unidcdf

Discrete uniform cumulative distribution function

Syntax

p = unidcdf(x,n)

p = unidcdf(x,n,"upper")

Description

p = unidcdf(x,n) returns the cumulative distribution function (cdf) of the discrete uniform distribution with the maximum value n, evaluated at the values in x.

p = unidcdf(x,n,"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 and Plot Discrete Uniform Cumulative Distribution Function

Compute the cumulative distribution function (cdf) of the discrete uniform distribution with the maximum value n=10, for values between 0 and 10.

x = 0:0.01:10;
n = 10;
p = unidcdf(x,n);

Plot the cdf.

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

Estimate Population Size

Use the discrete uniform distribution to estimate the size n of a collection of items that have serial numbers from 1 to n, based on a sample of observed serial numbers. This example is based on the famous German tank problem [1].

Suppose you observe k=5 items at random from the collection, and the highest observed serial number maxSerial is 40.

k = 5;  % Number of observed items
maxSerial = 40;  % Highest observed serial number

Compute the probability p of observing the highest serial number maxSerial in a sample of k items drawn without replacement from a population of size n ≥ maxSerial.

n = maxSerial:1:100;
p = 1 - unidcdf(maxSerial,n).^k;

Plot the probability as a function of n, and add a red dashed line corresponding to p=0.95.

plot(n,p)
grid on
yline(0.95,"r--")
xlabel("n")
ylabel("p")

As shown in the plot, the 95% confidence upper bound for the size of the collection is 73.

Input Arguments

x — Values at which to evaluate discrete uniform cdf

nonnegative scalar | array of nonnegative scalars

Values at which to evaluate the discrete uniform cdf, specified as a nonnegative scalar or an array of nonnegative scalars. The values of x do not need to be integers.

To evaluate the cdf at multiple values, specify x as an array. To evaluate the cdfs of multiple distributions, specify n as an array. If x and n are both arrays, they must have the same size. If only x or n is an array, unidcdf expands the other input argument into a constant array of the same size as the array input. Each element in p is the cdf value of the distribution specified by the corresponding element in n, evaluated at the corresponding element in x.

Data Types: single | double

n — Maximum value in discrete uniform distribution

positive integer | array of positive integers

Maximum value in the discrete uniform distribution, specified as a positive integer or an array of positive integers.

To evaluate the cdf at multiple values, specify x as an array. To evaluate the cdfs of multiple distributions, specify n as an array. If x and n are both arrays, they must have the same size. If only x or n is an array, unidcdf expands the other input argument into a constant array of the same size as the array input. Each element in p is the cdf value of the distribution specified by the corresponding element in n, evaluated at the corresponding element in x.

Data Types: single | double

Output Arguments

p — Discrete uniform cdf values

numeric scalar | numeric array

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

More About

Discrete Uniform cdf

The discrete uniform cdf is

$$p=F(x|N)=\frac{\text{Floor}\left(x\right)}{N}{I}_{(1,\mathrm{...},N)}(x)$$

The indicator function $$I_{(0,\mathrm{...},N)}(x)$$ ensures that x adopts only values in the range [0, N]. The result p is the probability that a single observation from the discrete uniform distribution with maximum N will be a positive integer less than or equal to x.

For more information, see Uniform Distribution (Discrete).

Alternative Functionality

• unidcdf is a function specific to the discrete uniform 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. Note that the distribution-specific function unidcdf 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.

References

[1] Lee, Anthony, and Steven J. Miller. Generalizing the German Tank Problem. arXiv:2210.15339. Preprint, arXiv, October 27, 2022.

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