n}(p)={\frac {n}{pq}}$ (for fixed $n$) |

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.