Noncentral t-distribution

In probability and statistics, the noncentral t -distribution is a generalization of Student's t-distribution. It is useful for calculating confidence intervals over non–central statistical parameters such as "the 10th percentile of X", given only sample data.

Occurrence and Specification of the Noncentral t-distribution
If $$X$$ is a normally distributed random variable with unit variance and mean $$\mu$$ and $$Y$$ is a chi-square random variable with $$\nu$$ degrees of freedom that's statistically independent of $$X$$, then

T=\frac{X}{\sqrt{Y/\nu}} $$ is a noncentral t-distributed random variable with $$\nu$$ degrees of freedom and noncentrality parameter $$\mu$$.

The probability density function for the noncentral t-distribution is

f(t) =\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} $$
 * $$\times\int\limits_0^\infty

x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx $$ where $$\nu>0$$. The support of this density is the reals.

The mean and variance of the noncentral t-distribution are

\mbox{E}\left[T\right]= \begin{cases} \mu\sqrt{\frac{\nu}{2}}\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)} &\nu>1\\ \mbox{Does not exist} &\nu\le1\\ \end{cases} $$ and

\mbox{Var}\left[T\right]= \begin{cases} \frac{\nu(1+\mu^2)}{\nu-2} -\frac{\mu^2\nu}{2} \left(\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)}\right)^2 &\nu>2\\ \mbox{Does not exist} &\nu\le2\\ \end{cases}. $$

Example Plots
Noncentral t-distributions with two degrees of freedom ($$\nu=2$$) and various noncentrality parameters are shown in the following figure. In the $$\mu=0$$ case, the noncentral t-distribution becomes the t-distribution.



Special cases
When $$ \mu=0 $$, the noncentral t-distribution becomes the t-distribution.

Related distributions

 * $$ Z $$ has a noncentral F-distribution if $$ Z=T^2 $$ where $$ T $$ has a noncentral t-distribution.
 * $$ Z $$ has a normal distribution if $$ Z=\lim_{\nu\to\infty}T $$ where $$ T $$ has a noncentral t-distribution.