Define a continuous probability distribution:

Probability density function:

Cumulative distribution function:

The 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:

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:

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:

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: