

   TThhee SSttuuddeenntt tt DDiissttrriibbuuttiioonn

        dt(x, df)
        pt(q, df, ncp=0)
        qt(p, df)
        rt(n, df)

   AArrgguummeennttss::

        x,q: vector of quantiles.

          p: vector of probabilities.

          n: number of observations to generate.

         df: degrees of freedom.

        ncp: non-centrality parameter delta; currently `ncp <=
             37.62'.

   DDeessccrriippttiioonn::

        These functions provide information about the t distri-
        bution with `df' degrees of freedom (and optional non-
        centrality parameter `ncp').  `dt' gives the density,
        `pt' gives the distribution function, `qt' gives the
        quantile function and `rt' generates random deviates.

   DDeettaaiillss::

        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.

        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 `pchisq'.

        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.

   RReeffeerreenncceess::

        Lenth, R. V. (1989). Algorithm AS 243 - Cumulative dis-
        tribution function of the non-central t distribution,
        Appl. Statist. 38, 185-189.

   SSeeee AAllssoo::

        `df' for the F distribution.

   EExxaammpplleess::

        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))
        image(tt,ncp,ptn, zlim=c(0,1),main=t.tit <- "Non-central t - Probabilities")
        persp(tt,ncp,ptn, zlim=0:1, r=2, phi=20, theta=200, main=t.tit)

