QuantityDistribution
QuantityDistribution[dist,unit]
represents a distribution dist of quantities with unit specified by unit.
QuantityDistribution[dist,{unit_{1},unit_{2},…}]
represents a multivariate distribution with units {unit_{1},unit_{2},…}.
Details
 QuantityDistribution is typically created by using quantities in parametric, derived distributions or estimated from data with quantities.
 QuantityDistribution will automatically attempt to parse an unknown unit string to its canonical form. »
 QuantityDistribution[dist,unit] is equivalent to TransformedDistribution[Quantity[x,unit],x dist]. »
 The arguments to distribution functions are assumed to have units compatible with unit. »
 The values of distribution functions for univariate distributions have the following units:

CDF unitless InverseCDF unit SurvivalFunction unitless InverseSurvivalFunction unit  The values of PDF and HazardFunction have the following units: »

continuous univariate dist unit^{1} continuous multivariate dist unit_{1}^{1} unit_{2}^{1} … discrete dist unitless  Moments of univariate QuantityDistribution[dist,unit] have the following units:

Moment[…,r] unit^{ r} CentralMoment[…,r] unit^{ r} Cumulant[…,r] unit^{ r} FactorialMoment[…,r] unit^{ r}  Moments of multivariate QuantityDistribution[dist,unit] have the following units:

Moment[…,{r_{1},r_{2},…}] unit_{1}^{r1} unit_{2}^{r2} … CentralMoment[…,{r_{1},r_{2},…}] unit_{1}^{r1} unit_{2}^{r2} … Cumulant[…,{r_{1},r_{2},…}] unit_{1}^{r1} unit_{2}^{r2} … FactorialMoment[…,{r_{1},r_{2},…}] unit_{1}^{r1} unit_{2}^{r2} …  Momentgenerating functions require their arguments to be quantities with units that are reciprocal to the units of the QuantityDistribution.
 Sampling from QuantityDistribution gives Quantity or QuantityArray.
 QuantityDistribution can be used with such functions as Mean, CDF, RandomVariate, QuantityMagnitude, and Expectation.
Examples
open allclose allBasic Examples (3)
Scope (29)
Basic Uses (11)
Find the mean time of service for a device with WeibullDistributionmodeled lifetime:
Find the median time of service:
Compute the cumulative distribution function for a quantity distribution:
The argument of CDF is assumed to be a quantity of time:
Evaluate the cumulative distribution function for a few time values:
Find the cumulative distribution function for the argument in minutes:
Compute quantile of a QuantityDistribution:
Sampling from QuantityDistribution gives Quantity or QuantityArray:
Sample 238 insurance claim sizes according to the chosen model:
Compute loglikelihood for distribution with quantities on data with quantities:
Data can have any compatible units:
PDF for continuous distribution with quantities:
The probability density function has a reciprocal unit:
This is consistent with substituting quantities into the symbolic expression for PDF:
The PDF for discrete distribution with quantities is unitless:
The HazardFunction for continuous distributions with quantities has reciprocal units:
Estimate the distribution of solar radiant energy reaching Earth's surface from synthetic data:
QuantityMagnitude and QuantityUnit of a quantity distribution:
Extract the distribution of the magnitude:
Use UnitConvert to convert distribution to compatible units:
Distribution of voltage with random resistance and current:
Use UnitSimplify to simplify units to "Volts":
Construction (6)
Define distribution for a quantity of length:
Define the same distribution using a quantity scale parameter:
Define joint distribution of height and weight:
Define the same distribution using quantity parameters:
Convert the distribution into different units:
Parsing of unknown unit strings is automatically attempted:
Quantities can be used to specify units in QuantityDistribution:
Magnitude of the input quantity is ignored:
Use TransformedDistribution to define distribution of random quantities:
Estimation (4)
Estimate distribution from quantity data:
Fit a normal distribution in meters to the data:
Fit a normal distribution in feet to the data:
Estimate parameters of a multivariate distribution:
Fit a multivariate distribution in the units of the data:
Fit a multivariate distribution in compatible units:
Estimate distribution from the quantity data:
The data is bimodal, suggesting a mixture of bivariate normal distributions might be a good fit:
EstimatedDistribution gives QuantityDistribution with fitted parameters:
FindDistributionParameters gives rules with Quantity values when possible:
Substituting these parameters into the model results in QuantityDistribution:
Estimate distribution from quantity data:
Specify starting values of relevant parameters using quantities:
Parameters of magnitude distribution dist in QuantityDistribution[dist,units] are numeric:
Derived Distributions (8)
Truncate a quantity distribution:
Truncation of QuantityDistribution gives another QuantityDistribution:
Censor a quantity distribution:
Censoring of QuantityDistribution gives another QuantityDistribution:
Compute the mean of the censored quantity distribution:
Define mixture of quantity distributions:
Mixture of compatible QuantityDistributions gives another QuantityDistribution:
Define a spliced distribution with quantity distributions:
Splicing several QuantityDistributions gives another QuantityDistribution:
Sample from the spliced distribution and visualize the histogram:
ProductDistribution of quantity distributions:
Product distribution evaluates to QuantityDistribution:
Create a parameter mixture with quantity distribution:
Parameter mixture of QuantityDistribution evaluates to QuantityDistribution:
Define distribution of the maximum of random quantities:
The distribution of the order statistics gives QuantityDistribution:
QuantityDistribution of a QuantityDistribution:
The result is consistent with the behavior of Quantity under composition:
Applications (8)
The average speed of cars traveling from Champaign, Illinois, to Chicago, Illinois, is well described as a triangular random variable:
Find the expected time of travel:
Define the distribution of a cube's sidelength measurements:
Plot the distribution density:
Compute the mean and the dispersion measures of the cube's volume:
Plot the PDF of the cube's volume:
Recorded body weights in kilograms:
Plot the histogram of the data:
Fit NormalDistribution to the data:
Compare the histogram with the estimated PDF:
Express the estimated distribution in pounds:
The waiting time a customer spends in a restaurant is believed to be exponentially distributed with an average wait time of 5 minutes:
Find the probability that the customer will have to wait more than 10 minutes:
The lifetime of a component has WeibullDistribution with shape and scale parameters of 2 and 997.5 hours, respectively. Find the probability that the component survives 300 hours:
Find the probability that the component is still working after 500 hours, given that it has survived 300 hours:
Find the mean time to failure:
Estimate the distribution of the daily mean temperature in Chicago in the summer of 2015:
Fit the distribution to PERTDistribution:
Find the estimated distribution for temperature in degrees Fahrenheit:
Velocity density function along any direction of a gas molecule follows a normal distribution with mean 0 and standard deviation . The standard deviation for molecular hydrogen at 573 K is:
The distribution of the speeds of molecules in a hydrogen gas at 573 K is given:
Find the probability that a hydrogen molecule has speed at least 4000 meters per second:
Find the average speed of such a molecule:
Compare the ratios of the average speed and the RMS speed to the most probable speed:
Simulate the speed of 100 hydrogen molecules in the above conditions:
The acceleration of gravity can be measured by measuring a pendulum's period and its length using . The uncertainty in the average of five repeated measurements of the period is modeled with a BatesDistribution:
The pendulum's length has been measured using a ruler with resolution of 1 mm, so its uncertainty is modeled with a UniformDistribution:
The uncertainty in the measurement of the acceleration of gravity:
Compare to the linear approximation:
Compute the average acceleration using the exact and the linearized distributions:
Compute the scales of uncertainty:
Find the sampling estimate of the 90% confidence interval for the measured acceleration:
Properties & Relations (6)
QuantityDistribution with dimensionless units will autoevaluate to the magnitude distribution:
Use QuantityMagnitude and QuantityUnit to extract distribution and units:
QuantityDistribution[dist,unit] is equivalent to TransformedDistribution:
Skewness and kurtosis of a QuantityDistribution are unitless:
Joint distribution of mass and acceleration:
The unit of the moment of order (1,1) is the product of units of each component:
The moment can be interpreted as the expected value of force:
PDF of a continuous distribution with units integrates to 1 over its domain:
Reciprocal units of the density function are canceled by the unit of the measure to give unitless antiderivative:
Possible Issues (4)
The dimensionality of the distribution and units must agree:
Specify two units for two dimensions:
The identical unit for all dimensions:
The unit conversion may map the data outside of the support of the distribution:
Estimate a quantity distribution in kilograms:
Estimate using the units of the data:
Then convert the distribution:
Setting fixed values for parameter estimation is unit dependent:
Convert the whole quantity distribution instead:
Use converted units in call to estimation:
The magnitude of the factorial moment of a QuantityDistribution coincides with the factorial moment of the magnitude distribution:
Factorial moment expression is not homogeneous in the location and scale parameters, hence direct substitution of quantity parameters results in an error: