# student_t: Scaled and shifted Student's t distribution In ggdist: Visualizations of Distributions and Uncertainty

## Description

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

 ```1 2 3 4 5 6 7``` ```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` vector of quantiles. `df` degrees of freedom (> 0, maybe non-integer). ```df = Inf``` is allowed. `mu` Location parameter (median) `sigma` Scale parameter `log` logical; if TRUE, probabilities p are given as log(p). `q` vector of quantiles. `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. `log.p` logical; if TRUE, probabilities p are given as log(p). `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.

`parse_dist()` and parsing distribution specs and the `stat_dist_slabinterval()` family of stats for visualizing them.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```library(dplyr) library(ggplot2) library(forcats) 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_dist_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)) ```