# ProductDistribution

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.

# Examples

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## Basic Examples(3)

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

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

Define a multivariate distribution with continuous and discrete components:

## Scope(26)

### Basic Uses(7)

Define a product of two independent continuous distributions:

The PDF is the product of the component PDFs:

Product of discrete distributions:

The PDF is the product of the component PDFs:

Define a product distribution in which three components are repeated:

Probability density function for the four-dimensional product distribution:

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:

Define a general product distribution with few repeated components:

Compare to a random sample:

Product of multivariate continuous distributions:

Probability density function:

Verify that the integral of the PDF is 1:

Product of multivariate discrete distributions:

Compute the variance of the distribution:

Compare with the values obtained by using a random sample:

### Parametric Distributions(6)

Create a bivariate normal distribution with independent components:

Probability density function:

Compare to BinormalDistribution:

Define a two-dimensional Laplace distribution:

Probability density function:

Mean and variance:

Define product distribution of independent PoissonDistribution:

Probability density function:

Covariance:

The MultivariatePoissonDistribution does not have independent components:

The assumptions:

Create the product distribution of two independent examples of StudentTDistribution:

Generate random sample:

Goodness-of-fit test:

Compute properties with symbolic parameters:

Distribution functions:

Special moments:

Moments with closed forms for symbolic order:

Other moments can be obtained numerically:

Generating functions:

Find marginals of MultinormalDistribution:

Find product distribution of the marginal distributions:

Probability density function of :

is a MultinormalDistribution with a diagonal covariance matrix:

### Nonparametric Distributions(3)

Define the product of SmoothKernelDistribution:

Compare to the product of original distributions:

Create a sample from and define SmoothKernelDistribution for this sample:

Compare all three distributions:

Define a product of EmpiricalDistribution:

Plot the probability density function and cumulative distribution function:

Define a product distribution with HistogramDistribution:

Probability density function:

### Derived Distributions(10)

Define a product with a CensoredDistribution:

MarginalDistribution chooses the components of ProductDistribution:

Compose product distribution from marginals:

Probability density function:

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

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

Create the product distribution from a MixtureDistribution:

Probability density function:

Mean and variance:

Find the product distribution of minimum and maximum OrderDistribution:

Probability density function:

Plot density function for fixed :

Define a product distribution of a ParameterMixtureDistribution:

Product distribution is used as an input for a TransformedDistribution:

Find the product distribution of a TransformedDistribution:

Probability density function:

Find the product distribution of a TruncatedDistribution:

Variance depends on the truncation interval:

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

Find the product distribution of a TruncatedDistribution:

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

Truncation influences the direction and value of skewness:

Product of QuantityDistribution evaluates to QuantityDistribution:

Find moments:

Convert the distribution to kilograms:

## Applications(8)

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:

The estimated distribution has correlation similar to the sample:

Force independent estimates by estimating the marginal distributions:

Create product distribution:

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:

Show the region for the overlapping event:

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

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

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

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

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

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:

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

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:

Obtain the numerical value of the probability directly:

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:

## Properties & Relations(7)

Marginal distributions are simply related to the component distributions:

One-dimensional marginal distributions:

Two-dimensional marginal distributions:

A product copula represents a product distribution:

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

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

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

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

Similarly for the variance:

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

## Neat Examples(1)

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

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).

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2023_productdistribution, author="Wolfram Research", title="{ProductDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/ProductDistribution.html}", note=[Accessed: 25-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_productdistribution, organization={Wolfram Research}, title={ProductDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/ProductDistribution.html}, note=[Accessed: 25-September-2023 ]}