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ProductDistribution

ProductDistribution
represents the joint distribution with independent component distributions , , ....
  • The probability density for ProductDistribution is given by where is the PDF of , is the PDF of , etc.
  • The notation indicates that is repeated n times.
  • The distributions can be any combination of univariate, multivariate, continuous, or discrete distributions.
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:
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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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:
Create a bivariate normal distribution with independent components:
Probability density function:
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:
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 denstiy function:
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:
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:
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:
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:
Iso-probability density levels for a three-dimensional product distribution:
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