# Td-class

##### Class "Td"

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 \(\nu > 2\)).
C.f. `rt`

- Keywords
- distribution

##### Note

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

is defined as a the distribution of
\(T_{\nu}(\delta) := \frac{U + \delta}{\chi_{\nu}/\sqrt{\nu}}\)
where \(U\) and \(\chi_{\nu}\) are independent random
variables, \(U \sim {\cal N}(0,1)\), and
\(\chi^2_\nu\)
is chi-squared, see `rchisq`

.

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.
\( N(\mu,\sigma^2)\).
Then \(T\) is distributed as non-centrally \(t\) with
`df`

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

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

##### 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`

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

```
# NOT RUN {
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.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
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 2.7.0, License: LGPL-3*