Cauchy {stats} R Documentation

## The Cauchy Distribution

### Description

Density, distribution function, quantile function and random generation for the Cauchy distribution with location parameter `location` and scale parameter `scale`.

### Usage

```dcauchy(x, location = 0, scale = 1, log = FALSE)
pcauchy(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qcauchy(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rcauchy(n, location = 0, scale = 1)
```

### Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `location, scale` location and scale parameters. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

### Details

If `location` or `scale` are not specified, they assume the default values of `0` and `1` respectively.

The Cauchy distribution with location l and scale s has density

f(x) = 1 / (π s (1 + ((x-l)/s)^2))

for all x.

### Value

`dcauchy`, `pcauchy`, and `qcauchy` are respectively the density, distribution function and quantile function of the Cauchy distribution. `rcauchy` generates random deviates from the Cauchy.

The length of the result is determined by `n` for `rcauchy`, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than `n` are recycled to the length of the result. Only the first elements of the logical arguments are used.

### Source

`dcauchy`, `pcauchy` and `qcauchy` are all calculated from numerically stable versions of the definitions.

`rcauchy` uses inversion.

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 16. Wiley, New York.

Distributions for other standard distributions, including `dt` for the t distribution which generalizes `dcauchy(*, l = 0, s = 1)`.
```dcauchy(-1:4)