# InverseCDF

InverseCDF[dist,q]

gives the inverse of the cumulative distribution function for the distribution dist as a function of the variable q.

# Details • The inverse CDF at q is also referred to as the q quantile of a distribution.
• For a continuous distribution dist the inverse CDF at q is the value x such that CDF[dist,x]q.
• For a discrete distribution dist the inverse CDF at q is the smallest integer x such that CDF[dist,x]q.
• The value q can be symbolic or any number between 0 and 1.

# Examples

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## Basic Examples(2)

The inverse CDF for a continuous univariate distribution:

The inverse CDF for a discrete univariate distribution:

## Scope(12)

### Parametric Distributions(5)

Obtain exact numeric results:

Obtain a machine-precision result:

Obtain a result at any precision for a continuous distribution:

Obtain a symbolic expression for the InverseCDF:

Plot the InverseCDF:

### Derived Distributions(3)

InverseCDF for a truncated distribution:

Quadratic transformation of an exponential distribution:

Censored distribution:

### Nonparametric Distributions(2)

InverseCDF for nonparametric distributions:

Compare with the value for the underlying parametric distribution:

Plot the InverseCDF for a histogram distribution:

### Random Processes(2)

InverseCDF for the SliceDistribution of a random process:

Find the InverseCDF of TemporalData at some time t=0.5:

Find the InverseCDF for a range of times together with all the simulations:

## Generalizations & Extensions(2)

Specify the argument using units of percent or permil:

## Applications(4)

Generate a random number from a distribution:

Find the quartiles of a distribution:

Generate random data for a distribution:

Compute mean of a distribution by integrating inverse CDF function:

Compute mean of 5:3 order statistics of the distribution:

## Properties & Relations(7)

InverseCDF is equivalent to Quantile for univariate distributions:

InverseCDF[,p] is continuous and strictly increasing for 0p1 and continuous:

InverseCDF[,p] is piecewise constant and increasing for 0p1 and discrete:

The function is continuous from the left, and discontinuous from the right:

InverseCDF[,p] is left-continuous and increasing for 0p1 and mixed:

InverseCDF[,CDF[,x]]x for a continuous distribution :

CDF[,InverseCDF[,p]]p for a continuous distribution :

InverseCDF[,CDF[,x]]x for a discrete distribution :

CDF[,InverseCDF[,p]]p for a discrete distribution :

TransformedDistribution[InverseCDF[,p],pUniformDistribution[]] is :

This can be used to generate random variates:

## Possible Issues(2)

Symbolic closed forms do not exist for some distributions:

Numerical evaluation works:

Substitution of invalid values into symbolic outputs gives results that are not meaningful:

When giving the input as argument complete checking is done: