Binomial distribution

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