This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ParameterMixtureDistribution

 ParameterMixtureDistribution represents a parameter mixture distribution where the parameter is distributed according to the weight distribution wdist. ParameterMixtureDistributionrepresents a parameter mixture distribution where the parameter has weight distribution , has weight distribution , etc.
• The probability density for value is given by Expectation[PDF[dist[], x], wdist].
• The domain of the weight distribution wdist needs to be a subset or equal to the parameter domain expected for .
• Parameters can be discrete or continuous.
Specify a parameter mixture distribution:
Parameter mixture distributions work like any other distributions:
Find a parameter mixture of a multivariate distribution:
Specify a parameter mixture distribution:
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Parameter mixture distributions work like any other distributions:
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Find a parameter mixture of a multivariate distribution:
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 Scope   (35)
Find the parameter mixture of ExponentialDistribution with uniformly distributed weight:
Probability density function:
The weight domain must satisfy the parameter assumptions of the main distribution:
Pick a weight distribution supported on the positive reals:
The resulting parameter mixture CDF:
Use a discrete weight for a discrete parameter:
Compare the probability density functions:
Compare probabilities of obtaining a value less than 4 from each distribution:
Vary only one parameter:
Cumulative distribution function:
Mean and variance:
Use a multivariate distribution as a weight to vary more than one parameter:
Use more than one weight distribution to vary multiple parameters:
Visualize the probability density function:
Estimate parameters in a parameter mixture distribution:
Create a random sample for a choice of weight distribution parameters:
Find the distribution parameters:
Compare the histogram of the sample with the PDF of the estimated distribution:
Define a parameter mixture with a continuous univariate weight distribution:
Define a parameter mixture with a discrete univariate weight distribution:
Find the probability density function:
Moment has closed form for a symbolic order:
Define a parameter mixture with a discrete univariate weight distribution:
Probability density function:
Define a parameter mixture with a multivariate weight distribution:
Probability density function:
Verify that the integral of the PDF is 1:
Use more than one weight distribution:
Mean:
Variance:
Define a parameter mixture of a continuous univariate distribution:
Cumulative distribution function:
Find a mixture of a continuous univariate distribution:
Probability density function:
Define a parameter mixture of a discrete univariate distribution:
Generate a random sample:
Find the mean and variance:
Find a parameter mixture of a discrete univariate distribution:
Probability density function:
Mean and variance:
For a fixed value of , Moment has closed form for symbolic order:
Define a parameter mixture of a bivariate continuous distribution:
Use a random sample and a smooth histogram to visualize the density function:
Define a parameter mixture of a bivariate distribution:
Probability density function:
Mean:
Covariance matrix:
Find a parameter mixture distribution of a multivariate discrete distribution:
Generate a random sample:
Histograms for each component:
LaplaceDistribution can be represented as a parametric mixture:
Use an EmpiricalDistribution as a weight:
Probability density function:
Use a HistogramDistribution as a weight:
Probability density function:
Define a parameter mixture with a SmoothKernelDistribution as a weight:
Plot the probability density function:
Plot the cumulative distribution function:
Use a ProductDistribution as a weight distribution in a parameter mixture:
Use the histogram of a random sample from to visualize the PDF:
Find a parameter mixture distribution of a TransformedDistribution:
Visualize the density function using a random sample:
Find a parameter mixture using a TransformedDistribution as a weight distribution:
Probability density function:
Use a MixtureDistribution as a weight distribution in a parameter mixture:
Probability density function:
Mean and variance:
Vary weights in a MixtureDistribution according to a probability distribution:
Probability density function:
Compare to the MixtureDistribution with fixed weights at the average values:
Both density functions are equal:
Define a parameter mixture of a TruncatedDistribution:
Mean and variance:
Define a parameter distribution of a TruncatedDistribution:
Visualize the probability density function using random sample:
Use a TruncatedDistribution as a weight distribution in a parameter mixture:
Probability density function:
Mean:
Find a parameter mixture distribution of an OrderDistribution:
Probability density function:
Use an OrderDistribution as a weight distribution:
Probability density function:
Parameter mixtures with BetaDistribution as a weight:
Parameter mixtures involving PoissonDistribution:
Parameter mixtures involving RayleighDistribution:
 Applications   (7)
SuzukiDistribution is defined as a parameter mixture of RayleighDistribution and LogNormalDistribution:
KDistribution can be represented as a parameter mixture of RayleighDistribution and GammaDistribution:
The time it takes a bank teller to serve a customer follows ExponentialDistribution with average time having LindleyDistribution with mean 3. Find the service time distribution:
Probability density function:
The average service time in minutes:
Simulate the service time for the next 30 customers:
In an optical communication system, transmitted light generates current at the receiver. The number of electrons follows the parametric mixture of a Poisson distribution and another distribution, depending on the type of light. If the source uses coherent laser light of intensity , then the electron-count distribution is Poisson:
If the source uses thermal illumination, then the Poisson parameter follows ExponentialDistribution with parameter and the electron count distribution can be determined:
These two distributions are distinguishable and allow you to determine the type of source:
The Voigt spectral line profile results from combining a Doppler profile and a Lorentzian profile:
Compute the density function:
Plot the density function:
Compute the profile half-width:
Define modified normal distribution:
Plot modified normal distribution densities for several values of and compare them with standard normal density:
Define multivariate Polya distribution:
Probability density function:
The PDF of a parameter mixture can be computed using Expectation:
Parameter mixture with a discrete weight, assuming a finite number of values can be represented as a MixtureDistribution:
Compare PDFs:
Parameter mixture with a discrete weight, assuming a countable number of values can be approximated by a MixtureDistribution:
Compare approximations for different quantiles as cut-offs:
Approximating a parameter mixture with a continuous weight by a MixtureDistribution:
Compare PDFs:
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