Documentation

# `stats`::`cauchyPDF`

Probability density function of the Cauchy distribution

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## Syntax

```stats::cauchyPDF(`a`, `b`)
```

## Description

`stats::cauchyPDF(a, b)` returns a procedure representing the probability density function

of the Cauchy distribution with median a and scale parameter b > 0.

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

• If `x` is a floating-point number and `a` and `b` can be converted to suitable floating-point numbers, then `f(x)` returns a floating-point number.

• In all other cases, the symbolic expression ```b/PI * 1/((x-a)^2 + b^2)``` is returned.

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

## Environment Interactions

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

## Examples

### Example 1

We calculate the Cauchy density with a = 2 and at various points:

```f := stats::cauchyPDF(2, 3/4): f(-infinity), f(9/10), f(0.9), f(2), f(infinity)```

`delete f:`

### Example 2

We use symbolic arguments:

```f := stats::cauchyPDF(a, b): f(x), f(2), f(2.0)```

When a and b evaluate to numbers, the function f starts to produce numerical values:

```a := PI: b:= 1/8: f(2), f(2.0)```

`delete f, a, b:`

## Parameters

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