binoinv

Binomial inverse cumulative distribution function

Syntax

x = binoinv(y,n,p)

Description

x = binoinv(y,n,p) returns the smallest integer x such that the binomial cdf evaluated at x is equal to or greater than y. You can think of y as the probability of observing x successes in n independent trials where p is the probability of success in each trial. Each value in x is a positive integer less than or equal to n.

Examples

Inverse of Binomial Distribution cdf

Find an interval that contains 95% of the values from a binomial distribution with 100 trials and a 0.6 probability of success in each trial.

n = 100;
p = 0.6;
x = binoinv([0.025 0.975],n,p)

x = 1×2

    50    69

There is a 2.5% probability of getting less than or equal to 50 successes, and 97.5% probability of getting less than or equal to 69 successes. In other words, in 95% of the trials, the number of successes will be between 50 to 69.

Input Arguments

y — Probability values at which to evaluate inverse of cdf

scalar value in [0,1] | array of scalar values

Probability values at which to evaluate the inverse of the cdf (icdf), specified as a scalar value or an array of scalar values, where each element is in the range [0,1].

To evaluate the icdf at multiple values, specify y using an array. To evaluate the icdfs of multiple distributions, specify n and p using arrays. If one or more of the input arguments y, n, and p are arrays, then the array sizes must be the same. In this case, binoinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in n and p, evaluated at the corresponding element in y.

Example: [0.1,0.5,0.9]

Data Types: single | double

n — Number of trials

positive integer | array of positive integers

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

To evaluate the icdfs of multiple distributions, specify n and p using arrays. If one or more of the input arguments y, n, and p are arrays, then the array sizes must be the same. In this case, binoinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in n and p, evaluated at the corresponding element in y.

Example: [10 20 50 100]

Data Types: single | double

p — Probability of success for each trial

scalar value from interval [0 1] | array of scalar values from interval [0 1]

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].

To evaluate the icdfs of multiple distributions, specify n and p using arrays. If one or more of the input arguments y, n, and p are arrays, then the array sizes must be the same. In this case, binoinv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in n and p, evaluated at the corresponding element in y.

Example: [0.01 0.1 0.5 0.7]

Data Types: single | double

Output Arguments

x — icdf values

scalar value | array of scalar values

icdf values, evaluated at the probability values in y, returned as a scalar value or an array of scalar values. x is the same size as y, n, and p after any necessary scalar expansion. Each element in x is the icdf value of the distribution specified by the corresponding elements in n and p, evaluated at the corresponding element in y.

Algorithms

The software uses various methods to approximate the inverse of the binomial cumulative distribution function:

1. It applies an asymptotic inversion using the complementary error function under valid conditions specified in [1].

2. For values of n greater than or equal to 20 and p less than or equal to 0.05, a Poisson approximation is used if the asymptotic method is not suitable.

3. If neither of the previous methods are applicable, the software uses the Cornish-Fisher [2] expansion truncated at the first order to include skewness.

After computing the approximate inverse, the software finds the exact inverse by searching in the vicinity of the approximated value.

References

[1] Gil, A., J. Segura, and N. M. Temme. “Asymptotic Inversion of the Binomial and Negative Binomial Cumulative Distribution Functions.” ETNA - Electronic Transactions on Numerical Analysis 52 (2020): 270–80, Section 3, Pg 6.

[2] Cornish, E. A., and R. A. Fisher. “Moments and Cumulants in the Specification of Distributions.” Revue de l’Institut International de Statistique / Review of the International Statistical Institute 5, no. 4 (January 1938): 307.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

The generated code does not compute the binomial inverse using the method described in Algorithms. Instead, it evaluates the inverse by checking the binomial cdf at incremental values of x starting from x=0 till the corresponding binomial cdf value is equal to or greater than the specified y.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2025a: Improved performance of binomial inverse algorithm for large number of trials

The binoinv function shows improved performance when computing the binomial inverse cdf for number of trials greater than 100. For example, the binomial inverse computation in this code is about 14x faster than in the previous release:

function timingTest()
    nval = 500;
    pval = 0.6;
    yval = 0.6;
    x = binoinv(yval,nval,pval);
end

timeit(@timingTest)

The approximate execution times are:

• R2025a: 0.0015 s

• R2024b: 0.022 s

The code was timed on a Windows^® 11, Intel^® Xeon^® E5-1650 v3 CPU @ 3.5 GHz test system by using the timeit function.