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# stats::cauchyRandom

Generate a random number generator for Cauchy deviates

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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.

## See Also

### MuPAD Functions

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