betacdf

Beta cumulative distribution function

Syntax

p = betacdf(x,a,b)

p = betacdf(x,a,b,"upper")

Description

p = betacdf(x,a,b) returns the cumulative distribution function (cdf) of the beta distribution with the shape parameters in a and b, evaluated at the values in x.

p = betacdf(x,a,b,"upper") returns the complement of the cdf, evaluated at the values in x, using an algorithm that more accurately computes the extreme upper-tail probabilities as compared to subtracting the lower-tail value from 1.

Examples

Compute Beta Cumulative Distribution Function

Compute the cumulative distribution function (cdf) for a beta distribution with the shape parameters a and b, evaluated at the values between 0 and 1.

x = 0:0.01:1;
a = 2;
b = 2;
p = betacdf(x,a,b);

Plot the cdf.

plot(x,p)
grid on
xlabel("x")
ylabel("p")

Compute Extreme Upper-Tail Probabilities for Beta Distribution

Determine the probability of sampling a number greater than 0.9999 from a beta distribution with the shape parameters a=2 and b=2.

p1 = betacdf(0.9999,2,2)

p1 = 
1.0000

This result shows that the probability of sampling a number less than or equal to 0.9999 is so close to 1 that subtracting the number from 1 gives 0.

To approximate the extreme upper-tail probability with greater precision, compute the complement of the beta cdf directly.

p2 = betacdf(0.9999,2,2,"upper")

p2 = 
2.9998e-08

The output indicates a small probability of sampling a number greater than 0.9999.

Input Arguments

x — Values at which to evaluate beta cdf

nonnegative scalar value | array of nonnegative scalar values

Values at which to evaluate the beta cdf, specified as a nonnegative scalar value or an array of nonnegative scalar values.

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

Data Types: single | double

a — First shape parameter

positive scalar value | array of positive scalar values

First shape parameter, specified as a positive scalar value or an array of positive scalar values.

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

Data Types: single | double

b — Second shape parameter

positive scalar value | array of positive scalar values

Second shape parameter, specified as a positive scalar value or an array of positive scalar values.

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

Data Types: single | double

Output Arguments

p — Beta cdf values

scalar value | array of scalar values

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

More About

Beta cdf

The beta cdf is

$$p=F\left(x|a,b\right)=\frac{1}{B(a,b)}{\displaystyle \underset{0}{\overset{x}{\int }}{t}^{a-1}{(1-t)}^{b-1}dt}$$

for 0 ≤ x ≤ 1, where B( · ) is the Beta function. The result p is the probability that a single observation from a beta distribution with shape parameters a and b falls in the interval [0,x].

For more information, see Beta Distribution.

Alternative Functionality

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

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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