Statistics: BinomialDistribution

Class Definition: `BinomialDistribution`

statistics: BinomialDistribution

Binomial probability distribution object.

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

The binomial distribution is a discrete probability distribution that models the number of successes in a sequence of N independent trials, each with a probability of success p.

There are several ways to create a BinomialDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor BinomialDistribution (N, p) to create a binomial distribution with fixed parameter values N and p.
Use the static method BinomialDistribution.fit (x, ntrials, alpha) to fit a distribution to data x using the same input arguments as the binofit 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 binomial distribution can be found at https://en.wikipedia.org/wiki/Binomial_distribution

See also: fitdist, makedist, binocdf, binoinv, binopdf, binornd, binofit, binolike, binostat

Source Code: BinomialDistribution

Properties

A positive integer value characterizing the number of trials in the binomial distribution. You can access the N property using dot name assignment.

Example: 1


 ## Create a binomial distribution with default parameters
 pd = makedist ("Binomial")

 ## Query parameter 'N' (number of trials)
 pd.N

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

 ## Use this to initialize or modify the number of trials in a binomial
 ## distribution. The number of trials must be a positive integer.

pd =
  BinomialDistribution

  binomial distribution
        N =   1
        p = 0.5

ans = 1
pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.5

Example: 2


 ## Create a binomial distribution object by calling its constructor
 pd = BinomialDistribution (10, 0.3)

 ## Query parameter 'N'
 pd.N

 ## This demonstrates direct construction with specific parameters, useful
 ## for defining a binomial distribution with known number of trials.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

ans = 10

A scalar value in the range $[0, 1]$ characterizing the probability of success in each trial of the binomial distribution. You can access the p property using dot name assignment.

Example: 1


 ## Create a binomial distribution with default parameters
 pd = makedist ("Binomial")

 ## Query parameter 'p' (probability of success)
 pd.p

 ## Set parameter 'p'
 pd.p = 0.4

 ## Use this to initialize or modify the probability of success in each
 ## trial. The probability must be between 0 and 1.

pd =
  BinomialDistribution

  binomial distribution
        N =   1
        p = 0.5

ans = 0.5000
pd =
  BinomialDistribution

  binomial distribution
        N =   1
        p = 0.4

Example: 2


 ## Create a binomial distribution object by calling its constructor
 pd = BinomialDistribution (10, 0.3)

 ## Query parameter 'p'
 pd.p

 ## This shows how to set the success probability directly via the
 ## constructor, ideal for modeling specific success rates.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

ans = 0.3000

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 $2×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $2×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $2×1$ 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 N and p properties.

A $2×2$ numeric matrix containing the variance-covariance of the parameter estimates. Diagonal elements contain the variance of each estimated parameter and non-diagonal elements contain the covariance between the parameter estimates. The covariance 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 all elements of the variance-covariance are zero. This property is read-only.

A $1×2$ logical vector specifying which parameters are fixed and which are estimated. true values correspond to fixed parameters, false values correspond to parameter estimates. This property is read-only.

A $1×2$ numeric vector specifying the truncation interval for the probability distribution. First element contains the lower boundary, 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 : an empty array, since BinomialDistribution does not allow censoring.
frequency : 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

BinomialDistribution: p = cdf (pd, x)
BinomialDistribution: 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 Binomial distribution
 x = 0:10;
 pd1 = makedist ("Binomial", "N", 10, "p", 0.2);
 pd2 = makedist ("Binomial", "N", 10, "p", 0.5);
 pd3 = makedist ("Binomial", "N", 10, "p", 0.8);
 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 ({"N = 10, p = 0.2", "N = 10, p = 0.5", "N = 10, p = 0.8"}, ...
          "location", "southeast")
 title ("Binomial CDF")
 xlabel ("Number of successes")
 ylabel ("Cumulative probability")

 ## Use this to compute and visualize the cumulative distribution function
 ## for different binomial distributions, showing how probability accumulates.

BinomialDistribution: 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 Binomial distribution
 p = 0.001:0.001:0.999;
 pd1 = makedist ("Binomial", "N", 10, "p", 0.2);
 pd2 = makedist ("Binomial", "N", 10, "p", 0.5);
 pd3 = makedist ("Binomial", "N", 10, "p", 0.8);
 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 ({"N = 10, p = 0.2", "N = 10, p = 0.5", "N = 10, p = 0.8"}, ...
          "location", "northwest")
 title ("Binomial iCDF")
 xlabel ("Probability")
 ylabel ("Number of successes")

 ## This demonstrates the inverse CDF (quantiles) for binomial distributions,
 ## useful for finding the number of successes corresponding to given
 ## probabilities.

BinomialDistribution: r = iqr (pd)

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

Example: 1


 ## Compute the interquartile range for a Binomial distribution
 pd = makedist ("Binomial", "N", 20, "p", 0.3)
 iqr_value = iqr (pd)

 ## Use this to calculate the interquartile range, which measures the spread
 ## of the middle 50% of the distribution.

pd =
  BinomialDistribution

  binomial distribution
        N =  20
        p = 0.3

iqr_value = 2

BinomialDistribution: m = mean (pd)

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

Example: 1


 ## Compute the mean for different Binomial distributions
 pd1 = makedist ("Binomial", "N", 10, "p", 0.2);
 pd2 = makedist ("Binomial", "N", 10, "p", 0.5);
 mean1 = mean (pd1)
 mean2 = mean (pd2)

 ## This shows how to compute the expected number of successes for binomial
 ## distributions with different parameters.

mean1 = 2
mean2 = 5

BinomialDistribution: m = median (pd)

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

Example: 1


 ## Compute the median for different Binomial distributions
 pd1 = makedist ("Binomial", "N", 10, "p", 0.2);
 pd2 = makedist ("Binomial", "N", 10, "p", 0.5);
 median1 = median (pd1)
 median2 = median (pd2)

 ## Use this to find the median number of successes, which splits the
 ## distribution into two equal probability halves.

median1 = 2
median2 = 5

BinomialDistribution: 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 Binomial distribution
 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 rand ("seed", 22);
 data = random (pd, 100, 1);
 pd_fitted = fitdist (data, "Binomial", "ntrials", 10)
 nlogL = negloglik (pd_fitted)

 ## This is useful for assessing the fit of a binomial distribution to data,
 ## lower values indicate a better fit.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

pd_fitted =
  BinomialDistribution

  binomial distribution
        N =    10
        p = 0.317   [0.288235, 0.346847]

nlogL = -181.52

BinomialDistribution: ci = paramci (pd)
BinomialDistribution: 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 Binomial
 ## distribution
 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 rand ("seed", 22);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "Binomial", "ntrials", 10)
 ci = paramci (pd_fitted, "Alpha", 0.05)

 ## Use this to obtain confidence intervals for the estimated parameters,
 ## providing a range of plausible values for p given the data.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

pd_fitted =
  BinomialDistribution

  binomial distribution
        N =     10
        p = 0.3027   [0.293704, 0.311811]

ci =

   0.2937   0.3118

BinomialDistribution: 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 Binomial distribution
 x = 0:10;
 pd1 = makedist ("Binomial", "N", 10, "p", 0.2);
 pd2 = makedist ("Binomial", "N", 10, "p", 0.5);
 pd3 = makedist ("Binomial", "N", 10, "p", 0.8);
 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 ({"N = 10, p = 0.2", "N = 10, p = 0.5", "N = 10, p = 0.8"}, ...
          "location", "north")
 title ("Binomial PDF")
 xlabel ("Number of successes")
 ylabel ("Probability")

 ## This visualizes the probability mass function for binomial distributions,
 ## showing the likelihood of different numbers of successes.

BinomialDistribution: plot (pd)
BinomialDistribution: plot (pd, Name, Value)
BinomialDistribution: 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. When `pd` is a continuous distribution object, option is ignored.
`"Parent"`		An axes graphics object for 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 Binomial distribution with fixed parameters N = 20 and p = 0.4,
 ## and plot its PDF.

 pd = makedist ("Binomial", "N", 20, "p", 0.4)
 plot (pd)
 title ("Fixed Binomial distribution with N = 20 and p = 0.4")

pd =
  BinomialDistribution

  binomial distribution
        N =  20
        p = 0.4

Example: 2


 ## Generate a data set of 100 random samples from a Binomial distribution
 ## with parameters N = 10 and p = 0.3. Fit a Binomial distribution to this
 ## data and plot its CDF superimposed over an empirical CDF of the data

 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 rand ("seed", 22);
 data = random (pd, 100, 1);
 pd = fitdist (data, "Binomial", "ntrials", 10)
 plot (pd, "PlotType", "cdf", "Discrete", true)
 title (sprintf ("Fitted Binomial distribution with N = %d and p = %0.2f", ...
                  pd.N, pd.p))
 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.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

pd =
  BinomialDistribution

  binomial distribution
        N =    10
        p = 0.317   [0.288235, 0.346847]

Example: 3


 ## Generate a data set of 200 random samples from a Binomial distribution
 ## with parameters N = 10 and p = 0.3. Display a probability plot for the
 ## Binomial distribution fit to the data.

 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 rand ("seed", 22);
 data = random (pd, 200, 1);
 pd = fitdist (data, "Binomial", "ntrials", 10)
 plot (pd, "PlotType", "probability", "Discrete", true)
 title (sprintf (["Probability plot of fitted Binomial distribution with " ...
                  "N = %d and p = %0.2f"], pd.N, pd.p));
 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 binomial model is appropriate.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

pd =
  BinomialDistribution

  binomial distribution
        N =    10
        p = 0.301   [0.280951, 0.321634]

BinomialDistribution: [nlogL, param] = proflik (pd, pnum)
BinomialDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)
BinomialDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
BinomialDistribution: [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 binomial distribution, pnum = 1 selects the parameter N and pnum = 2 selects the parameter p.

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 used-defined range of parameter values.

Example: 1


 ## Compute and plot the profile likelihood for the probability parameter of
 ## a fitted Binomial distribution
 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 rand ("seed", 22);
 data = random (pd, 1000, 1);
 pd_fitted = fitdist (data, "Binomial", "ntrials", 10)
 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

 ## Use this to analyze the profile likelihood of the success probability
 ## (p), helping to understand the uncertainty in parameter estimates.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

pd_fitted =
  BinomialDistribution

  binomial distribution
        N =     10
        p = 0.3027   [0.293704, 0.311811]

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

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

When called with a single size argument, binornd 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 Binomial distribution
 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 rand ("seed", 22);
 samples = random (pd, 100, 1);
 hist (samples, 0:10)
 title ("Histogram of 100 random samples from Binomial(N=10, p=0.3)")
 xlabel ("Number of successes")
 ylabel ("Frequency")

 ## This generates random samples from a binomial distribution, useful for
 ## simulating experiments with a fixed number of trials and success
 ## probability.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

BinomialDistribution: s = std (pd)

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

Example: 1


 ## Compute the standard deviation for a Binomial distribution
 pd = makedist ("Binomial", "N", 20, "p", 0.3)
 std_value = std (pd)

 ## Use this to calculate the standard deviation, which measures the
 ## variability in the number of successes.

pd =
  BinomialDistribution

  binomial distribution
        N =  20
        p = 0.3

std_value = 2.0494

BinomialDistribution: 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 Binomial distribution, with parameters N = 10 and
 ## p = 0.3, truncated at [2, 8] intervals. Generate 10000 random samples
 ## from this truncated distribution and superimpose a histogram scaled
 ## accordingly

 pd = makedist ("Binomial", "N", 10, "p", 0.3)
 t = truncate (pd, 2, 8)
 rand ("seed", 22);
 data = random (t, 10000, 1);

 ## Histogram data for range 2 to 8
 edges = 1.5:1:8.5;
 centers = 2:8;
 counts = histc (data, edges);
 counts = counts(1:end-1);  # Remove extra edge bin
 probs = counts / sum(counts);  # Normalize to get probabilities

 ## Plot histogram bars
 bar (centers, probs, 0.5, "facecolor", [0.6 0.6 1]);
 hold on;

 ## PMF of the truncated distribution
 pmf = pdf (t, centers);
 plot (centers, pmf, 'r-', "linewidth", 2);

 title ("Binomial distribution (N = 10, p = 0.3) truncated at [2, 8]");
 xlabel ("x");
 ylabel ("Probability");
 legend ("Histogram", "Truncated PMF");
 hold off

 ## This demonstrates truncating a binomial distribution to a specific range
 ## and visualizing the resulting distribution with random samples.

pd =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3

t =
  BinomialDistribution

  binomial distribution
        N =  10
        p = 0.3
  Truncated to the interval [2, 8]

BinomialDistribution: v = var (pd)

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

Example: 1


 ## Compute the variance for a Binomial distribution
 pd = makedist ("Binomial", "N", 20, "p", 0.3)
 var_value = var (pd)

 ## Use this to calculate the variance, which quantifies the spread of the
 ## number of successes in the distribution.

pd =
  BinomialDistribution

  binomial distribution
        N =  20
        p = 0.3

var_value = 4.2000

Categories &

Functions List

Clustering

Clustering

Classification Classes

Classification Classes

Clustering Classes

Clustering Classes

Regression Classes

Regression Classes

Data Manipulation

Data Manipulation

Descriptive Statistics

Descriptive Statistics

Distribution Classes

Distribution Classes

Distribution Fitting

Distribution Fitting

Distribution Functions

Distribution Functions

Distribution Statistics

Distribution Statistics

Distribution Wrappers

Distribution Wrappers

Experimental Design

Experimental Design

Machine Learning

Machine Learning

Model Fitting

Model Fitting

Hypothesis Testing

Hypothesis Testing

I/O

I/O

Plotting

Plotting

Regression

Regression

Transforms

Transforms

Class Definition: BinomialDistribution

Properties

N

Example: 1

Example: 2

p

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

Methods

cdf

Example: 1

icdf

Example: 1

iqr

Example: 1

mean

Example: 1

median

Example: 1

negloglik

Example: 1

paramci

Example: 1

pdf

Example: 1

plot

Example: 1

Example: 2

Example: 3

proflik

Class Definition: `BinomialDistribution`

`N`

`p`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

`icdf`

`iqr`

`mean`

`median`

`negloglik`

`paramci`

`pdf`

`plot`

`proflik`

`random`

`std`

`truncate`

`var`