Univariate Continuous Distributions
Compute, fit, and generate samples from real-valued distributions
A univariate continuous distribution is a probability distribution with a single continuous random variable, which can assume any value. Statistics and Machine Learning Toolbox™ offers several ways to work with univariate continuous distributions:
Create a distribution object and use distribution object functions.
Use distribution-specific functions with specified distribution parameters. The functions can accept parameters of multiple distributions.
Use the generic distribution functions with the specified distribution name and corresponding parameters.
For more information, see Working with Probability Distributions.
Apps
| Distribution Fitter | Fit probability distributions to data |
Tools
| Probability Distribution Function Tool | Interactive density and distribution plots |
| randtool | Interactive random number generation |
Objects
Functions
Topics
Chi-Square Distributions
- Chi-Square Distribution
The chi-square distribution is commonly used in hypothesis testing, particularly the chi-square test for goodness of fit. - Noncentral Chi-Square Distribution
The noncentral chi-square distribution is a more general case of the chi-square distribution, with applications in thermodynamics and signal processing.
Extreme Value Distributions
- Extreme Value Distribution
Extreme value distributions are often used to model the smallest or largest value in a large set of independent, identically distributed random values representing measurements or observations. - Generalized Extreme Value Distribution
The generalized extreme value distribution is often used to model the smallest or largest value in a large set of independent, identically distributed random values representing measurements or observations.
F Distributions
- F Distribution
The F distribution is often used in the analysis of variance, as in the F-test. - Noncentral F Distribution
The noncentral F distribution is a more general case of the F distribution and is used to calculate the power of a hypothesis test relative to a particular alternative.
Logistic Distributions
- Logistic Distribution
The logistic distribution is used for growth models and in logistic regression. - Loglogistic Distribution
The loglogistic distribution is a probability distribution whose logarithm has a logistic distribution.
Normal Distributions
- Normal Distribution
Learn about the normal distribution. The normal distribution is a two-parameter (mean and standard deviation) family of curves. Central Limit Theorem states that the normal distribution models the sum of independent samples from any distribution as the sample size goes to infinity. - Half-Normal Distribution
The half-normal distribution is a special case of the folded normal and truncated normal distributions. - Lognormal Distribution
The lognormal distribution is a probability distribution whose logarithm has a normal distribution.
t Distributions
- Noncentral t Distribution
The noncentral t distribution is a more general case of the Student’s t distribution and is used to calculate the power of the t test. - Student's t Distribution
The Student’s t distribution is a family of curves depending on a single parameter ν (the degrees of freedom). - t Location-Scale Distribution
The t location-scale distribution is useful for modeling data distributions with heavier tails (more prone to outliers) than the normal distribution.
Uniform Distributions
- Uniform Distribution (Continuous)
The uniform distribution (also called the rectangular distribution) is notable because it has a constant probability distribution function between its two bounding parameters. - Loguniform Distribution
The loguniform distribution (also called the reciprocal distribution) has a density that is proportional to the reciprocal of the variable value within its two bounding parameters.
Weibull Distribution
- Weibull Distribution
The Weibull distribution is an appropriate analytical tool for modeling the breaking strength of materials. Current usage also includes reliability and lifetime modeling. - Three-Parameter Weibull Distribution
Find maximum likelihood estimates (MLEs) for the three-parameter Weibull distribution with scale, shape, and location parameters.
Other Distributions
- Beta Distribution
The beta distribution describes a family of curves that are nonzero only on the interval [0,1]. - Birnbaum-Saunders Distribution
The Birnbaum-Saunders distribution was originally proposed as a lifetime model for materials subject to cyclic patterns of stress and strain, where the ultimate failure of the material comes from the growth of a prominent flaw. - Burr Type XII Distribution
The Burr type XII distribution is a three-parameter family of distributions on the positive real line. - Exponential Distribution
The exponential distribution is used to model events that occur randomly over time. The main application area is in studies of lifetimes. - Gamma Distribution
The gamma distribution models sums of exponentially distributed random variables. - Inverse Gaussian Distribution
Also known as the Wald distribution, the inverse Gaussian is used to model nonnegative positively skewed data. - Kernel Distribution
A kernel distribution is a nonparametric representation of the probability density function of a random variable. - Nakagami Distribution
In communications theory, Nakagami distributions are used to model scattered signals that reach a receiver by multiple paths. - Generalized Pareto Distribution
The generalized Pareto distribution is used to model extreme events from a distribution. - Pearson Distribution
The Pearson distribution is a flexible, four-parameter distribution that has an arbitrary mean, standard deviation, skewness, and kurtosis. - Piecewise Linear Distribution
The piecewise linear distribution creates a nonparametric representation of the cumulative distribution function (cdf) by linearly connecting the known cdf values from the sample data. - Rayleigh Distribution
The Rayleigh distribution is a special case of the Weibull distribution, with applications in communications theory. - Rician Distribution
In communications theory, Rician distributions are used to model scattered signals that reach a receiver by multiple paths. - Stable Distribution
Stable distributions are a class of probability distributions suitable for modeling heavy tails and skewness. - Triangular Distribution
The triangular distribution provides a simplistic representation of the probability distribution when limited sample data is available.
How To
- Maximum Likelihood Estimation
Themlefunction computes maximum likelihood estimates (MLEs) for a distribution specified by its name and for a custom distribution specified by its probability density function (pdf), log pdf, or negative log likelihood function. - Negative Loglikelihood Functions
Find maximum likelihood estimates using negative loglikelihood functions. - Compare Multiple Distribution Fits
This example shows how to fit multiple probability distribution objects to the same set of sample data, and obtain a visual comparison of how well each distribution fits the data. - Model Data with the Generalized Extreme Value Distribution
The extreme value distribution is used to model the largest or smallest value from a group or block of data. - Fit Probability Distribution Objects to Grouped Data
This example shows how to fit probability distribution objects to grouped sample data, and create a plot to visually compare the pdf of each group. - Fit Distributions to Grouped Data Using ksdensity
Fit kernel distributions to grouped sample data using theksdensityfunction. - Fit a Nonparametric Distribution with Pareto Tails
Fit a nonparametric probability distribution to sample data using Pareto tails to smooth the distribution in the tails. - Modeling Tail Data with the Generalized Pareto Distribution
Fit tail data to the Generalized Pareto distribution by maximum likelihood estimation. - Generate Cauchy Random Numbers Using Student's t
Use the Student's t distribution to generate random numbers from a standard Cauchy distribution. - Represent Cauchy Distribution Using t Location-Scale
This example shows how to use the t location-scale probability distribution object to work with a Cauchy distribution with nonstandard parameter values. - Generate Random Numbers Using the Triangular Distribution
Create a triangular probability distribution object based on sample data, and generate random numbers for use in a simulation. - Generate Random Numbers Using Uniform Distribution Inversion
Generate random numbers using the uniform distribution inversion method. - Generating Data Using Flexible Families of Distributions
The Pearson and Johnson systems are flexible parametric families of distributions that provide good matches for a wide range of data shapes. - Fit Custom Distributions
Fit a custom distribution to univariate data by using themlefunction. - Nonparametric and Empirical Probability Distributions
Estimate a probability density function or a cumulative distribution function from sample data. - Nonparametric Estimates of Cumulative Distribution Functions and Their Inverses
Estimate the cumulative distribution function (cdf) from data in a nonparametric or semiparametric way.
