The most general representation of the noncentral *t* distribution is quite complicated. Johnson and Kotz [61] give a formula for the probability
that a noncentral *t* variate falls in the range
[–*u*, *u*].

$$P\left(-u<x<u|\nu ,\delta \right)={\displaystyle \sum _{j=0}^{\infty}\left(\frac{{\left(\frac{1}{2}{\delta}^{2}\right)}^{j}}{j!}{e}^{\frac{-{\delta}^{2}}{2}}\right)}I\left(\frac{{u}^{2}}{\nu +{u}^{2}}|\frac{1}{2}+j,\frac{\nu}{2}\right)$$

*I*(*x*|*ν*,*δ*)
is the incomplete beta function with parameters *ν* and *δ*. *δ* is
the noncentrality parameter, and *ν* is the
number of degrees of freedom.

The noncentral *t* distribution is a generalization
of Student's *t* distribution.

Student's *t* distribution with *n* –
1 degrees of freedom models the *t*-statistic

$$t=\frac{\overline{x}-\mu}{s/\sqrt{n}}$$

where
is
the sample mean and *s* is the sample standard
deviation of a random sample of size *n* from a
normal population with mean *μ*. If the population
mean is actually *μ*_{0},
then the *t*-statistic has a noncentral *t* distribution
with noncentrality parameter

$$\delta =\frac{{\mu}_{0}-\mu}{\sigma /\sqrt{n}}$$

The noncentrality parameter is the normalized difference between *μ*_{0} and *μ*.

The noncentral *t* distribution gives the
probability that a *t* test will correctly reject
a false null hypothesis of mean *μ* when
the population mean is actually *μ*_{0};
that is, it gives the power of the *t* test. The
power increases as the difference *μ*_{0} – *μ* increases,
and also as the sample size *n* increases.

Compute the pdf of a noncentral *t* distribution with degrees of freedom `V = 10`

and noncentrality parameter `DELTA = 1`

. For comparison, also compute the pdf of a *t* distribution with the same degrees of freedom.

x = (-5:0.1:5)'; nct = nctpdf(x,10,1); t = tpdf(x,10);

Plot the pdf of the noncentral *t* distribution and the pdf of the *t* distribution on the same figure.

plot(x,nct,'b-','LineWidth',2) hold on plot(x,t,'g--','LineWidth',2) legend('nct','t')

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