# TDist

##### The Student t Distribution

Density, distribution function, quantile function and random
generation for the t distribution with `df`

degrees of freedom
(and optional non-centrality parameter `ncp`

).

- Keywords
- distribution

##### Usage

```
dt(x, df, ncp, log = FALSE)
pt(q, df, ncp, lower.tail = TRUE, log.p = FALSE)
qt(p, df, ncp, lower.tail = TRUE, log.p = FALSE)
rt(n, df, ncp)
```

##### 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. - df
- degrees of freedom ($> 0$, maybe non-integer).
`df = Inf`

is allowed. - ncp
- non-centrality parameter $delta$;
currently except for
`rt()`

, only for`abs(ncp) <= 37.62<="" code="">. If omitted, use the central t distribution.`

- 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, $P[X > x]$.

##### Details

The $t$ distribution with `df`

$= n$ degrees of
freedom has density
$$
f(x) = \frac{\Gamma ((\nu+1)/2)}{\sqrt{\pi \nu} \Gamma (\nu/2)}
(1 + x^2/\nu)^{-(\nu+1)/2}%
$$
for all real $x$.
It has mean $0$ (for $n > 1$) and
variance $n/(n-2)$ (for $n > 2$).

The general *non-central* $t$
with parameters $(df, Del)$ `= (df, ncp)`

is defined as the distribution of
$T(df, Del) := (U + Del) / \sqrt(V/df) $
where $U$ and $V$ are independent random
variables, $U ~ N(0,1)$ and
$V ~ \chi^2(df)$ (see Chisquare).

The most used applications are power calculations for $t$-tests:
Let $T= (mX - m0) / (S/sqrt(n))$
where
$mX$ is the `mean`

and $S$ the sample standard
deviation (`sd`

) of $X_1, X_2, \dots, X_n$ which are
i.i.d. $N(\mu, \sigma^2)$
Then $T$ is distributed as non-central $t$ with
`df`

$= n - 1$
degrees of freedom and **n**on-**c**entrality **p**arameter
`ncp`

$ = (\mu - m0) * sqrt(n)/\sigma$.

##### Value

`dt`

gives the density,
`pt`

gives the distribution function,
`qt`

gives the quantile function, and
`rt`

generates random deviates.Invalid arguments will result in return value `NaN`

, with a warning.The length of the result is determined by `n`

for
`rt`

, 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.
##### Note

Supplying `ncp = 0`

uses the algorithm for the non-central
distribution, which is not the same algorithm used if `ncp`

is
omitted. This is to give consistent behaviour in extreme cases with
values of `ncp`

very near zero.

The code for non-zero `ncp`

is principally intended to be used
for moderate values of `ncp`

: it will not be highly accurate,
especially in the tails, for large values.

##### Source

The central `dt`

is computed via an accurate formula
provided by Catherine Loader (see the reference in `dbinom`

). For the non-central case of `dt`

, C code contributed by
Claus EkstrĂ¸m based on the relationship (for
$x != 0$) to the cumulative distribution. For the central case of `pt`

, a normal approximation in the
tails, otherwise via `pbeta`

. For the non-central case of `pt`

based on a C translation of Lenth, R. V. (1989). *Algorithm AS 243* ---
Cumulative distribution function of the non-central $t$ distribution,
*Applied Statistics* **38**, 185--189. This computes the lower tail only, so the upper tail suffers from
cancellation and a warning will be given when this is likely to be
significant. For central `qt`

, a C translation of Hill, G. W. (1970) Algorithm 396: Student's t-quantiles.
*Communications of the ACM*, **13(10)**, 619--620. altered to take account of Hill, G. W. (1981) Remark on Algorithm 396, *ACM Transactions on
Mathematical Software*, **7**, 250--1. The non-central case is done by inversion.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole. (Except non-central versions.)

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
*Continuous Univariate Distributions*, volume 2, chapters 28 and 31.
Wiley, New York.

##### See Also

Distributions for other standard distributions, including
`df`

for the F distribution.

##### Examples

`library(stats)`

```
require(graphics)
1 - pt(1:5, df = 1)
qt(.975, df = c(1:10,20,50,100,1000))
tt <- seq(0, 10, len = 21)
ncp <- seq(0, 6, len = 31)
ptn <- outer(tt, ncp, function(t, d) pt(t, df = 3, ncp = d))
t.tit <- "Non-central t - Probabilities"
image(tt, ncp, ptn, zlim = c(0,1), main = t.tit)
persp(tt, ncp, ptn, zlim = 0:1, r = 2, phi = 20, theta = 200, main = t.tit,
xlab = "t", ylab = "non-centrality parameter",
zlab = "Pr(T <= t)")
plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32),
main = "Non-central t - Density", yaxs = "i")
```

*Documentation reproduced from package stats, version 3.2.5, License: Part of R 3.2.5*