# ProbabilityDistribution

ProbabilityDistribution[pdf,{x,xmin,xmax}]

represents the continuous distribution with PDF pdf in the variable x where the pdf is taken to be zero for and .

ProbabilityDistribution[pdf,{x,xmin,xmax,1}]

represents the discrete distribution with PDF pdf in the variable x where the pdf is taken to be zero for and .

ProbabilityDistribution[pdf,{x,},{y,},]

represents a multivariate distribution with PDF pdf in the variables x, y, , etc.

ProbabilityDistribution[{"CDF",cdf},]

represents a probability distribution with CDF given by cdf.

ProbabilityDistribution[{"SF",sf},]

represents a probability distribution with survival function given by sf.

ProbabilityDistribution[{"HF",hf},]

represents a probability distribution with hazard function given by hf.

# Examples

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## Basic Examples(1)

Define a continuous probability distribution:

Probability density function:

Cumulative distribution function:

The mean and variance:

## Scope(13)

### Distribution Specification(8)

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:

Define probability distribution by its hazard function:

Compute survival probability:

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:

### Continuous Univariate Distributions(2)

Define a two-sided exponential distribution:

Probability density function:

Cumulative distribution function:

Quantile function:

Moments:

Define a distribution with PDF given in terms of DiracDelta:

Cumulative distribution function:

Quantile function:

### Discrete Univariate Distributions(1)

A discrete distribution with hypergeometric term PDF:

Probability density function:

Cumulative distribution function:

Mean and variance:

### Continuous Multivariate Distributions(1)

A bivariate triangular distribution:

Probability density function:

Cumulative distribution function:

Mean and variance:

### Discrete Multivariate Distributions(1)

A discrete bivariate rectangular distribution:

Probability density function:

Cumulative distribution function:

Mean and variance:

## Options(2)

### Assumptions(1)

Specify assumptions:

### Method(1)

Normalize a continuous probability distribution:

Verify that the PDF of the distribution is normalized to unity:

Normalize a multivariate probability distribution:

## Applications(14)

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:

Compute the probability that a random variable is within one standard deviation from the mean:

Probability of being within two standard deviations from the mean:

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:

Find each MarginalDistribution:

If dist is the joint distribution of the vector {x,y}, 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 distribution on a square:

Each marginal distribution is the uniform distribution on the interval from to :

Verify that random variate is equal in distribution 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:

Compute properties for the slice distribution at time of an inhomogeneous Poisson process with intensity function :

The mean is the integral of the intensity function up to time :

Compare to the continuous time version of the process:

## Properties & Relations(3)

The first argument of ProbabilityDistribution is the PDF by default:

The integral of the PDF over the distribution domain needs to be unity:

ProbabilityDistribution decomposes into absolutely continuous and discrete parts:

PDF can be given as InterpolatingFunction:

## Possible Issues(5)

Probability density function used to define the distribution is assumed to be valid:

The specified PDF is invalid since it is not non-negative and not normalized to 1:

Sampling from this distribution may generate variates outside the distribution domain:

The PDF of this distribution is not normalized to unity:

Normalize the distribution:

Automatically normalize:

Normalization will not change the sign of the PDF:

Normalization will not be meaningful when the integral is not defined:

Use TransformedDistribution to create a discrete probability distribution with noninteger support:

Wolfram Research (2010), ProbabilityDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ProbabilityDistribution.html (updated 2015).

#### Text

Wolfram Research (2010), ProbabilityDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/ProbabilityDistribution.html (updated 2015).

#### CMS

Wolfram Language. 2010. "ProbabilityDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/ProbabilityDistribution.html.

#### APA

Wolfram Language. (2010). ProbabilityDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ProbabilityDistribution.html

#### BibTeX

@misc{reference.wolfram_2022_probabilitydistribution, author="Wolfram Research", title="{ProbabilityDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/ProbabilityDistribution.html}", note=[Accessed: 05-October-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2022_probabilitydistribution, organization={Wolfram Research}, title={ProbabilityDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/ProbabilityDistribution.html}, note=[Accessed: 05-October-2022 ]}