poisscdf

Poisson cumulative distribution function

Syntax

y = poisscdf(x,lambda)

y = poisscdf(x,lambda,'upper')

Description

y = poisscdf(x,lambda) computes the Poisson cumulative distribution function at each of the values in x using the rate parameters in lambda.

x and lambda can be scalars, vectors, matrices, or multidimensional arrays that all have the same size. If only one argument is a scalar, poisscdf expands it to a constant array with the same dimensions as the other argument.

y = poisscdf(x,lambda,'upper') returns the complement of the Poisson cumulative distribution function at each value in x, using an algorithm that computes the extreme upper tail probabilities more accurately.

Examples

Compute and Plot Poisson Cumulative Distribution Function

Compute and plot the Poisson cumulative distribution function for the specified range of integer values and average rate.

A computer hard disk manufacturing facility performs random tests of individual hard disks. The policy is to shut down the manufacturing process if an inspector finds more than four bad sectors on a disk. Assuming that on average a disk has two bad sectors, find the probability of a manufacturing process shutdown after the first inspection.

1 - poisscdf(4,2)

ans = 0.0527

Compute the probabilities a manufacturing process shutdown after the first inspection if on average a disk has 0, 1, 2, ..., 10 bad sectors.

lambda = 0:10;
y = 1 - poisscdf(4,lambda);

Plot the results.

scatter(lambda,y,'Marker',"o")
grid on

Compute Extreme Upper Tail Probabilities

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

A computer hard disk manufacturing facility performs random tests of individual hard disks. Assuming that on average a disk has 10 bad sectors, find the probability that a disk has more than 100 bad sectors.

format long
1 - poisscdf(100,10)

ans = 
     0

This result shows that poisscdf(100,10) 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 Poisson cumulative distribution function directly instead of computing the difference.

poisscdf(100,10,'upper')

ans = 
     5.339405460719755e-64

Input Arguments

x — Values at which to evaluate Poisson cdf

scalar value | array of scalar values

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

Example: [0,1,3,4]

Data Types: single | double

lambda — Rate parameters

positive value | array of positive values

Rate parameters, specified as a positive value or array of positive values. The rate parameter indicates the average number of events in a given time interval.

Example: 2

Data Types: single | double

Output Arguments

y — Poisson cdf values

scalar value | array of scalar values

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

More About

Poisson Cumulative Distribution Function

The Poisson cumulative distribution function lets you obtain the probability of an event occurring within a given time or space interval less than or equal to x times if on average the event occurs λ times within that interval.

The Poisson cumulative distribution function for the given values x and λ is

$$p=F(x|\lambda )=e^{-\lambda }{\displaystyle \sum _{i=0}^{floor(x)}\frac{{\lambda }^i}{i!}}.$$

Alternative Functionality

• poisscdf is a function specific to Poisson 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 PoissonDistribution probability distribution object and pass the object as an input argument. Note that the distribution-specific function poisscdf 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.

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).