3.5.3.1.21 Nctcdf

Definition:

\[prob = nctcdf( t, df, delta, maxiter)\] computes the lower tail probability for the non-central Student's t-distribution.

The lower tail probability of the non-central Student's t-distribution with \(\nu\) degrees of freedom and non-centrality parameter \(\delta\),\(P(T\leq t)\) is defined by: \(P(T\leq t)=C_\nu \int_0^\infty (\frac 1{\sqrt{2\pi }}\int_{-\infty }^{\alpha u-\delta }e^{-x^2}dx)u^{v-1}e^{-u^2/2}du,\nu >0\)
with

\[C_\nu =\frac 1{\Gamma (\frac 12\nu )2^{(\nu -2)/2}},\alpha =\frac t{\sqrt{\nu }}\]

Parameters:

t (intput,double)

The deviate from the Student's t-distribution with \(\nu\) degrees of freedom, t.

\(-\infty <t<+\infty\).

df (intput,double)

The degrees of freedom of the Student's t-distribution,\(\nu\).\(df\geq 0\).

delta (intput,double)

The non-centrality parameter of the Student's t-distribution,\(\delta\).

maxiter (intput,int)

Optional. Maximum number of iterations, default value is 100.

prob (output,double)

The probability.