| Td-class {distr} | R Documentation |
The t distribution with df = n degrees of
freedom has density
f(x) = Gamma((n+1)/2) / (sqrt(n pi) Gamma(n/2)) (1 + x^2/n)^-((n+1)/2)
for all real x.
It has mean 0 (for n > 1) and
variance n/(n-2) (for n > 2).
C.f. rt
Objects can be created by calls of the form Td(df).
This object is a t distribution.
img:"Reals": The domain of this distribution has got dimension 1
and the name "Real Space". param:"TParameter": the parameter of this distribution (df),
declared at its instantiation r:"function": generates random numbers (calls function rt)d:"function": density function (calls function dt)p:"function": cumulative function (calls function pt)q:"function": inverse of the cumulative function (calls function qt)
Class "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".
signature(.Object = "Td"): initialize method signature(object = "Td"): returns the slot df of the parameter of the distribution signature(object = "Td"): modifies the slot df of the parameter of the distribution
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,...,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.
Thomas Stabla Thomas.Stabla@uni-bayreuth.de,
Florian Camphausen Florian.Camphausen@uni-bayreuth.de,
Peter Ruckdeschel Peter.Ruckdeschel@uni-bayreuth.de,
Matthias Kohl Matthias.Kohl@stamats.de
TParameter-class
AbscontDistribution-class
Reals-class
rt
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.