ProbabilityDistribution

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:

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:

Find each MarginalDistribution:

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: