# binocdf

Binomial cumulative distribution function

## Description

example

y = binocdf(x,n,p) computes a binomial cumulative distribution function at each of the values in x using the corresponding number of trials in n and the probability of success for each trial in p.

x, n, and p can be vectors, matrices, or multidimensional arrays of the same size. Alternatively, one or more arguments can be scalars. The binocdf function expands scalar inputs to constant arrays with the same dimensions as the other inputs.

example

y = binocdf(x,n,p,'upper') returns the complement of the binomial cumulative distribution function at each value in x, using an algorithm that computes the extreme upper tail probabilities more accurately than the default algorithm.

## Examples

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Compute and plot the binomial cumulative distribution function for the specified range of integer values, number of trials, and probability of success for each trial.

A baseball team plays 100 games in a season and has a 50-50 chance of winning each game. Find the probability of the team winning more than 55 games in a season.

format long
1 - binocdf(55,100,0.5)
ans =
0.135626512036917

Find the probability of the team winning between 50 and 55 games in a season.

binocdf(55,100,0.5) - binocdf(49,100,0.5)
ans =
0.404168106656672

Compute the probabilities of the team winning more than 55 games in a season if the chance of winning each game ranges from 10% to 90%.

chance = 0.1:0.05:0.9;
y = 1 - binocdf(55,100,chance);

Plot the results.

scatter(chance,y)
grid on

Compute the complement of the binomial cumulative distribution function with more accurate upper tail probabilities.

A baseball team plays 100 games in a season and has a 50-50 chance of winning each game. Find the probability of the team winning more than 95 games in a season.

format long
1 - binocdf(95,100,0.5)
ans =
0

This result shows that the probability is so close to 1 (within eps) that subtracting it from 1 gives 0. To approximate the extreme upper tail probabilities better, compute the complement of the binomial cumulative distribution function directly instead of computing the difference.

binocdf(95,100,0.5,'upper')
ans =
3.224844447881779e-24

Alternatively, use the binopdf function to find the probabilities of the team winning 96, 97, 98, 99, and 100 games in a season. Find the sum of these probabilities by using the sum function.

sum(binopdf(96:100,100,0.5),'all')
ans =
3.224844447881779e-24

## Input Arguments

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Values at which to evaluate the binomial cdf, specified as an integer or an array of integers. All values of x must belong to the interval [0 n], where n is the number of trials.

Example: [0 1 3 4]

Data Types: single | double

Number of trials, specified as a positive integer or an array of positive integers.

Example: [10 20 50 100]

Data Types: single | double

Probability of success for each trial, specified as a scalar value or an array of scalar values. All values of p must belong to the interval [0 1].

Example: [0.01 0.1 0.5 0.7]

Data Types: single | double

## Output Arguments

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Binomial cdf values, returned as a scalar value or an array of scalar values. Each element in y is the binomial cdf value of the distribution evaluated at the corresponding element in x.

Data Types: single | double

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### Binomial Cumulative Distribution Function

The binomial cumulative distribution function lets you obtain the probability of observing less than or equal to x successes in n trials, with the probability p of success on a single trial.

The binomial cumulative distribution function for a given value x and a given pair of parameters n and p is

$y=F\left(x|n,p\right)=\sum _{i=0}^{x}\left(\begin{array}{c}n\\ i\end{array}\right){p}^{i}{\left(1-p\right)}^{\left(n-i\right)}{I}_{\left(0,1,...,n\right)}\left(i\right).$

The resulting value y is the probability of observing up to x successes in n independent trials, where the probability of success in any given trial is p. The indicator function ${I}_{\left(0,1,...,n\right)}\left(i\right)$ ensures that x only adopts values of 0,1,...,n.

## Alternative Functionality

• binocdf is a function specific to binomial distribution. Statistics and Machine Learning Toolbox™ also offers the generic function cdf, which supports various probability distributions. To use cdf, specify the probability distribution name and its parameters. Alternatively, create a BinomialDistribution probability distribution object and pass the object as an input argument. Note that the distribution-specific function binocdf is faster than the generic function cdf.

• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.