TransformedDistribution
TransformedDistribution[expr,xdist]
represents the transformed distribution of expr where the random variable x follows the distribution dist.
TransformedDistribution[expr,{x1,x2,…}dist]
represents the transformed distribution of expr where {x1,x2,…} follows the multivariate distribution dist.
TransformedDistribution[expr,xproc]
represents the transformed distribution where expr contains expressions of the form x[t], referring the value at time t from the random process proc.
TransformedDistribution[expr,{x1dist1,x2dist2 ,…}]
represents a transformed distribution where x1, x2, … are independent and follow the distributions dist1, dist2, ….
Details and Options
- xdist can be entered as x dist dist or x \[Distributed] dist.
- TransformedDistribution will simplify to known special distributions whenever possible.
- Assumptions on parameters can be specified using the options Assumptions->assum.
- TransformedDistribution can be used with such functions as Mean, CDF, RandomVariate, etc.
Examples
open allclose allBasic Examples (3)
Scope (61)
Basic Uses (6)
Compare the PDFs with the probability density function of the original distribution:
Generate random numbers following shifted distribution:
Use Assumptions to specify conditions on a parameter in the transformation:
Define a nonlinear transformation of a discrete distribution:
Probability density function is defined on integer square roots:
Find the distribution of the sum of two different variables:
Compare the resulting distribution with the summands:
The mean of is the sum of the means:
Find the distribution of the product:
Quantity Uses (4)
Consistent use of Quantity in the transformation function yields QuantityDistribution:
Define a transformation with Quantity to obtain QuantityDistribution:
Transformations of QuantityDistribution:
Use Quantity[x,u1]QuantityDistribution[dist,u2] to indicate that x is the magnitude of the random variable relative to unit u1:
Transformations (9)
The domain has been automatically chosen so it is a probability distribution:
Create a piecewise continuous distribution:
Transformation composed of few functions:
Compare with the original distribution:
Find the distribution of the maximum of two different distributions:
Cumulative distribution function and survival function:
Notice it is larger than the means of both original distributions:
Find the distribution of a product of powers of two independent distributions:
Visualize distribution by smooth histogram and histogram based on a random sample:
Add two bivariate distributions:
Visualize the distribution of the sum:
Scale a bivariate distribution:
Visualize the probability density function:
Create a multivariate distribution given its marginals:
It is the same as using product kernel in copula construction:
Plot the distribution function:
Dimension-reducing transformation of a multivariate distribution:
Parametric Distributions (7)
Prove a relation between distributions:
Create a heavy-tail distribution using exponential transformation:
The moments exist only for the orders less than :
Transformation of two identically distributed independent variables:
Add two discrete independent distributions:
Cumulative distribution function:
Create an arbitrary two-dimensional distribution:
The components are uncorrelated:
Define a bivariate discrete distribution:
Nonparametric Distributions (3)
Shift an EmpiricalDistribution:
Compare cumulative distribution functions:
Scale a HistogramDistribution:
Compare probability density functions:
Define a transformed SmoothKernelDistribution:
Derived Distributions (9)
Complex transformations can be done in steps:
The direct calculation may take longer than calculation in steps:
Split the transformation to find the probability density function:
Find a transformation of a MixtureDistribution:
The mean is shifted by the same amount as the distribution:
Find a transformation of a ParameterMixtureDistribution:
Cumulative distribution function:
Standard deviation is scaled by the same factor as the distribution:
Find a transformation of a TruncatedDistribution:
Find a transformation of a CensoredDistribution:
Plot the probability density function:
Find a transformation of an OrderDistribution:
The mean is not the exponent of the mean of the original distribution:
Find a transformation of a MarginalDistribution:
Transform a CopulaDistribution:
Define a transformation of a ProductDistribution:
Random Processes (4)
Define transformations on the values of a random process:
This is equivalent to the exponential transformation of SliceDistribution:
A time ordering is implied for multiple distinct time stamps:
Use Assumptions to give an explicit ordering:
TransformedDistribution supports coincident use of both processes and distributions:
Find the slice distribution of a product of a process and a distribution:
Automatic Simplifications (19)
Continuous Distributions (9)
Special transformations of NormalDistribution:
Special transformations of ExponentialDistribution:
Special transformations of UniformDistribution:
Special transformations between SinghMaddalaDistribution and DagumDistribution:
Special transformation of ChiSquareDistribution:
Special transformations of StudentTDistribution:
Special transformation of BetaDistribution:
Special transformations of BinormalDistribution:
Special transformation of ParetoDistribution:
Discrete Distributions (7)
Special transformations of BernoulliDistribution:
Special transformation of BorelTannerDistribution:
Special transformations of GeometricDistribution:
Special transformations of PoissonDistribution:
Special transformation of PoissonConsulDistribution:
Special transformation of PolyaAeppliDistribution:
Special transformations of SkellamDistribution:
Multivariate Distributions (3)
The multinormal distribution is closed under affine transformation:
Multivariate Student distribution is closed under affine transformations:
Transformation creating LogMultinormalDistribution:
Options (1)
Assumptions (1)
Compute the PDF for an affine transformation of a Weibull distribution:
Use Assumptions to specify the condition :
Applications (8)
Two points are chosen randomly and independently from the interval , according to a uniform distribution. Compute the expected distance between the two points:
Two archers shoot at a target. The distance of each shot from the center of the target is uniformly distributed from 0 to 10 inches, independent of the other shot. Find the probability that the losing shot is more than 5 inches away from the target:
Romeo and Juliet have a date at a given time, and each, independently, will be late by an amount of minutes that is exponentially distributed with parameter . Find the distribution of the difference between their times of arrival:
Probability that they miss each other by at least t minutes:
A driver travels with an average speed of 65 mph for a distance of 120 miles. Assuming the speed has normal distribution with standard deviation of 3 mph and there was no road work, find the distribution of time it takes the driver to cover the distance:
Plot the probability density function:
Find the median travel time in hours:
The Young modulus and the shear modulus of a bar have been measured as and , respectively. Assuming a symmetric triangular distribution for measurement uncertainty, and that respective coverage intervals have 90% coverage probability, determine the uncertainty of Poisson's ratio :
Confirm that measurements are contained in given intervals with 90% probability:
Use TransformedDistribution to define the distribution for uncertainty of Poisson's ratio:
Compare to the linear approximation:
Find the mean ratio using the exact and the approximate distributions:
Find the standard deviation of the ratio using the exact and the approximate distributions:
Compute the Poisson's ratio measurement density function:
Visualize the density function and compare it to the normal approximation:
Concentration-time curve for the circulation of a medication injected in a bloodstream is described by lagged normal distribution:
Compute the first several moments:
Plot the distribution density:
Find the distribution of the distance between the origin and the points placed according to DirichletDistribution on a plane:
Plot the probability density function:
Find the mean distance to the origin:
Use TransformedDistribution to create a discrete probability distribution with noninteger support:
Properties & Relations (8)
TransformedDistribution uses local names for the variables in the input:
Hence subsequent computations can be done with the original variable name:
The support of the PDF may change under a transformation:
Applying the identity transformation to a distribution leaves it unchanged:
Components of the identity transformation give marginal distributions:
Compute the probability of an event for a transformed distribution:
Substituting transformation into the event:
Compute the expectation of an expression for a transformed distribution:
Substituting transformation into the expression:
CensoredDistribution is a special case of TransformedDistribution:
OrderDistribution is a special case of TransformedDistribution:
In particular, the extreme cases correspond to Min and Max:
SliceDistribution relates TransformedProcess to TransformedDistribution:
Possible Issues (3)
Let be a sum of random variates . Distribution of may be different from distribution of :
Distribution of the sum of two independent identically distributed variates may be different from that of :
Compare distribution densities:
Autoevaluation may fail for complicated expressions:
Evaluating TransformedDistribution in steps may allow special rules to be recognized:
Text
Wolfram Research (2010), TransformedDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformedDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "TransformedDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/TransformedDistribution.html.
APA
Wolfram Language. (2010). TransformedDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TransformedDistribution.html