# Td-class

From distr v2.6
0th

Percentile

##### 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 non-centrality parameter 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

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.

TParameter-class, AbscontDistribution-class, Reals-class, rt

##### Aliases
• Td-class
• Td
• initialize,Td-method
##### 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

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