This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ProbabilityDistribution

 ProbabilityDistribution represents the continuous distribution with PDF pdf in the variable x where the pdf is taken to be zero for and . ProbabilityDistributionrepresents the discrete distribution with PDF pdf in the variable x where the pdf is taken to be zero for and . ProbabilityDistributionrepresents a multivariate distribution with PDF pdf in the variables x, y, ..., etc. ProbabilityDistributionrepresents a probability distribution with CDF given by cdf. ProbabilityDistributionrepresents a probability distribution with survival function given by sf.
• For a multivariate ProbabilityDistribution definition, all variables need to be either discrete or continuous; no mixed cases can occur.
Define a continuous probability distribution:
Probability density function:
Cumulative distribution function:
The mean and variance:
Define a continuous probability distribution:
Probability density function:
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Cumulative distribution function:
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The mean and variance:
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 Scope   (11)
Define a univariate continuous probability distribution:
Probability density function:
Define a univariate discrete probability distribution:
Cumulative distribution function:
Define a multivariate continuous distribution:
Verify that the integral of the PDF over the domain of the distribution is 1:
Define a multivariate discrete distribution:
Compute the expectation for an expression in this distribution:
Formula distribution specified by its CDF:
Mean and variance for the distribution:
Formula distribution specified by its survival function:
Kurtosis for the distribution:
Compare with the value obtained by using a random sample from the distribution:
Specify assumptions on a parameter in the definition of a formula distribution:
Probability density function:
Verify that the integral of the PDF is 1 under the given assumptions:
Define a two-sided exponential distribution:
Probability density function:
Cumulative distribution function:
Quantile function:
Moments:
A discrete distribution with hypergeometric term PDF:
Probability density function:
Cumulative distribution function:
Mean and variance:
A bivariate triangular distribution:
Probability density function:
Cumulative distribution function:
Mean and variance:
A discrete bivariate rectangular distribution:
Probability density function:
Cumulative distribution function:
Mean and variance:
 Applications   (13)
Define a continuous univariate distribution using its probability density function:
Obtain the cumulative distribution function for this distribution:
Study the statistical properties of the distribution:
Find the probability of an event:
Compute a conditional expectation:
Find the percentage of values between and in a probability distribution:
Between and :
Package it up as a function using NProbability:
Estimate the value of a parameter in ProbabilityDistribution:
Muth distribution is related to GompertzMakehamDistribution and has a PDF:
However, the third parameter of a GompertzMakehamDistribution is required to be positive:
Define a new distribution:
Probability density function:
Hazard function:
A double-sided power distribution is used in economics:
Probability density function:
Skewness:
Kurtosis:
Moment ratio diagram:
Create a uniform distribution over the unit disk:
If dist is the joint distribution of the vector , then x and y are not independent:
In a reliability study the CDF for the lifetime distribution is given by with and . What is the mean time to failure (MTTF) for the system? MTTF is also known as the mean:
Hence the mean time to failure is:
Change point distribution is characterized by a two-value hazard function:
Hazard function:
The probability density function is discontinuous at :
The limiting case is ExponentialDistribution:
The second limit:
Define a joint probability density function for two variables and :
Determine the value of the normalization factor :
The joint probability distribution is given by:
Compute the probability of an event in this distribution:
Obtain the numerical value of the probability directly:
The waiting times for buying tickets and for buying popcorn at a movie theater are independent and both follow an 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:
Define the distribution corresponding to an electron's radial density in a hydrogen atom:
Generate random numbers from an instance of this distribution:
Compare a sample histogram to the distribution density plot:
Find the mean radius and its standard deviation:
Define a joint probability on a square:
Each marginal is a uniform distribution:
Verify that random variate is distributionally equivalent to the sum of independent uniforms, using characteristic functions:
This is equal to the product of characteristic functions of marginals, i.e. :
This is possible because and are uncorrelated, albeit dependent:
The first argument of ProbabilityDistribution is the PDF by default:
The integral of the PDF over the distribution domain needs to be unity:
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