# 3.5.3.2.18 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 .

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.