WOLFRAM

ProductDistribution[dist1,dist2,]

represents the joint distribution with independent component distributions dist1, dist2, .

Details

  • The probability density for ProductDistribution[dist1,dist2,] is given by where is the PDF of dist1, is the PDF of dist2, etc.
  • The notation {disti,n} indicates that disti is repeated n times.
  • The distributions disti can be any combination of univariate, multivariate, continuous, or discrete distributions.
  • ProductDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

Examples

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Basic Examples  (3)Summary of the most common use cases

Define a two-dimensional distribution for independent normal random variables:

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Define a two-dimensional distribution for independent identically distributed components:

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Define a multivariate distribution with continuous and discrete components:

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Scope  (26)Survey of the scope of standard use cases

Basic Uses  (7)

Define a product of two independent continuous distributions:

The PDF is the product of the component PDFs:

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Product of discrete distributions:

The PDF is the product of the component PDFs:

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Define a product distribution in which three components are repeated:

Probability density function for the four-dimensional product distribution:

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Product distribution with both continuous and discrete components:

Draw a random sample from this distribution:

Estimate the distribution parameters for the components using the random sample:

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Define a general product distribution with few repeated components:

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Compare to a random sample:

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Product of multivariate continuous distributions:

Probability density function:

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Verify that the integral of the PDF is 1:

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Product of multivariate discrete distributions:

Compute the variance of the distribution:

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Compare with the values obtained by using a random sample:

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Parametric Distributions  (6)

Create a bivariate normal distribution with independent components:

Probability density function:

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Compare to BinormalDistribution:

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Define a two-dimensional Laplace distribution:

Probability density function:

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Mean and variance:

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Define product distribution of independent PoissonDistribution:

Probability density function:

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Covariance:

The MultivariatePoissonDistribution does not have independent components:

The assumptions:

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Create the product distribution of two independent examples of StudentTDistribution:

Generate random sample:

Fit a MultivariateTDistribution:

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Goodness-of-fit test:

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Compute properties with symbolic parameters:

Distribution functions:

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Special moments:

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Moments with closed forms for symbolic order:

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Other moments can be obtained numerically:

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Generating functions:

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Find marginals of MultinormalDistribution:

Find product distribution of the marginal distributions:

Probability density function of :

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is a MultinormalDistribution with a diagonal covariance matrix:

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Nonparametric Distributions  (3)

Define the product of SmoothKernelDistribution:

Compare to the product of original distributions:

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Create a sample from and define SmoothKernelDistribution for this sample:

Compare all three distributions:

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Define a product of EmpiricalDistribution:

Plot the probability density function and cumulative distribution function:

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Define a product distribution with HistogramDistribution:

Probability density function:

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Derived Distributions  (10)

Define a product with a CensoredDistribution:

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MarginalDistribution chooses the components of ProductDistribution:

Compose product distribution from marginals:

Probability density function:

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It is the same as for binormal distribution with no correlation:

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The components of product distribution are assumed to be independent, hence the original distribution cannot be recovered when is not zero:

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Create the product distribution from a MixtureDistribution:

Probability density function:

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Mean and variance:

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Find the product distribution of minimum and maximum OrderDistribution:

Probability density function:

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Plot density function for fixed :

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Define a product distribution of a ParameterMixtureDistribution:

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Product distribution is used as an input for a TransformedDistribution:

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Find the product distribution of a TransformedDistribution:

Probability density function:

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Find the product distribution of a TruncatedDistribution:

Variance depends on the truncation interval:

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Compare the PDF to the product of distributions that are not truncated:

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Find the product distribution of a TruncatedDistribution:

Compare the PDF with the product distribution of two Poisson distributions:

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Truncation influences the direction and value of skewness:

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Product of QuantityDistribution evaluates to QuantityDistribution:

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Find moments:

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Convert the distribution to kilograms:

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Applications  (8)Sample problems that can be solved with this function

Generate an uncorrelated sample from a binormal distribution:

The sample is slightly correlated, even though the original distribution is not:

Estimate the distribution from data:

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The estimated distribution has correlation similar to the sample:

Force independent estimates by estimating the marginal distributions:

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Create product distribution:

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The resulting distribution has no correlation:

Two people try to meet at a certain place between 5pm and 5:30pm. Each person arrives at a time uniformly distributed in the time interval independently of each other and stays for five minutes. Find the probability that they meet:

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Show the region for the overlapping event:

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Two six-sided dice are thrown independently of each other. Find the density of the sum:

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Find the density of the sum when three dice are thrown independently:

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Find the probability that the values lie outside a circle of radius 7, in a square of side 14:

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Generate random samples of size 100 from a standard normal distribution:

The sampling distribution for the mean is given by NormalDistribution[0,1/10]:

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A lottery sells 10 tickets for $1 per ticket. Each time there is only one winning ticket. A gambler has $5 to spend. Find his probability of winning if he buys 5 tickets in 5 different lotteries:

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His probability of winning is greater if he buys 5 tickets in the same lottery:

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The waiting times for buying tickets and for buying popcorn at a movie theater are independent and they both follow the exponential distribution. The average waiting time for buying a ticket is 10 minutes and the average waiting time for buying popcorn is 5 minutes. Find the probability that a moviegoer waits for a total of less than 25 minutes before taking his or her seat:

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Obtain the numerical value of the probability directly:

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A factory produces cylindrically shaped roller bearings. The diameters of the bearings are normally distributed with mean 5 cm and standard deviation 0.01 cm. The lengths of the bearings are normally distributed with mean 7 cm and standard deviation 0.01 cm. Assuming that the diameter and the length are independently distributed, find the probability that a bearing has either diameter or length that differs from the mean by more than 0.02 cm.

The joint distribution of the diameters and lengths is given by:

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Properties & Relations  (7)Properties of the function, and connections to other functions

Marginal distributions are simply related to the component distributions:

One-dimensional marginal distributions:

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Two-dimensional marginal distributions:

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A product copula represents a product distribution:

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The PDF is the product of the PDFs of the component distributions:

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The CDF is the product of the CDFs of the component distributions:

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The generating functions are products of generating functions of component distributions:

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The components of the mean vector are the means of the component distributions:

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Similarly for the variance:

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A MultinormalDistribution is a product distribution when the covariance matrix is diagonal:

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Neat Examples  (1)Surprising or curious use cases

Iso-probability density levels for a three-dimensional product distribution:

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

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_productdistribution, author="Wolfram Research", title="{ProductDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/ProductDistribution.html}", note=[Accessed: 10-July-2025 ]}

@misc{reference.wolfram_2025_productdistribution, author="Wolfram Research", title="{ProductDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/ProductDistribution.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_productdistribution, organization={Wolfram Research}, title={ProductDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/ProductDistribution.html}, note=[Accessed: 10-July-2025 ]}

@online{reference.wolfram_2025_productdistribution, organization={Wolfram Research}, title={ProductDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/ProductDistribution.html}, note=[Accessed: 10-July-2025 ]}