Statistics: PoissonDistribution

Class Definition: `PoissonDistribution`

statistics: PoissonDistribution

Poisson probability distribution object.

A PoissonDistribution object consists of parameters, a model description, and sample data for a Poisson probability distribution.

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence. It is defined by the rate parameter lambda.

There are several ways to create a PoissonDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor PoissonDistribution (lambda) to create a Poisson distribution with fixed parameter value lambda.
Use the static method PoissonDistribution.fit (x, freq) to fit a distribution to the data in x using the same input arguments as the poissfit function.

It is highly recommended to use fitdist and makedist functions to create probability distribution objects, instead of the class constructor or the aforementioned static method.

Further information about the Poisson distribution can be found at https://en.wikipedia.org/wiki/Poisson_distribution

Source Code: PoissonDistribution

Properties

A positive scalar value characterizing the rate of the Poisson distribution. You can access the lambda property using dot name assignment.

Example: 1


 ## Create a Poisson distribution from data
 rand ("seed", 21);
 data = poissrnd (5, 100, 1);
 pd = fitdist (data, "Poisson");

 ## Query parameter 'lambda' (rate parameter)
 pd.lambda

 ## Set parameter 'lambda'
 pd.lambda = 10

 ## Use this to initialize or modify the rate parameter of a Poisson distribution.
 ## The rate parameter must be a positive real scalar, representing both the mean
 ## and variance, ideal for count data modeling.

ans = 4.9100
pd =
  PoissonDistribution

  Poisson distribution
   lambda = 10

Example: 2


 ## Create a Poisson distribution object by calling its constructor
 pd = PoissonDistribution (7.5)

 ## Query parameter 'lambda'
 pd.lambda

 ## This shows direct construction with a specific rate parameter, useful for
 ## theoretical analysis or when lambda is estimated from prior knowledge.

pd =
  PoissonDistribution

  Poisson distribution
   lambda = 7.5

ans = 7.5000

A character vector specifying the name of the probability distribution object. This property is read-only.

A scalar integer value specifying the number of parameters characterizing the probability distribution. This property is read-only.

A $1×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $1×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $1×1$ numeric vector containing the values of the distribution parameters. This property is read-only. You can change the distribution parameters by assigning new values to the lambda property.

A $1×1$ numeric matrix containing the variance of the parameter estimate. This matrix is only meaningful when the distribution was fitted to data. If the distribution object was created with fixed parameters, or a parameter of a fitted distribution is modified, then the variance is zero. This property is read-only.

A logical scalar specifying whether the parameter is fixed or estimated. A true value corresponds to a fixed parameter, a false value corresponds to a parameter estimate. This property is read-only.

A $1×2$ numeric vector specifying the truncation interval for the probability distribution. The first element contains the lower boundary, the second element contains the upper boundary. This property is read-only. You can only truncate a probability distribution with the truncate method.

A logical scalar value specifying whether a probability distribution is truncated or not. This property is read-only.

A scalar structure containing the following fields:

data: a numeric vector containing the data used for distribution fitting.
cens: a numeric vector of logical values indicating censoring information corresponding to the elements of the data used for distribution fitting. If no censoring vector was used for distribution fitting, then this field defaults to an empty array.
freq: a numeric vector of non-negative integer values containing the frequency information corresponding to the elements of the data used for distribution fitting. If no frequency vector was used for distribution fitting, then this field defaults to an empty array.

Methods

PoissonDistribution: p = cdf (pd, x)
PoissonDistribution: p = cdf (pd, x, "upper")

p = cdf (pd, x) computes the CDF of the probability distribution object, pd, evaluated at the values in x.

p = cdf (…, "upper") returns the complement of the CDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1


 ## Plot various CDFs from the Poisson distribution
 x = 0:20;
 data1 = poissrnd (2, 10000, 1);
 data2 = poissrnd (5, 10000, 1);
 data3 = poissrnd (10, 10000, 1);
 pd1 = fitdist (data1, "Poisson");
 pd2 = fitdist (data2, "Poisson");
 pd3 = fitdist (data3, "Poisson");
 p1 = cdf (pd1, x);
 p2 = cdf (pd2, x);
 p3 = cdf (pd3, x);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"lambda = 2", "lambda = 5", "lambda = 10"}, "location", "southeast")
 title ("Poisson CDF")
 xlabel ("values in x (non-negative integers)")
 ylabel ("Cumulative probability")

 ## Use this to compute and visualize the cumulative distribution function
 ## for different Poisson distributions, showing the probability of observing
 ## at most k events, useful in risk assessment or queueing theory.

PoissonDistribution: x = icdf (pd, p)

x = icdf (pd, p) computes the quantile (the inverse of the CDF) of the probability distribution object, pd, evaluated at the values in p.

Example: 1


 ## Plot various iCDFs from the Poisson distribution
 p = 0.001:0.001:0.999;
 data1 = poissrnd (2, 10000, 1);
 data2 = poissrnd (5, 10000, 1);
 data3 = poissrnd (10, 10000, 1);
 pd1 = fitdist (data1, "Poisson");
 pd2 = fitdist (data2, "Poisson");
 pd3 = fitdist (data3, "Poisson");
 x1 = icdf (pd1, p);
 x2 = icdf (pd2, p);
 x3 = icdf (pd3, p);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 legend ({"lambda = 2", "lambda = 5", "lambda = 10"}, "location", "northwest")
 title ("Poisson iCDF")
 xlabel ("Probability")
 ylabel ("values in x (non-negative integers)")

 ## This demonstrates the inverse CDF (quantiles) for Poisson distributions,
 ## useful for finding the number of events corresponding to a given probability,
 ## such as in inventory management for stock levels.

PoissonDistribution: r = iqr (pd)

r = iqr (pd) computes the interquartile range of the probability distribution object, pd.

Example: 1


 ## Compute the interquartile range for a Poisson distribution
 data = poissrnd (5, 10000, 1);
 pd = fitdist (data, "Poisson");
 iqr_value = iqr (pd)

 ## Use this to calculate the interquartile range, measuring the spread of the
 ## middle 50% of the distribution, helpful for understanding variability in count data.

iqr_value = 3

PoissonDistribution: m = mean (pd)

m = mean (pd) computes the mean of the probability distribution object, pd.

Example: 1


 ## Compute the mean for different Poisson distributions
 data1 = poissrnd (2, 10000, 1);
 data2 = poissrnd (5, 10000, 1);
 pd1 = fitdist (data1, "Poisson");
 pd2 = fitdist (data2, "Poisson");
 mean1 = mean (pd1)
 mean2 = mean (pd2)

 ## This shows how to compute the expected value for Poisson distributions,
 ## which equals lambda, representing the average number of events.

mean1 = 2.0276
mean2 = 5.0409

PoissonDistribution: m = median (pd)

m = median (pd) computes the median of the probability distribution object, pd.

Example: 1


 ## Compute the median for different Poisson distributions
 data1 = poissrnd (2, 10000, 1);
 data2 = poissrnd (5, 10000, 1);
 pd1 = fitdist (data1, "Poisson");
 pd2 = fitdist (data2, "Poisson");
 median1 = median (pd1)
 median2 = median (pd2)

 ## Use this to find the median count, which splits the distribution into
 ## two equal probability halves, robust for skewed count data.

median1 = 2
median2 = 5

PoissonDistribution: nlogL = negloglik (pd)

nlogL = negloglik (pd) computes the negative loglikelihood of the probability distribution object, pd.

Example: 1


 ## Compute the negative loglikelihood for a fitted Poisson distribution
 rand ("seed", 21);
 data = poissrnd (5, 100, 1);
 pd_fitted = fitdist (data, "Poisson");
 nlogL = negloglik (pd_fitted)

 ## This is useful for assessing the fit of a Poisson distribution to data,
 ## with lower values indicating better fit, often used in model selection.

nlogL = -209.83

PoissonDistribution: ci = paramci (pd)
PoissonDistribution: ci = paramci (pd, Name, Value)

ci = paramci (pd) computes the lower and upper boundaries of the 95% confidence interval for each parameter of the probability distribution object, pd.

ci = paramci (pd, Name, Value) computes the confidence intervals with additional options specified by Name-Value pair arguments listed below.

`Name`		`Value`
`"Alpha"`		A scalar value in the range $(0,1)$ specifying the significance level for the confidence interval. The default value 0.05 corresponds to a 95% confidence interval.
`"Parameter"`		A character vector or a cell array of character vectors specifying the parameter names for which to compute confidence intervals. By default, `paramci` computes confidence intervals for all distribution parameters.

paramci is meaningful only when pd is fitted to data, otherwise an empty array, [], is returned.

Example: 1


 ## Compute confidence intervals for parameters of a fitted Poisson distribution
 rand ("seed", 21);
 data = poissrnd (5, 1000, 1);
 pd_fitted = fitdist (data, "Poisson");
 ci = paramci (pd_fitted, "Alpha", 0.05)

 ## Use this to obtain confidence intervals for the estimated parameter (lambda),
 ## providing a range of plausible values based on the data.

ci =

   4.8239
   5.1001

PoissonDistribution: y = pdf (pd, x)

y = pdf (pd, x) computes the PDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1


 ## Plot various PDFs from the Poisson distribution
 x = 0:20;
 data1 = poissrnd (2, 10000, 1);
 data2 = poissrnd (5, 10000, 1);
 data3 = poissrnd (10, 10000, 1);
 pd1 = fitdist (data1, "Poisson");
 pd2 = fitdist (data2, "Poisson");
 pd3 = fitdist (data3, "Poisson");
 y1 = pdf (pd1, x);
 y2 = pdf (pd2, x);
 y3 = pdf (pd3, x);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 legend ({"lambda = 2", "lambda = 5", "lambda = 10"}, "location", "northeast")
 title ("Poisson PDF")
 xlabel ("values in x (non-negative integers)")
 ylabel ("Probability")

 ## This visualizes the probability mass function for Poisson distributions,
 ## showing the likelihood of exact count values.

PoissonDistribution: plot (pd)
PoissonDistribution: plot (pd, Name, Value)
PoissonDistribution: h = plot (…)

plot (pd) plots a probability density function (PDF) of the probability distribution object pd. If pd contains data, which have been fitted by fitdist, the PDF is superimposed over a histogram of the data.

plot (pd, Name, Value) specifies additional options with the Name-Value pair arguments listed below.

`Name`		`Value`
`"PlotType"`		A character vector specifying the plot type. `"pdf"` plots the probability density function (PDF). When `pd` is fit to data, the PDF is superimposed on a histogram of the data. `"cdf"` plots the cumulative density function (CDF). When `pd` is fit to data, the CDF is superimposed over an empirical CDF. `"probability"` plots a probability plot using a CDF of the data and a CDF of the fitted probability distribution. This option is available only when `pd` is fitted to data.
`"Discrete"`		A logical scalar to specify whether to plot the PDF or CDF of a discrete distribution object as a line plot or a stem plot, by specifying `false` or `true`, respectively. By default, it is `true` for discrete distributions and `false` for continuous distributions.
`"Parent"`		An axes graphics object for the plot. If not specified, the `plot` function plots into the current axes or creates a new axes object if one does not exist.

h = plot (…) returns a graphics handle to the plotted objects.

Example: 1


 ## Create a Poisson distribution with fixed parameter lambda=5 and plot its PDF.
 data = poissrnd (5, 10000, 1);
 pd = fitdist (data, "Poisson");
 plot (pd)
 title ("Fixed Poisson distribution with lambda = 5")

Example: 2


 ## Generate a data set of 100 random samples from a Poisson distribution with
 ## lambda=5. Fit a Poisson distribution to this data and plot its CDF superimposed
 ## over an empirical CDF.
 rand ("seed", 21);
 data = poissrnd (5, 100, 1);
 pd_fitted = fitdist (data, "Poisson");
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Poisson distribution with lambda = %0.2f";
 title (sprintf (txt, pd_fitted.lambda))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

 ## Use this to visualize the fitted CDF compared to the empirical CDF of the data,
 ## useful for assessing model fit in count data.

Example: 3


 ## Generate a data set of 200 random samples from a Poisson distribution with
 ## lambda=5. Display a probability plot for the Poisson distribution fit to the data.
 rand ("seed", 21);
 data = poissrnd (5, 200, 1);
 pd_fitted = fitdist (data, "Poisson");
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Poisson distribution with lambda = %0.2f");
 title (sprintf (txt, pd_fitted.lambda))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

 ## This creates a probability plot to compare the fitted distribution to the data,
 ## useful for checking if the Poisson model is appropriate for count data.

PoissonDistribution: [nlogL, param] = proflik (pd, pnum)
PoissonDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)
PoissonDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
PoissonDistribution: [nlogL, param] = proflik (pd, pnum, setparam, "Display", display)

[nlogL, param] = proflik (pd, pnum) returns a vector nlogL of negative loglikelihood values and a vector param of corresponding parameter values for the parameter in the position indicated by pnum. By default, proflik uses the lower and upper bounds of the 95% confidence interval and computes 100 equispaced values for the selected parameter. pd must be fitted to data.

[nlogL, param] = proflik (pd, pnum, "Display", "on") also plots the profile likelihood against the default range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam) defines a user-defined range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam, "Display", "on") also plots the profile likelihood against the user-defined range of the selected parameter.

For the Poisson distribution, pnum = 1 selects the parameter lambda.

When opted to display the profile likelihood plot, proflik also plots the baseline loglikelihood computed at the lower bound of the 95% confidence interval and estimated maximum likelihood. The latter might not be observable if it is outside of the user-defined range of parameter values.

Example: 1


 ## Compute and plot the profile likelihood for the rate parameter of a fitted
 ## Poisson distribution
 rand ("seed", 21);
 data = poissrnd (5, 1000, 1);
 pd_fitted = fitdist (data, "Poisson");
 [nlogL, param] = proflik (pd_fitted, 1, "Display", "on");

 ## Use this to analyze the profile likelihood of the rate parameter (lambda),
 ## helping to understand uncertainty in parameter estimates for count models.

PoissonDistribution: r = random (pd)
PoissonDistribution: r = random (pd, rows)
PoissonDistribution: r = random (pd, rows, cols, …)
PoissonDistribution: r = random (pd, [sz])

r = random (pd) returns a random number from the distribution object pd.

When called with a single size argument, poissrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Example: 1


 ## Generate random samples from a Poisson distribution
 rand ("seed", 21);
 samples = poissrnd (5, 500, 1);
 hist (samples, 20)
 title ("Histogram of 500 random samples from Poisson(lambda=5)")
 xlabel ("values in x (non-negative integers)")
 ylabel ("Frequency")

 ## This generates random count samples from a Poisson distribution, useful
 ## for simulating event occurrences like defect counts or traffic arrivals.

PoissonDistribution: s = std (pd)

s = std (pd) computes the standard deviation of the probability distribution object, pd.

Example: 1


 ## Compute the standard deviation for a Poisson distribution
 data = poissrnd (5, 10000, 1);
 pd = fitdist (data, "Poisson");
 std_value = std (pd)

 ## Use this to calculate the standard deviation, which equals sqrt(lambda),
 ## measuring variability in event counts.

std_value = 2.2371

PoissonDistribution: t = truncate (pd, lower, upper)

t = truncate (pd, lower, upper) returns a probability distribution t, which is the probability distribution pd truncated to the specified interval with lower limit, lower, and upper limit, upper. If pd is fitted to data with fitdist, the returned probability distribution t is not fitted, does not contain any data or estimated values, and it is as it has been created with the makedist function, but it includes the truncation interval.

Example: 1


 ## Plot the PDF of a Poisson distribution, with lambda=5, truncated at [2, 10]
 ## intervals. Generate 10000 random samples from this truncated distribution and
 ## superimpose a histogram scaled accordingly
 rand ("seed", 21);
 data_all = poissrnd (5, 20000, 1);
 data = data_all(data_all >= 2 & data_all <= 10);
 data = data(1:10000);

 pd = fitdist (data, "Poisson");
 t = truncate (pd, 2, 10);

 ## Plot histogram and truncated PDF
 plot (t)
 hold on
 [counts, centers] = hist (data, 20);
 bin_width = centers(2) - centers(1);
 counts_norm = counts / (sum (counts) * bin_width);
 bar (centers, counts_norm, 1);

 title ("Poisson distribution (lambda=5) truncated at [2, 10]")
 legend ("Truncated PDF", "Histogram")

 ## This demonstrates truncating a Poisson distribution to a specific range
 ## and visualizing the result with random samples, useful for bounded count data.

PoissonDistribution: v = var (pd)

v = var (pd) computes the variance of the probability distribution object, pd.

Example: 1


 ## Compute the variance for a Poisson distribution
 data = poissrnd (5, 10000, 1);
 pd = fitdist (data, "Poisson");
 var_value = var (pd)

 ## Use this to calculate the variance, which equals lambda, quantifying the
 ## spread in count data.

var_value = 4.9747

Examples

 pd_fixed = makedist ("Poisson", "lambda", 5)
 rand ("seed", 2);
 data = random (pd_fixed, 5000, 1);
 pd_fitted = fitdist (data, "Poisson")
 plot (pd_fitted)
 msg = "Fitted Poisson distribution with lambda = %0.2f";
 title (sprintf (msg, pd_fitted.lambda))

pd_fixed =
  PoissonDistribution

  Poisson distribution
   lambda = 5

pd_fitted =
  PoissonDistribution

  Poisson distribution
   lambda = 4.9748   [4.91298, 5.03662]

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Class Definition: PoissonDistribution

statistics: PoissonDistribution

Properties

lambda

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

Methods

cdf

PoissonDistribution: p = cdf (pd, x)

PoissonDistribution: p = cdf (pd, x, "upper")

Example: 1

icdf

PoissonDistribution: x = icdf (pd, p)

Example: 1

iqr

PoissonDistribution: r = iqr (pd)

Example: 1

mean

PoissonDistribution: m = mean (pd)

Example: 1

median

PoissonDistribution: m = median (pd)

Example: 1

negloglik

PoissonDistribution: nlogL = negloglik (pd)

Example: 1

paramci

PoissonDistribution: ci = paramci (pd)

PoissonDistribution: ci = paramci (pd, Name, Value)

Example: 1

Class Definition: `PoissonDistribution`

`lambda`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

`PoissonDistribution: p = cdf (pd, x)`

`PoissonDistribution: p = cdf (pd, x, "upper")`

`icdf`

`PoissonDistribution: x = icdf (pd, p)`

`iqr`

`PoissonDistribution: r = iqr (pd)`

`mean`

`PoissonDistribution: m = mean (pd)`

`median`

`PoissonDistribution: m = median (pd)`

`negloglik`

`PoissonDistribution: nlogL = negloglik (pd)`

`paramci`

`PoissonDistribution: ci = paramci (pd)`

`PoissonDistribution: ci = paramci (pd, Name, Value)`

`pdf`

`PoissonDistribution: y = pdf (pd, x)`

`plot`

`PoissonDistribution: plot (pd)`

`PoissonDistribution: plot (pd, Name, Value)`

`PoissonDistribution: h = plot (…)`

`proflik`

`PoissonDistribution: [nlogL, param] = proflik (pd, pnum)`

`PoissonDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)`

`PoissonDistribution: [nlogL, param] = proflik (pd, pnum, setparam)`

`PoissonDistribution: [nlogL, param] = proflik (pd, pnum, setparam, "Display", display)`

`random`

`PoissonDistribution: r = random (pd)`

`PoissonDistribution: r = random (pd, rows)`

`PoissonDistribution: r = random (pd, rows, cols, …)`

`PoissonDistribution: r = random (pd, [sz])`

`std`

`PoissonDistribution: s = std (pd)`

`truncate`

`PoissonDistribution: t = truncate (pd, lower, upper)`

`var`

`PoissonDistribution: v = var (pd)`