| Cauchy {distributions3} | R Documentation |
Create a Cauchy distribution
Description
Note that the Cauchy distribution is the student's t distribution with one degree of freedom. The Cauchy distribution does not have a well defined mean or variance. Cauchy distributions often appear as priors in Bayesian contexts due to their heavy tails.
Usage
Cauchy(location = 0, scale = 1)
Arguments
location |
The location parameter. Can be any real number. Defaults
to |
scale |
The scale parameter. Must be greater than zero (?). Defaults
to |
Details
We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.
In the following, let X be a Cauchy variable with mean
location = x_0 and scale = \gamma.
Support: R, the set of all real numbers
Mean: Undefined.
Variance: Undefined.
Probability density function (p.d.f):
f(x) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_0}{\gamma} \right)^2 \right]}
Cumulative distribution function (c.d.f):
F(t) = \frac{1}{\pi} \arctan \left( \frac{t - x_0}{\gamma} \right) +
\frac{1}{2}
Moment generating function (m.g.f):
Does not exist.
Value
A Cauchy object.
See Also
Other continuous distributions:
Beta(),
ChiSquare(),
Erlang(),
Exponential(),
FisherF(),
Frechet(),
GEV(),
GP(),
Gamma(),
Gumbel(),
LogNormal(),
Logistic(),
Normal(),
RevWeibull(),
StudentsT(),
Tukey(),
Uniform(),
Weibull()
Examples
set.seed(27)
X <- Cauchy(10, 0.2)
X
mean(X)
variance(X)
skewness(X)
kurtosis(X)
random(X, 10)
pdf(X, 2)
log_pdf(X, 2)
cdf(X, 2)
quantile(X, 0.7)
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))