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

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

).

```
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)
```

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`

.
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, \(P[X > x]\).

`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.

The \(t\) distribution with `df`

\(= \nu\) 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 \(\nu > 1\)) and
variance \(\frac{\nu}{\nu-2}\) (for \(\nu > 2\)).

The general *non-central* \(t\)
with parameters \((\nu, \delta)\) `= (df, ncp)`

is defined as the distribution of
\(T_{\nu}(\delta) := (U + \delta)/\sqrt{V/\nu}\)
where \(U\) and \(V\) are independent random
variables, \(U \sim {\cal N}(0,1)\) and
\(V \sim \chi^2_\nu\) (see Chisquare).

The most used applications are power calculations for \(t\)-tests:
Let \(T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}}\)
where
\(\bar{X}\) is the `mean`

and \(S\) the sample standard
deviation (`sd`

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

\({} = n-1\)
degrees of freedom and **n**on-**c**entrality **p**arameter
`ncp`

\({} = (\mu - \mu_0) \sqrt{n}/\sigma\).

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.

Distributions for other standard distributions, including
`df`

for the F distribution.

# NOT RUN { 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") # }