Density, distribution function, quantile function and random generation for the
scaled and shifted Student's t distribution, parameterized by degrees of freedom (`df`

),
location (`mu`

), and scale (`sigma`

).

## Usage

```
dstudent_t(x, df, mu = 0, sigma = 1, log = FALSE)
pstudent_t(q, df, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qstudent_t(p, df, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rstudent_t(n, df, mu = 0, sigma = 1)
```

## Arguments

- x, q
vector of quantiles.

- df
degrees of freedom (\(> 0\), maybe non-integer).

`df = Inf`

is allowed.- mu
Location parameter (median)

- sigma
Scale parameter

- log, log.p
logical; if TRUE, probabilities p are given as log(p).

- lower.tail
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required.

## Value

`dstudent_t`

gives the density`pstudent_t`

gives the cumulative distribution function (CDF)`qstudent_t`

gives the quantile function (inverse CDF)`rstudent_t`

generates random draws.

The length of the result is determined by `n`

for `rstudent_t`

, 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.

## See also

`parse_dist()`

and parsing distribution specs and the `stat_slabinterval()`

family of stats for visualizing them.

## Examples

```
library(dplyr)
library(ggplot2)
expand.grid(
df = c(3,5,10,30),
scale = c(1,1.5)
) %>%
ggplot(aes(y = 0, dist = "student_t", arg1 = df, arg2 = 0, arg3 = scale, color = ordered(df))) +
stat_slab(p_limits = c(.01, .99), fill = NA) +
scale_y_continuous(breaks = NULL) +
facet_grid( ~ scale) +
labs(
title = "dstudent_t(x, df, 0, sigma)",
subtitle = "Scale (sigma)",
y = NULL,
x = NULL
) +
theme_ggdist() +
theme(axis.title = element_text(hjust = 0))
```