# Td-class

##### Class "Td"

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 $n > 2$).
C.f. `rt`

- Keywords
- distribution

##### Note

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

is defined as a the distribution of
$
T(df,Del) := (U + Del) / (Chi(df) / sqrt(df)) $
where $U$ and $Chi(df)$ are independent random
variables, $U \~ N(0,1)$, and

$Chi(df)^2$
is chi-squared, see `rchisq`

.

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-centrally $t$ with
`df`

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

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

##### Objects from the Class

Objects can be created by calls of the form `Td(df)`

.
This object is a $t$ distribution.

##### Slots

`img`

- Object of class
`"Reals"`

: The domain of this distribution has got dimension 1 and the name "Real Space". `param`

- Object of class
`"TParameter"`

: the parameter of this distribution (df), declared at its instantiation `r`

- Object of class
`"function"`

: generates random numbers (calls function`rt`

) `d`

- Object of class
`"function"`

: density function (calls function`dt`

) `p`

- Object of class
`"function"`

: cumulative function (calls function`pt`

) `q`

- Object of class
`"function"`

: inverse of the cumulative function (calls function`qt`

) `.withArith`

- logical: used internally to issue warnings as to interpretation of arithmetics
`.withSim`

- logical: used internally to issue warnings as to accuracy
`.logExact`

- logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function
`.lowerExact`

- logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function
`Symmetry`

- object of class
`"DistributionSymmetry"`

; used internally to avoid unnecessary calculations.

##### Extends

Class `"AbscontDistribution"`

, directly.
Class `"UnivariateDistribution"`

, by class `"AbscontDistribution"`

.
Class `"Distribution"`

, by class `"AbscontDistribution"`

.

##### Methods

- initialize
`signature(.Object = "Td")`

: initialize method- df
`signature(object = "Td")`

: returns the slot df of the parameter of the distribution- df<-
`signature(object = "Td")`

: modifies the slot df of the parameter of the distribution- ncp
`signature(object = "Td")`

: returns the slot ncp of the parameter of the distribution- ncp<-
`signature(object = "Td")`

: modifies the slot ncp of the parameter of the distribution

##### Ad hoc methods

For R Version `<2.3.0< code=""> ad hoc methods are provided for slots `

`q`

, `r`

if `ncp!=0`

;
for R Version `>=2.3.0`

the methods from package stats are used.

##### See Also

`TParameter-class`

,
`AbscontDistribution-class`

,
`Reals-class`

,
`rt`

##### Examples

```
T <- Td(df = 1) # T is a t distribution with df = 1.
r(T)(1) # one random number generated from this distribution, e.g. -0.09697573
d(T)(1) # Density of this distribution is 0.1591549 for x = 1.
p(T)(1) # Probability that x < 1 is 0.75.
q(T)(.1) # Probability that x < -3.077684 is 0.1.
df(T) # df of this distribution is 1.
df(T) <- 2 # df of this distribution is now 2.
Tn <- Td(df = 1, ncp = 5)
# T is a noncentral t distribution with df = 1 and ncp = 5.
d(Tn)(1) ## from R 2.3.0 on ncp no longer ignored...
```

*Documentation reproduced from package distr, version 2.6, License: LGPL-3*