# Binomial distribution

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Consider an event—such as a coin coming down heads—that can be true or false in each trial—or each toss—with probability «p» -- it has a Bernoulli distribution. A binomial distribution describes the number of times an event is true -- e.g., the coin is heads -- in «n» independent trials—or tosses—where the event occurs with probability «p» on each trial.

## Functions

### Parameters

The binomial distribution is described by

- «n»: The number of trials
- «p»: The probability of success in each trial
- «k»: Number of successes
- «u» : The overall probability of
*k*successes in «n» trials (where*k*is the result). This is also known as the fractile or quartile level of the outcome. Must be*0 ≤ «u» ≤ 1*.

### Binomial(n, p)

The number of successes for a quantity described by a binomial distribution. This is the Poisson Distribution function. Use this to describe an uncertain variable that represents the number of successes in a repeated trial.

### ProbBinomial(k, n, p)

The Binomial distribution is a non-negative discrete distribution where the probability of outcome *k* is given by

- [math]\displaystyle{ P_{n,p}(k) = \left(\begin{array}{c}n\\k\end{array}\right) p^k (1-p)^{n-k} }[/math]

### CumBinomial(k, n, p)

The analytic cumulative probability function for a Poisson distribution. Returns the probability that the actual number of successes is less than or equal to «k».

### CumBinomialInv(u, n, p)

The analytic inverse cumulative probability function for a Poisson(n,p) distribution, also known as the *quantile function*. For 0<=u<1, this returns the number of successes, k, so that the probability of the outcome of «n» trials having «k» or fewer successes is less than or equal to «k».

#### Examples

An experiment is to be conducted that will consist of rolling a die 100 times and counting the number of times a 6 is rolled. With a 75% probability, the number of sixes that will be observe will be less than or equal to:

`CumBinomialInv(75%, 100, 1/6 )`

CumBinomialInv is the inverse of CumBinomial, so that for all *«n» ≥ 0, 0 ≤ k≤ «n»*, and *0 < «p» < 1*, the following holds (up to the available numeric precision):

`CumBinomialInv(CumBinomial(k, n, p), n, p) → k`

## Statistics

The distribution has the following statistics:

- Mean = [math]\displaystyle{ n p }[/math]
- Variance = [math]\displaystyle{ n p (1 - p) }[/math]
- Skewness = [math]\displaystyle{ {{1-2 p}\over{\sqrt{n p (1-p)}}} }[/math]
- Kurtosis = [math]\displaystyle{ {{1 - 6 p (1-p)}\over{n p (1-p)}} }[/math]

## See Also

- Multinomial -- A generalization of Binomial in which more than two outcomes are possible.
- NegativeBinomial -- the two other common discrete distributions on the non-negative integers
- Poisson distribution
- Normal distribution
- Parametric discrete distributions
- Distribution Densities Library

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