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ProductDistribution
✖
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.

Background & Context

Examples

open allclose all

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

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-oaujul
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-wl1j5j
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-2bes4e
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-c1k8e9
Out[2]=2

Define a multivariate distribution with continuous and discrete components:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-xflhoo
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-emtmk2
Out[2]=2

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

Basic Uses  (7)

Define a product of two independent continuous distributions:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-bd0bna

The PDF is the product of the component PDFs:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-bnlf3t
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-jh7k0
Out[5]=5

Product of discrete distributions:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-036md
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-honnf9

The PDF is the product of the component PDFs:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-bv4rd6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-fve4k
Out[4]=4

Define a product distribution in which three components are repeated:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-dkmuh8

Probability density function for the four-dimensional product distribution:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-dm0qt5
Out[2]=2

Product distribution with both continuous and discrete components:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-f39ma

Draw a random sample from this distribution:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-l85dq

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

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-hixzpg
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-m2i95
Out[4]=4

Define a general product distribution with few repeated components:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-badhkl
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-edixfa
Out[2]=2

Compare to a random sample:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-qmg1ym
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-endzq1
Out[4]=4

Product of multivariate continuous distributions:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-ja94by

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-mkvnt
Out[2]=2

Verify that the integral of the PDF is 1:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-pdtd7l
Out[3]=3

Product of multivariate discrete distributions:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-bcfmyi

Compute the variance of the distribution:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-pgmk4p
Out[2]=2

Compare with the values obtained by using a random sample:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-g77g85
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-g2zkw5
Out[4]=4

Parametric Distributions  (6)

Create a bivariate normal distribution with independent components:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-9mu7cy

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-d2nbm6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-fvnqcz
Out[3]=3

Compare to BinormalDistribution:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-kztnfj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-mml7ye
Out[5]=5

Define a two-dimensional Laplace distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-f1t2eh

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-z7tx4z
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-nt2325
Out[3]=3

Mean and variance:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-jowqet
Out[4]=4

Define product distribution of independent PoissonDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-m8zgqq

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-ib3ctr
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-co1t0e
Out[3]=3

Covariance:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-qz0yaa

The MultivariatePoissonDistribution does not have independent components:

In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-ogr5q3

The assumptions:

In[6]:=6
✖
https://wolfram.com/xid/0gfropdlbat2-gbmkcl
Out[6]=6

Create the product distribution of two independent examples of StudentTDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-in7t2x

Generate random sample:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-x8cczy

Fit a MultivariateTDistribution:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-cz7o6
Out[3]=3

Goodness-of-fit test:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-342rch
Out[4]=4

Compute properties with symbolic parameters:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-ouy3e9

Distribution functions:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-hvrwb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-deq9q1
Out[3]=3

Special moments:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-bwnuhe
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-eeaytt
Out[5]=5

Moments with closed forms for symbolic order:

In[6]:=6
✖
https://wolfram.com/xid/0gfropdlbat2-p4g3z0
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0gfropdlbat2-lm31az
Out[7]=7

Other moments can be obtained numerically:

In[8]:=8
✖
https://wolfram.com/xid/0gfropdlbat2-dyn9z1
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0gfropdlbat2-rp4zp
Out[9]=9

Generating functions:

In[10]:=10
✖
https://wolfram.com/xid/0gfropdlbat2-ec0m4l
Out[10]=10
In[11]:=11
✖
https://wolfram.com/xid/0gfropdlbat2-drtn1w
Out[11]=11
In[12]:=12
✖
https://wolfram.com/xid/0gfropdlbat2-duicxo
Out[12]=12

Find marginals of MultinormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-3z4s1n
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-qsw3lc

Find product distribution of the marginal distributions:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-8igz5g

Probability density function of :

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-m56eb
Out[4]=4

is a MultinormalDistribution with a diagonal covariance matrix:

In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-x5lvfh
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0gfropdlbat2-eot5gl
Out[6]=6

Nonparametric Distributions  (3)

Define the product of SmoothKernelDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-f5jq8p

Compare to the product of original distributions:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-ujc4kn
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-t9yvgs
Out[5]=5

Create a sample from and define SmoothKernelDistribution for this sample:

In[6]:=6
✖
https://wolfram.com/xid/0gfropdlbat2-8alr6s

Compare all three distributions:

In[7]:=7
✖
https://wolfram.com/xid/0gfropdlbat2-boj7dq
Out[7]=7

Define a product of EmpiricalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-6jo8z6

Plot the probability density function and cumulative distribution function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-foar19
Out[2]=2

Define a product distribution with HistogramDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-nm6thp

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-wcfjq4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-8bju5j
Out[3]=3

Derived Distributions  (10)

Define a product with a CensoredDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-hx4ndj
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-qqe21
Out[3]=3

MarginalDistribution chooses the components of ProductDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-6xc2uc

Compose product distribution from marginals:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-7mha58

Probability density function:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-htmm9q
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-yfisdf
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-oki7tl
Out[5]=5

Create the product distribution from a MixtureDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-curquo

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-1a0t0q
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-eomgd4
Out[3]=3

Mean and variance:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-kq7m18
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-22d7fy
Out[5]=5

Find the product distribution of minimum and maximum OrderDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-gq9pyc

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-3f16io
Out[2]=2

Plot density function for fixed :

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-soub24
Out[3]=3

Define a product distribution of a ParameterMixtureDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-nzn1f9
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-xg17js
Out[2]=2

Product distribution is used as an input for a TransformedDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-kxc5wr
Out[1]=1

Find the product distribution of a TransformedDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-e71hat

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-jlszdr
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-zu5ovw
Out[3]=3

Find the product distribution of a TruncatedDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-f3r90r

Variance depends on the truncation interval:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-7js88y
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-c6q7cm
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-fy7lvr
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-flp9rr
Out[5]=5

Find the product distribution of a TruncatedDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-q1tksp

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

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-h3h3hj
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-xmo9dc
Out[3]=3

Truncation influences the direction and value of skewness:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-tucktu
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-d149t9
Out[5]=5

Product of QuantityDistribution evaluates to QuantityDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-3fl8x5
Out[1]=1

Find moments:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-h363ms
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-883grg
Out[3]=3

Convert the distribution to kilograms:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-j24zu1
Out[4]=4

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

Generate an uncorrelated sample from a binormal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-7usz77

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

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-msj51f

Estimate the distribution from data:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-4c25vt
Out[3]=3

The estimated distribution has correlation similar to the sample:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-iaxsqx

Force independent estimates by estimating the marginal distributions:

In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-dxp68z
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0gfropdlbat2-nd9qux
Out[6]=6

Create product distribution:

In[7]:=7
✖
https://wolfram.com/xid/0gfropdlbat2-ffwla1
In[8]:=8
✖
https://wolfram.com/xid/0gfropdlbat2-7t19ek
Out[8]=8

The resulting distribution has no correlation:

In[9]:=9
✖
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:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-48p2gv
Out[1]=1

Show the region for the overlapping event:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-da8rm7
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-b32gvh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-bbj8cu
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-bgj1im
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-ihdti
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-iaoufc
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-tvmbi
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-bxnbwt
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-bb8pcp
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-zk9sz

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

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-f2hxd5
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-bhk5
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-ckwazd
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0gfropdlbat2-lc89
Out[6]=6

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-bm1kiu
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-bvbb39
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-c48k9f
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-gr2ml7
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-cxs0cb
Out[5]=5

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-ucczjm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-5vkn5h
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-daxc0i
Out[3]=3

Obtain the numerical value of the probability directly:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-y90xq
Out[4]=4

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-pprh9u
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-67t9j7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-zegqr5
Out[3]=3

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

Marginal distributions are simply related to the component distributions:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-btnllm

One-dimensional marginal distributions:

In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-d7whv0
Out[2]=2

Two-dimensional marginal distributions:

In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-v95zb
Out[3]=3

A product copula represents a product distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-k7czqt
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-5wnbd
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-j9ikfm
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-bg3bdk
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-s5njms
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-cy50co
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-phrml
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-cactay
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-el103m
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-fs3nu5
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-bl7tkw
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-hocjdf
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-uxl88n
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-fxlohc
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-o2xpj5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-fhppg3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-gp8mce
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-c5xtc8
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-getc7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-cr2pel
Out[3]=3

Similarly for the variance:

In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-izjl2y
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfropdlbat2-h1yayf
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-d7rclv
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-hkpgx0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfropdlbat2-bw4rd
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfropdlbat2-jxo4z7
Out[4]=4

Neat Examples  (1)Surprising or curious use cases

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

In[1]:=1
✖
https://wolfram.com/xid/0gfropdlbat2-cc1el
In[2]:=2
✖
https://wolfram.com/xid/0gfropdlbat2-l7wfie
Out[2]=2
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 ]}