Statistics: TriangularDistribution

Class Definition: `TriangularDistribution`

statistics: TriangularDistribution

Triangular probability distribution object.

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

The triangular distribution uses the following parameters.

`Parameter`	`Description`	`Support`
`A`	Lower limit	$-Inf < A < Inf$
`B`	Peak location	$A <= B <= C$
`C`	Upper limit	$C > A$

There are several ways to create a TriangularDistribution object.

Create a distribution with specified parameter values using the makedist function.
Use the constructor TriangularDistribution (A, B, C) to create a triangular distribution with specified parameter values A, B, and C.

It is highly recommended to use makedist function to create probability distribution objects, instead of the constructor.

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

See also: makedist, tricdf, triinv, tripdf, trirnd, tristat

Source Code: TriangularDistribution

The TriangularDistribution class contains the following properties:

A scalar value characterizing the lower limit of the triangular distribution. You can access the A property using dot name assignment.

Example: 1

Create a Triangular distribution with default parameters

 pd = makedist ("Triangular", "A", 0, "B", 0.5, "C", 1);

Query parameter 'A' (lower limit)

 pd.A

ans = 0

Set parameter 'A'

 pd.A = -1

pd =
  TriangularDistribution

  Triangular distribution
        A =  -1
        B = 0.5
        C =   1

Use this to initialize or modify the lower limit parameter of a Triangular distribution. The lower limit must be a real scalar less than the upper limit C, useful for defining the minimum possible value in scenarios like cost estimation or time to completion.

Example: 2

Create a Triangular distribution object by calling its constructor

 pd = TriangularDistribution (1, 2, 3);

Query parameter 'A'

 pd.A

ans = 1

This demonstrates direct construction with a specific lower limit, useful for modeling bounded data, such as project durations or resource requirements.

A scalar value characterizing the peak location of the triangular distribution. You can access the B property using dot name assignment.

Example: 1

Create a Triangular distribution with default parameters

 pd = makedist ("Triangular", "A", 0, "B", 0.5, "C", 1);

Query parameter 'B' (peak location)

 pd.B

ans = 0.5000

Set parameter 'B'

 pd.B = 0.7

pd =
  TriangularDistribution

  Triangular distribution
        A =   0
        B = 0.7
        C =   1

Use this to initialize or modify the peak location parameter of a Triangular distribution. The peak location must be a real scalar between A and C, representing the most likely value in applications like risk analysis.

Example: 2

Create a Triangular distribution object by calling its constructor

 pd = TriangularDistribution (1, 2, 3);

Query parameter 'B'

 pd.B

ans = 2

This shows how to set the peak location directly via the constructor, ideal for modeling the mode of triangularly distributed data, such as expected task completion times.

A scalar value characterizing the upper limit of the triangular distribution. You can access the C property using dot name assignment.

Example: 1

Create a Triangular distribution with default parameters

 pd = makedist ("Triangular", "A", 0, "B", 0.5, "C", 1);

Query parameter 'C' (upper limit)

 pd.C

ans = 1

Set parameter 'C'

 pd.C = 2

pd =
  TriangularDistribution

  Triangular distribution
        A =   0
        B = 0.5
        C =   2

Use this to initialize or modify the upper limit parameter of a Triangular distribution. The upper limit must be a real scalar greater than A, defining the maximum possible value in scenarios like budgeting or scheduling.

Example: 2

Create a Triangular distribution object by calling its constructor

 pd = TriangularDistribution (1, 2, 3);

Query parameter 'C'

 pd.C

ans = 3

This demonstrates setting the upper limit directly via the constructor, useful for modeling the maximum bound in applications like cost or time estimates.

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 $3×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $3×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $3×1$ 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 A, B, and C properties.

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.

The TriangularDistribution class offers the following public methods:

TriangularDistribution: p = cdf (pd, x)
TriangularDistribution: 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 Triangular distribution

 x = -1:0.01:6;
 pd1 = TriangularDistribution (0, 1, 2);
 pd2 = TriangularDistribution (1, 2, 3);
 pd3 = TriangularDistribution (2, 3, 4);
 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 ({"A=0, B=1, C=2", "A=1, B=2, C=3", "A=2, B=3, C=4"}, ...
         "location", "southeast")
 title ("Triangular CDF")
 xlabel ("Values in x (A <= x <= C)")
 ylabel ("Cumulative probability")

Use this to compute and visualize the cumulative distribution function for different Triangular distributions, showing how probability accumulates within the bounds, useful in risk assessment or forecasting.

TriangularDistribution: 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 Triangular distribution

 p = 0.001:0.001:0.999;
 pd1 = TriangularDistribution (0, 1, 2);
 pd2 = TriangularDistribution (1, 2, 3);
 pd3 = TriangularDistribution (2, 3, 4);
 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 ({"A=0, B=1, C=2", "A=1, B=2, C=3", "A=2, B=3, C=4"}, ...
         "location", "northwest")
 title ("Triangular iCDF")
 xlabel ("Probability")
 ylabel ("Values in x (A <= x <= C)")

This demonstrates the inverse CDF (quantiles) for Triangular distributions, useful for finding values corresponding to given probabilities, such as thresholds in project planning or quality control.

TriangularDistribution: r = iqr (pd)

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

Example: 1

Compute the interquartile range for a Triangular distribution

 pd = TriangularDistribution (0, 1, 2);
 iqr_value = iqr (pd)

iqr_value = 0.5858

Use this to calculate the interquartile range, which measures the spread of the middle 50% of the distribution, helpful for understanding central variability in bounded data like cost estimates.

TriangularDistribution: m = mean (pd)

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

Example: 1

Compute the mean for different Triangular distributions

 pd1 = TriangularDistribution (0, 1, 2);
 pd2 = TriangularDistribution (1, 2, 3);
 mean1 = mean (pd1)

mean1 = 1

 mean2 = mean (pd2)

mean2 = 2

This shows how to compute the expected value for Triangular distributions with different parameters, representing the average value in scenarios like budgeting or time estimation.

TriangularDistribution: m = median (pd)

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

Example: 1

Compute the median for different Triangular distributions

 pd1 = TriangularDistribution (0, 1, 2);
 pd2 = TriangularDistribution (1, 2, 3);
 median1 = median (pd1)

median1 = 1

 median2 = median (pd2)

median2 = 2

Use this to find the median value, which splits the distribution into two equal probability halves, robust for skewed triangular data in applications like scheduling.

TriangularDistribution: 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 Triangular distribution

 x = -1:0.01:6;
 pd1 = TriangularDistribution (0, 1, 2);
 pd2 = TriangularDistribution (1, 2, 3);
 pd3 = TriangularDistribution (2, 3, 4);
 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 ({"A=0, B=1, C=2", "A=1, B=2, C=3", "A=2, B=3, C=4"}, ...
         "location", "northeast")
 title ("Triangular PDF")
 xlabel ("Values in x (A <= x <= C)")
 ylabel ("Probability density")

This visualizes the probability density function for Triangular distributions, showing the likelihood within the bounds, useful for understanding data distribution in risk analysis.

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

plot (pd) plots a probability density function (PDF) of the probability distribution object pd.

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). `'cdf'` plots the cumulative density function (CDF).
`'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, this option is ignored.
`'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 Triangular distribution with fixed parameters A=0, B=1, C=2 and plot its PDF.

 pd = TriangularDistribution (0, 1, 2);
 plot (pd)
 title ("Triangular distribution with A=0, B=1, C=2")

Use this to visualize the PDF of a Triangular distribution with fixed parameters, useful for understanding the shape of the distribution.

Example: 2

Generate a data set of 100 random samples from a Triangular distribution with parameters A=0, B=1, C=2. Plot its CDF.

 rand ("seed", 21);
 data = trirnd (0, 1, 2, 100, 1);
 pd = makedist ("Triangular", "A", 0, "B", 1, "C", 2);
 plot (pd, "PlotType", "cdf")
 title ("Triangular distribution with A=0, B=1, C=2")
 legend ({"Fitted CDF"}, "location", "southeast")

Use this to visualize the CDF of a Triangular distribution, useful for assessing cumulative probabilities in bounded data scenarios.

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

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

When called with a single size argument, trirnd 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 Triangular distribution

 rand ("seed", 21);
 samples = trirnd (0, 1, 2, 500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Triangular(A=0, B=1, C=2)")
 xlabel ("Values in x (A <= x <= C)")
 ylabel ("Frequency")

This generates random samples from a Triangular distribution, useful for simulating bounded data like project costs or completion times.

TriangularDistribution: s = std (pd)

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

Example: 1

Compute the standard deviation for a Triangular distribution

 pd = TriangularDistribution (0, 1, 2);
 std_value = std (pd)

std_value = 0.4082

Use this to calculate the standard deviation, which measures the variability within the bounds of the Triangular distribution, useful for risk assessment.

TriangularDistribution: 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.

Example: 1

Plot the PDF of a Triangular distribution with parameters A=0, B=1, C=2, truncated at [0.5, 1.5] intervals. Generate 10000 random samples from this truncated distribution and superimpose a histogram scaled accordingly.

 rand ("seed", 21);
 data_all = trirnd (0, 1, 2, 20000, 1);
 data = data_all(data_all >= 0.5 & data_all <= 1.5);
 data = data(1:10000);
 pd = makedist ("Triangular", "A", 0, "B", 1, "C", 2);
 t = truncate (pd, 0.5, 1.5);
 [counts, centers] = hist (data, 20);
 bin_width = centers(2) - centers(1);
 bar (centers, counts / (sum (counts) * bin_width), 1);
 hold on;

Plot histogram and truncated PDF

 x = linspace (0.5, 5, 500);
 y = pdf (t, x);
 plot (x, y, "r", "linewidth", 2);
 title ("Triangular distribution (A=0, B=1, C=2) truncated at [0.5, 1.5]")
 legend ("Truncated PDF", "Histogram")

This demonstrates truncating a Triangular distribution to a specific range and visualizing the resulting distribution with random samples, useful for constrained scenarios like limited budgets.

TriangularDistribution: v = var (pd)

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

Example: 1

Compute the variance for a Triangular distribution

 pd = TriangularDistribution (0, 1, 2);
 var_value = var (pd)

var_value = 0.1667

Use this to calculate the variance, which quantifies the spread of values within the bounds of the Triangular distribution, useful for uncertainty analysis.

Examples

 pd_fixed = makedist ('Triangular', 'A', 0, 'B', 1, 'C', 2);
 rand ('seed', 2);
 data = random (pd_fixed, 5000, 1);
 A = min (data);
 C = mean (data);
 B = max (data);
 [counts, centers] = hist (data, 50);
 bin_width = centers(2) - centers(1);
 normalized_counts = counts / (sum (counts) * bin_width);
 bar (centers, normalized_counts, 1);
 hold on;
 x = linspace (A, B, 100);
 y = (2 * (x - A) / (C - A) .* (x <= C)) + (2 * (B - x) / (B - C) .* (x > C));
 plot (x, y, 'r-', 'LineWidth', 2);
 msg = sprintf ("Fitted Triangular distribution with A = %0.2f, C = %0.2f, B = %0.2f", A, C, B);
 title (msg);
 hold off;

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Class Definition: TriangularDistribution

statistics: TriangularDistribution

A

Example: 1

Example: 2

B

Example: 1

Example: 2

C

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

Truncation

IsTruncated

cdf

TriangularDistribution: p = cdf (pd, x)

TriangularDistribution: p = cdf (pd, x, 'upper')

Example: 1

icdf

TriangularDistribution: x = icdf (pd, p)

Example: 1

iqr

TriangularDistribution: r = iqr (pd)

Example: 1

mean

TriangularDistribution: m = mean (pd)

Example: 1

median

TriangularDistribution: m = median (pd)

Example: 1

pdf

TriangularDistribution: y = pdf (pd, x)

Example: 1

plot

Class Definition: `TriangularDistribution`

`A`

`B`

`C`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`Truncation`

`IsTruncated`

`cdf`

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

`TriangularDistribution: p = cdf (pd, x, 'upper')`

`icdf`

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

`iqr`

`TriangularDistribution: r = iqr (pd)`

`mean`

`TriangularDistribution: m = mean (pd)`

`median`

`TriangularDistribution: m = median (pd)`

`pdf`

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

`plot`

`TriangularDistribution: plot (pd)`

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

`TriangularDistribution: h = plot (…)`

`random`

`TriangularDistribution: r = random (pd)`

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

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

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

`std`

`TriangularDistribution: s = std (pd)`

`truncate`

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

`var`

`TriangularDistribution: v = var (pd)`