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MixtureDistribution[{w1,…,wn},{dist1,…,distn}]

represents a mixture distribution whose CDF is given as a sum of the CDFs of the component distributions disti, each with weight wi.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Parametric Distributions  
Nonparametric Distributions  
Derived Distributions  
Automatic Simplifications  
Applications  
Properties & Relations  
Neat Examples  
See Also
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MixtureDistribution

MixtureDistribution[{w1,…,wn},{dist1,…,distn}]

represents a mixture distribution whose CDF is given as a sum of the CDFs of the component distributions disti, each with weight wi.

Details

  • The cumulative distribution function for value is proportional to , where is the CDF for disti.
  • The distributions disti need to be all continuous or all discrete, and have the same dimensionality.
  • The weights wi can be any non-negative real numbers.
  • MixtureDistribution can be used with such functions as Mean, CDF, and RandomVariate, etc.

Examples

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Basic Examples  (3)

Define a mixture of two continuous distributions:

Define a mixture of two discrete distributions:

Define a multivariate mixture:

Scope  (30)

Basic Uses  (10)

A mixture with numeric weights:

Cumulative distribution function:

A mixture with symbolic weights:

Probability density function:

The weights control the contribution by each distribution:

Two univariate continuous distributions:

The mixture combines the densities according to their weights:

Two bivariate continuous distributions:

The mixture combines the densities according to their weights:

Two univariate discrete distributions:

Probability density function:

Plot a density function for different weights:

Mean and variance:

Two multivariate discrete distributions:

Probability density function:

Generate random numbers:

Several univariate continuous distributions:

Moments:

Factorial moments:

Central moments:

Cumulants:

Several univariate discrete distributions:

Generating functions:

Estimate weights in a mixture:

Parametric Distributions  (5)

Define a mixture of two different continuous distributions:

Probability density function:

Hazard function:

In the limit the exponential distribution component dominates:

Define a mixture of two distributions with different supports:

Probability density function for a few values of the weight:

Define a mixture of two different univariate discrete distributions:

Probability density function:

Cumulative distribution function:

Moments can be obtained numerically:

Define a mixture of two different multivariate discrete distributions:

Probability density function:

Covariance:

Define a mixture distribution of multivariate uniform distributions:

Cumulative distribution function:

Nonparametric Distributions  (3)

Define a mixture with SmoothKernelDistribution:

The mixture combines the densities according to their weights:

Define a mixture with EmpiricalDistribution:

The mixture combines the cumulative distribution functions according to their weights:

Plot the cumulative distribution function:

Define a mixture with HistogramDistribution:

The mixture combines the densities according to their weights:

Derived Distributions  (10)

Define a mixture distribution with components given by MixtureDistribution:

The PDF is piecewise continuous:

The mean is a convex combination of the means of the components:

Find which components cause the mean of the mixture to be indeterminate:

Find a mixture distribution of the OrderDistribution of the minimum and the maximum:

Compare the probability density functions:

The mean of the mixture distribution:

Compare to the average of the means of order distributions:

Find the mixture distribution of a TruncatedDistribution:

The probability density function is not continuous:

The mean can be computed explicitly:

Find the probability density function of the mixture distribution with a ProductDistribution:

Define a mixture distribution with a TransformedDistribution:

Probability density function:

Define a mixture distribution of a MarginalDistribution:

Characteristic function:

Define a mixture with a CensoredDistribution:

Probability density function:

A mixture with a ParameterMixtureDistribution:

PDFs of scaled mixture components and mixture distribution:

Define a mixture distribution with a CopulaDistribution:

Mixture of compatible QuantityDistribution evaluates to QuantityDistribution:

Evaluate the mean:

Plot the PDF:

Automatic Simplifications  (2)

One component mixture simplifies to the input distribution:

A mixture with zero weights will reduce the number of input distributions:

A mixture with one zero weight will return unevaluated:

Applications  (7)

Find the percentage of values between and :

Between and :

Package it up as a function using NProbability:

Determine the maximal variance of a mixture:

The heights of females in the United States follow normal distribution with mean 64 inches and standard deviation of 2 inches, while the heights of males in the United States follow normal distribution with mean 70 inches and standard deviation of 2 inches. If the population ratio of males to females is 1.1, then the heights of the whole population have the following bimodal distribution:

Simulate a typical distribution of heights in a town of population 100:

Find the probability that a person is at least 73 inches tall:

A binary transmission is sent with 0 coded as a voltage signal and 1 as a voltage signal. 1 is sent with probability but the signal is corrupted by white noise. Find the PDF of the received signal:

Simulate transmission at the receiver for p=0.4 and v=1:

To distinguish between the two signals, the voltage difference must be bigger:

MixtureDistribution can be used to create multimodal models:

The magnitudes of earthquakes in the United States in the selected years have two modes:

Find an estimated distribution from possible mixtures of two normal distributions:

Compare the histogram to the PDF of the estimated distribution:

Find the probability of an earthquake of magnitude 7 or higher:

Find the mean earthquake magnitude:

Simulate magnitudes of the next 30 earthquakes:

The average city and highway mileage for midsize cars follows a binormal distribution:

Show the distribution of city and highway mileage:

Assuming 65% of the driving is done in the city, the mileage follows a MixtureDistribution:

Find the average mileage:

Gaussian mixture model is commonly used for the purpose of image segmentation. Image is represented as an array of pixels. A pixel is a scalar (or vector) which shows the intensity (or color):

Visualize the distribution of pixel values via Histogram:

Fit the pixel values to a three-component Gaussian mixture with EstimatedDistribution:

Label each pixel with maximum a posterior probability (MAP) estimate:

Visualize the result:

Properties & Relations  (8)

A mixture with weights w is equivalent to one with weights w/Total[w]:

Compare PDFs:

The PDF of a mixture is a convex combination of the PDF of its components:

The CDF of a mixture is a convex combination of the CDF of its components:

The moments of a mixture are a convex combination of the moments of its components:

A moment of general order:

A ParameterMixtureDistribution with a discrete weight, assuming a finite number of values, can be represented as a mixture distribution:

Compare PDFs:

A ParameterMixtureDistribution with a discrete weight, assuming a countable number of values, can be approximated by a mixture distribution:

Compare approximations for different quantiles as cut-offs:

Approximating a ParameterMixtureDistribution with a continuous weight by a mixture distribution:

Compare PDFs:

A KernelMixtureDistribution is a MixtureDistribution derived from data:

Neat Examples  (3)

A mixture of two binormal distributions:

A variety of distributional shapes from Gaussian mixtures:

A multivariate Gaussian mixture:

Wolfram Research (2010), MixtureDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MixtureDistribution.html (updated 2016).

Text

Wolfram Research (2010), MixtureDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MixtureDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "MixtureDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/MixtureDistribution.html.

APA

Wolfram Language. (2010). MixtureDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MixtureDistribution.html

BibTeX

@misc{reference.wolfram_2025_mixturedistribution, author="Wolfram Research", title="{MixtureDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/MixtureDistribution.html}", note=[Accessed: 04-February-2026]}

BibLaTeX

@online{reference.wolfram_2025_mixturedistribution, organization={Wolfram Research}, title={MixtureDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/MixtureDistribution.html}, note=[Accessed: 04-February-2026]}