ProductDistribution
✖
ProductDistribution
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
- ProductDistribution[dist1,dist2,…,distn] represents a multivariate statistical distribution whose
marginal distribution (MarginalDistribution) is precisely distj, and for which the component distributions dist1,dist2,…,distn are independent. In particular, the probability density function (PDF) of a general product distribution ProductDistribution[dist1,dist2,…,distn] is precisely
, where
is the PDF of distj. While all product distributions share these properties, the characteristics and behavior of specific product distributions depend on their marginals dist1,dist2,…,distn.
- The component distributions dist1,dist2,…,distn may be continuous or discrete, univariate or multivariate, and may consist of any and all combinations of standard named distributions (e.g. BinomialDistribution, NormalDistribution, HypergeometricDistribution, etc.) and modifications (via TransformedDistribution, CensoredDistribution, ProductDistribution, CopulaDistribution, etc.) thereof. Moreover, each component distribution distj may be either symbolic (e.g. NormalDistribution[μ,σ]) or numeric (e.g. NormalDistribution[0,1]), and the shorthand ProductDistribution[{dist1,k1},{dist2,k2},… ,{distn,kn}] may be used to indicate that the j
marginal distj is repeated kj times.
- A program for beginning a systematic study of product distributions was proposed in the 1940s. Despite work throughout the 1950s and early 1960s toward that end, the first thorough treatment of the topic was a 1966 paper by Springer and Thompson. Since then, methods have been improved upon so that products of specially defined (e.g. piecewise) probability distributions can be studied both theoretically and algorithmically, and from a practical standpoint, product distributions have proven to be of tantamount importance in fields such as machine learning and finance. Product distributions have also been studied extensively using Monte Carlo theory and other numerical methods.
- Many relationships exist between ProductDistribution and various other distributions. ProductDistribution is a special case of CopulaDistribution in the sense that ProductDistribution[dist1,dist2,…,distn] is equivalent to CopulaDistribution["Product",{dist1,dist2,…,distn}]. KDistribution is defined to be a product of variates distributed according to GammaDistribution, the product of two instances of LogNormalDistribution is again LogNormalDistribution, and the product of BetaDistribution with GammaDistribution is again GammaDistribution. MultinormalDistribution (and hence BinormalDistribution) with a diagonal covariance matrix is also an example of ProductDistribution whose marginals are NormalDistribution.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Define a two-dimensional distribution for independent normal random variables:

https://wolfram.com/xid/0gfropdlbat2-oaujul

https://wolfram.com/xid/0gfropdlbat2-wl1j5j

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

https://wolfram.com/xid/0gfropdlbat2-2bes4e

https://wolfram.com/xid/0gfropdlbat2-c1k8e9

Define a multivariate distribution with continuous and discrete components:

https://wolfram.com/xid/0gfropdlbat2-xflhoo

https://wolfram.com/xid/0gfropdlbat2-emtmk2

Scope (26)Survey of the scope of standard use cases
Basic Uses (7)
Define a product of two independent continuous distributions:

https://wolfram.com/xid/0gfropdlbat2-bd0bna
The PDF is the product of the component PDFs:

https://wolfram.com/xid/0gfropdlbat2-bnlf3t


https://wolfram.com/xid/0gfropdlbat2-jh7k0

Product of discrete distributions:

https://wolfram.com/xid/0gfropdlbat2-036md

https://wolfram.com/xid/0gfropdlbat2-honnf9
The PDF is the product of the component PDFs:

https://wolfram.com/xid/0gfropdlbat2-bv4rd6


https://wolfram.com/xid/0gfropdlbat2-fve4k

Define a product distribution in which three components are repeated:

https://wolfram.com/xid/0gfropdlbat2-dkmuh8
Probability density function for the four-dimensional product distribution:

https://wolfram.com/xid/0gfropdlbat2-dm0qt5

Product distribution with both continuous and discrete components:

https://wolfram.com/xid/0gfropdlbat2-f39ma
Draw a random sample from this distribution:

https://wolfram.com/xid/0gfropdlbat2-l85dq
Estimate the distribution parameters for the components using the random sample:

https://wolfram.com/xid/0gfropdlbat2-hixzpg

https://wolfram.com/xid/0gfropdlbat2-m2i95

Define a general product distribution with few repeated components:

https://wolfram.com/xid/0gfropdlbat2-badhkl

https://wolfram.com/xid/0gfropdlbat2-edixfa


https://wolfram.com/xid/0gfropdlbat2-qmg1ym

https://wolfram.com/xid/0gfropdlbat2-endzq1

Product of multivariate continuous distributions:

https://wolfram.com/xid/0gfropdlbat2-ja94by

https://wolfram.com/xid/0gfropdlbat2-mkvnt

Verify that the integral of the PDF is 1:

https://wolfram.com/xid/0gfropdlbat2-pdtd7l

Product of multivariate discrete distributions:

https://wolfram.com/xid/0gfropdlbat2-bcfmyi
Compute the variance of the distribution:

https://wolfram.com/xid/0gfropdlbat2-pgmk4p

Compare with the values obtained by using a random sample:

https://wolfram.com/xid/0gfropdlbat2-g77g85

https://wolfram.com/xid/0gfropdlbat2-g2zkw5

Parametric Distributions (6)
Create a bivariate normal distribution with independent components:

https://wolfram.com/xid/0gfropdlbat2-9mu7cy

https://wolfram.com/xid/0gfropdlbat2-d2nbm6


https://wolfram.com/xid/0gfropdlbat2-fvnqcz

Compare to BinormalDistribution:

https://wolfram.com/xid/0gfropdlbat2-kztnfj


https://wolfram.com/xid/0gfropdlbat2-mml7ye

Define a two-dimensional Laplace distribution:

https://wolfram.com/xid/0gfropdlbat2-f1t2eh

https://wolfram.com/xid/0gfropdlbat2-z7tx4z


https://wolfram.com/xid/0gfropdlbat2-nt2325


https://wolfram.com/xid/0gfropdlbat2-jowqet

Define product distribution of independent PoissonDistribution:

https://wolfram.com/xid/0gfropdlbat2-m8zgqq

https://wolfram.com/xid/0gfropdlbat2-ib3ctr


https://wolfram.com/xid/0gfropdlbat2-co1t0e


https://wolfram.com/xid/0gfropdlbat2-qz0yaa

The MultivariatePoissonDistribution does not have independent components:

https://wolfram.com/xid/0gfropdlbat2-ogr5q3


https://wolfram.com/xid/0gfropdlbat2-gbmkcl

Create the product distribution of two independent examples of StudentTDistribution:

https://wolfram.com/xid/0gfropdlbat2-in7t2x

https://wolfram.com/xid/0gfropdlbat2-x8cczy
Fit a MultivariateTDistribution:

https://wolfram.com/xid/0gfropdlbat2-cz7o6


https://wolfram.com/xid/0gfropdlbat2-342rch

Compute properties with symbolic parameters:

https://wolfram.com/xid/0gfropdlbat2-ouy3e9

https://wolfram.com/xid/0gfropdlbat2-hvrwb


https://wolfram.com/xid/0gfropdlbat2-deq9q1


https://wolfram.com/xid/0gfropdlbat2-bwnuhe


https://wolfram.com/xid/0gfropdlbat2-eeaytt

Moments with closed forms for symbolic order:

https://wolfram.com/xid/0gfropdlbat2-p4g3z0


https://wolfram.com/xid/0gfropdlbat2-lm31az

Other moments can be obtained numerically:

https://wolfram.com/xid/0gfropdlbat2-dyn9z1


https://wolfram.com/xid/0gfropdlbat2-rp4zp


https://wolfram.com/xid/0gfropdlbat2-ec0m4l


https://wolfram.com/xid/0gfropdlbat2-drtn1w


https://wolfram.com/xid/0gfropdlbat2-duicxo

Find marginals of MultinormalDistribution:

https://wolfram.com/xid/0gfropdlbat2-3z4s1n

https://wolfram.com/xid/0gfropdlbat2-qsw3lc
Find product distribution of the marginal distributions:

https://wolfram.com/xid/0gfropdlbat2-8igz5g
Probability density function of :

https://wolfram.com/xid/0gfropdlbat2-m56eb

is a MultinormalDistribution with a diagonal covariance matrix:

https://wolfram.com/xid/0gfropdlbat2-x5lvfh


https://wolfram.com/xid/0gfropdlbat2-eot5gl

Nonparametric Distributions (3)
Define the product of SmoothKernelDistribution:

https://wolfram.com/xid/0gfropdlbat2-f5jq8p
Compare to the product of original distributions:

https://wolfram.com/xid/0gfropdlbat2-ujc4kn

https://wolfram.com/xid/0gfropdlbat2-t9yvgs

Create a sample from and define SmoothKernelDistribution for this sample:

https://wolfram.com/xid/0gfropdlbat2-8alr6s
Compare all three distributions:

https://wolfram.com/xid/0gfropdlbat2-boj7dq

Define a product of EmpiricalDistribution:

https://wolfram.com/xid/0gfropdlbat2-6jo8z6
Plot the probability density function and cumulative distribution function:

https://wolfram.com/xid/0gfropdlbat2-foar19

Define a product distribution with HistogramDistribution:

https://wolfram.com/xid/0gfropdlbat2-nm6thp

https://wolfram.com/xid/0gfropdlbat2-wcfjq4


https://wolfram.com/xid/0gfropdlbat2-8bju5j

Derived Distributions (10)
Define a product with a CensoredDistribution:

https://wolfram.com/xid/0gfropdlbat2-hx4ndj

https://wolfram.com/xid/0gfropdlbat2-qqe21

MarginalDistribution chooses the components of ProductDistribution:

https://wolfram.com/xid/0gfropdlbat2-6xc2uc
Compose product distribution from marginals:

https://wolfram.com/xid/0gfropdlbat2-7mha58

https://wolfram.com/xid/0gfropdlbat2-htmm9q

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

https://wolfram.com/xid/0gfropdlbat2-yfisdf

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

https://wolfram.com/xid/0gfropdlbat2-oki7tl

Create the product distribution from a MixtureDistribution:

https://wolfram.com/xid/0gfropdlbat2-curquo

https://wolfram.com/xid/0gfropdlbat2-1a0t0q


https://wolfram.com/xid/0gfropdlbat2-eomgd4


https://wolfram.com/xid/0gfropdlbat2-kq7m18


https://wolfram.com/xid/0gfropdlbat2-22d7fy

Find the product distribution of minimum and maximum OrderDistribution:

https://wolfram.com/xid/0gfropdlbat2-gq9pyc

https://wolfram.com/xid/0gfropdlbat2-3f16io

Plot density function for fixed :

https://wolfram.com/xid/0gfropdlbat2-soub24

Define a product distribution of a ParameterMixtureDistribution:

https://wolfram.com/xid/0gfropdlbat2-nzn1f9

https://wolfram.com/xid/0gfropdlbat2-xg17js

Product distribution is used as an input for a TransformedDistribution:

https://wolfram.com/xid/0gfropdlbat2-kxc5wr

Find the product distribution of a TransformedDistribution:

https://wolfram.com/xid/0gfropdlbat2-e71hat

https://wolfram.com/xid/0gfropdlbat2-jlszdr


https://wolfram.com/xid/0gfropdlbat2-zu5ovw

Find the product distribution of a TruncatedDistribution:

https://wolfram.com/xid/0gfropdlbat2-f3r90r
Variance depends on the truncation interval:

https://wolfram.com/xid/0gfropdlbat2-7js88y


https://wolfram.com/xid/0gfropdlbat2-c6q7cm

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

https://wolfram.com/xid/0gfropdlbat2-fy7lvr

https://wolfram.com/xid/0gfropdlbat2-flp9rr

Find the product distribution of a TruncatedDistribution:

https://wolfram.com/xid/0gfropdlbat2-q1tksp
Compare the PDF with the product distribution of two Poisson distributions:

https://wolfram.com/xid/0gfropdlbat2-h3h3hj

https://wolfram.com/xid/0gfropdlbat2-xmo9dc

Truncation influences the direction and value of skewness:

https://wolfram.com/xid/0gfropdlbat2-tucktu


https://wolfram.com/xid/0gfropdlbat2-d149t9

Product of QuantityDistribution evaluates to QuantityDistribution:

https://wolfram.com/xid/0gfropdlbat2-3fl8x5


https://wolfram.com/xid/0gfropdlbat2-h363ms


https://wolfram.com/xid/0gfropdlbat2-883grg

Convert the distribution to kilograms:

https://wolfram.com/xid/0gfropdlbat2-j24zu1

Applications (8)Sample problems that can be solved with this function
Generate an uncorrelated sample from a binormal distribution:

https://wolfram.com/xid/0gfropdlbat2-7usz77
The sample is slightly correlated, even though the original distribution is not:

https://wolfram.com/xid/0gfropdlbat2-msj51f

Estimate the distribution from data:

https://wolfram.com/xid/0gfropdlbat2-4c25vt

The estimated distribution has correlation similar to the sample:

https://wolfram.com/xid/0gfropdlbat2-iaxsqx

Force independent estimates by estimating the marginal distributions:

https://wolfram.com/xid/0gfropdlbat2-dxp68z


https://wolfram.com/xid/0gfropdlbat2-nd9qux


https://wolfram.com/xid/0gfropdlbat2-ffwla1

https://wolfram.com/xid/0gfropdlbat2-7t19ek

The resulting distribution has no correlation:

https://wolfram.com/xid/0gfropdlbat2-fwjik

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:

https://wolfram.com/xid/0gfropdlbat2-48p2gv

Show the region for the overlapping event:

https://wolfram.com/xid/0gfropdlbat2-da8rm7

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

https://wolfram.com/xid/0gfropdlbat2-b32gvh


https://wolfram.com/xid/0gfropdlbat2-bbj8cu

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

https://wolfram.com/xid/0gfropdlbat2-bgj1im


https://wolfram.com/xid/0gfropdlbat2-ihdti

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

https://wolfram.com/xid/0gfropdlbat2-iaoufc

https://wolfram.com/xid/0gfropdlbat2-tvmbi


https://wolfram.com/xid/0gfropdlbat2-bxnbwt

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

https://wolfram.com/xid/0gfropdlbat2-bb8pcp

https://wolfram.com/xid/0gfropdlbat2-zk9sz
The sampling distribution for the mean is given by NormalDistribution[0,1/10]:

https://wolfram.com/xid/0gfropdlbat2-f2hxd5

https://wolfram.com/xid/0gfropdlbat2-bhk5


https://wolfram.com/xid/0gfropdlbat2-ckwazd


https://wolfram.com/xid/0gfropdlbat2-lc89

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:

https://wolfram.com/xid/0gfropdlbat2-bm1kiu

https://wolfram.com/xid/0gfropdlbat2-bvbb39


https://wolfram.com/xid/0gfropdlbat2-c48k9f

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

https://wolfram.com/xid/0gfropdlbat2-gr2ml7


https://wolfram.com/xid/0gfropdlbat2-cxs0cb

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:

https://wolfram.com/xid/0gfropdlbat2-ucczjm


https://wolfram.com/xid/0gfropdlbat2-5vkn5h


https://wolfram.com/xid/0gfropdlbat2-daxc0i

Obtain the numerical value of the probability directly:

https://wolfram.com/xid/0gfropdlbat2-y90xq

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:

https://wolfram.com/xid/0gfropdlbat2-pprh9u

https://wolfram.com/xid/0gfropdlbat2-67t9j7


https://wolfram.com/xid/0gfropdlbat2-zegqr5

Properties & Relations (7)Properties of the function, and connections to other functions
Marginal distributions are simply related to the component distributions:

https://wolfram.com/xid/0gfropdlbat2-btnllm
One-dimensional marginal distributions:

https://wolfram.com/xid/0gfropdlbat2-d7whv0

Two-dimensional marginal distributions:

https://wolfram.com/xid/0gfropdlbat2-v95zb

A product copula represents a product distribution:

https://wolfram.com/xid/0gfropdlbat2-k7czqt

https://wolfram.com/xid/0gfropdlbat2-5wnbd

https://wolfram.com/xid/0gfropdlbat2-j9ikfm


https://wolfram.com/xid/0gfropdlbat2-bg3bdk


https://wolfram.com/xid/0gfropdlbat2-s5njms

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

https://wolfram.com/xid/0gfropdlbat2-cy50co

https://wolfram.com/xid/0gfropdlbat2-phrml


https://wolfram.com/xid/0gfropdlbat2-cactay


https://wolfram.com/xid/0gfropdlbat2-el103m

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

https://wolfram.com/xid/0gfropdlbat2-fs3nu5

https://wolfram.com/xid/0gfropdlbat2-bl7tkw


https://wolfram.com/xid/0gfropdlbat2-hocjdf


https://wolfram.com/xid/0gfropdlbat2-uxl88n

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

https://wolfram.com/xid/0gfropdlbat2-fxlohc

https://wolfram.com/xid/0gfropdlbat2-o2xpj5


https://wolfram.com/xid/0gfropdlbat2-fhppg3


https://wolfram.com/xid/0gfropdlbat2-gp8mce

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

https://wolfram.com/xid/0gfropdlbat2-c5xtc8

https://wolfram.com/xid/0gfropdlbat2-getc7


https://wolfram.com/xid/0gfropdlbat2-cr2pel


https://wolfram.com/xid/0gfropdlbat2-izjl2y


https://wolfram.com/xid/0gfropdlbat2-h1yayf

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

https://wolfram.com/xid/0gfropdlbat2-d7rclv

https://wolfram.com/xid/0gfropdlbat2-hkpgx0


https://wolfram.com/xid/0gfropdlbat2-bw4rd


https://wolfram.com/xid/0gfropdlbat2-jxo4z7

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
]}
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
]}