# Documentation

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# `stats`::`cauchyRandom`

Generate a random number generator for Cauchy deviates

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## Syntax

```stats::cauchyRandom(`a`, `b`, <`Seed = n`>)
```

## Description

`stats::cauchyRandom(a, b)` returns a procedure that produces Cauchy deviates (random numbers) with median a and scale parameter b > 0.

The procedure `f := stats::cauchyRandom(a, b)` can be called in the form `f()`. The return value of `f()` is either a floating-point number or a symbolic expression:

If `a` can be converted to a real floating point number and `b` to a positive real floating point number, then `f()` returns a real floating point number.

In all other cases, f() returns the symbolic call `stats::cauchyRandom(a, b)()`.

Numerical values of `a` and `b` are only accepted, if they are real and b is positive.

The values `X = f()` are distributed randomly according to the the Cauchy distribution with parameters `a` and `b`. For any real `x`, the probability that Xx is given by

.

Without the option `Seed = n`, an initial seed is chosen internally. This initial seed is set to a default value when MuPAD® is started. Thus, each time MuPAD is started or re-initialized with the `reset` function, random generators produce the same sequences of numbers.

### Note

In contrast to the function `random`, the generators produced by `stats::cauchyRandom` do not react to the environment variable `SEED`.

For efficiency, it is recommended to produce sequences of K random numbers via

`f := stats::cauchyRandom(a, b): f() \$ k = 1..K;`
rather than by
`stats::cauchyRandom(a, b)() \$k = 1..K;`
The latter call produces a sequence of generators each of which is called once. Also note that
`stats::cauchyRandom(a, b, Seed = n)() \$ k = 1..K;`
does not produce a random sequence, because a sequence of freshly initialized generators would be created each of them producing the same number.

## Environment Interactions

The function is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

We generate Cauchy deviates with parameters a = 2 and :

`f := stats::cauchyRandom(2, 3/4): f() \$ k = 1..4`

`delete f:`

### Example 2

With symbolic parameters, no random floating-point numbers can be produced:

`f := stats::cauchyRandom(a, b): f()`

When a and b evaluate to suitable real numbers, the generator starts to produce random numbers:

`a := -PI: b := 1/2: f() \$ k = 1..4`

`delete f, a, b:`

### Example 3

We use the option `Seed` = `n` to reproduce a sequence of random numbers:

`f := stats::cauchyRandom(PI, 3, Seed = 1): f() \$ k = 1..4`

`g := stats::cauchyRandom(PI, 3, Seed = 1): g() \$ k = 1..4`

`f() = g(), f() = g()`

`delete f, g:`

## Parameters

 `a` The median: an arithmetical expression representing a real value `b` The scale parameter: an arithmetical expression representing a positive real value

## Options

 `Seed` Option, specified as `Seed = n` Initializes the random generator with the integer seed `n`. `n` can also be the option `CurrentTime`, to make the seed depend on the current time. This option serves for generating generators that return predictable sequences of pseudo-random numbers. The generator is initialized with the seed `n` which may be an arbitrary integer. Several generators with the same initial seed produce the same sequence of numbers. When this option is used, the parameters `a` and `b` must be convertible to suitable floating-point numbers at the time when the random generator is generated.

## Algorithms

The implemented algorithm for the computation of the Cauchy deviates uses the quantile function of the Cauchy distribution applied to uniformly distributed random numbers between 0 and 1.