# CDF

CDF[dist,x]

gives the cumulative distribution function for the distribution dist evaluated at x.

CDF[dist,{x1,x2,}]

gives the multivariate cumulative distribution function for the distribution dist evaluated at {x1,x2,}.

CDF[dist]

gives the CDF as a pure function.

# Details • CDF[dist,x] gives the probability that an observed value will be less than or equal to x.
• CDF[dist,x] is equivalent to Probability[ξx,ξdist].
• CDF[dist,{x1,,xn}] is equivalent to Probability[ξ1x1ξnxn,{ξ1,,ξn}dist].
• CDF[dist,x] is equivalent to 1-SurvivalFunction[dist,x].

# Examples

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## Basic Examples(4)

The CDF of a univariate continuous distribution:

The CDF of a univariate discrete distribution:

The CDF of a bivariate continuous distribution:

The CDF for a multivariate Poisson distribution:

## Scope(21)

### Parametric Distributions(4)

Obtain exact numeric results:

Obtain a machine-precision result:

Obtain a result at any precision for a continuous distribution:

Obtain a result at any precision for a discrete distribution with inexact parameters:

### Nonparametric Distributions(4)

CDF for nonparametric distributions:

Plot the CDF for a histogram distribution:

Closed-form expression for the CDF of a kernel mixture distribution:

Plot of the CDF of a bivariate smooth kernel distribution:

### Derived Distributions(10)

Product of independent distributions:

Component mixture distribution:

Quadratic transformation of a discrete distribution:

Censored distribution:

Truncated distribution:

Parameter mixture distribution:

Copula distribution:

Formula distribution defined by its PDF:

Defined by its CDF:

Defined by its SurvivalFunction:

Marginal distribution:

The CDF for QuantityDistribution assumes the argument is a Quantity with compatible units:

This allows for direct quantity substitution:

Compare with the direct use of the quantity argument:

### Random Processes(3)

Find the CDF for a SliceDistribution of a discrete-state random process:

A continuous-state random process:

Find the multiple time-slice CDF for a discrete-state process:

A multi-slice for a continuous-state process:

Find the CDF for the StationaryDistribution of a discrete-state random process:

## Generalizations & Extensions(1)

Multivariate distributions:

## Applications(5)

Plot the CDF for a standard normal distribution:

Plot the CDF for a binomial distribution:

Compute the probability of for a distribution with 20 degrees of freedom:

Compute the probability of for the same distribution:

Compute the probability of :

Perform a probability integral transform on data by mapping the CDF over it:

The transformed data is uniformly distributed if the original data came from the chosen distribution:

Comparing transformed data to a uniform distribution and comparing original data to original distribution should give identical results for all applicable tests:

Define a general survival distribution function (SDF) as used in actuarial science:

Compare with the expression given by SurvivalFunction:

Define the force of mortality (FM):

Compare with the expression given by HazardFunction:

## Properties & Relations(12)

The probability of for a univariate distribution is given by its CDF:

The probability of for a multivariate distribution is given by its CDF:

A univariate CDF is 0 at and 1 at :

A multivariate CDF has value 0 at and 1 at :

The CDF is the integral of the PDF for continuous distributions :

The CDF is the sum of the PDF for discrete distributions :

CDF and InverseCDF are inverses for continuous distributions:

Compositions of CDF and InverseCDF give step functions for a discrete distribution:

CDF and Quantile are inverses for continuous distributions:

The sum of the CDF and the survival function is 1:

ProbabilityPlot generates a parametric plot of the empirical CDF vs estimated CDF:

CDF is a right-continuous function with left limits:

## Possible Issues(2)

Symbolic closed forms do not exist for some distributions:

Numerical evaluation works:

Substitution of invalid values into symbolic formulas can give results that are not meaningful:

When CDF is given an explicit value as an argument, it does complete checking and does not produce invalid results:

## Neat Examples(1)

CDF for a bivariate censored distribution: