We know that density for a student-t distribution is given as
$$\frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left[1+\frac{\lambda(x-\mu)^2}{\nu}\right]^{-\frac{\nu+1}{2}}$$
with $\text{E}(X) = \mu$, $\text{var}(X) = \frac{1}{\lambda}\frac{\nu}{\nu-2}$
where the three parameters are shape factor $\nu$, location $\mu$ and dispersion $\lambda$.
Now if I standardize X to Z as $\frac{X-\text{E}(X)}{\text{Std}(X)}$, what is the distribution of $Z$. If I standardize my input data to $Z$ as above, what is the density function for $f_Z(z)$?
I came across this question in Stephen Taylor's Asset Dynamics book. I am a bit confused after reading this question. As I understand, $\frac{X-\mu}{\sigma}$ where $\sigma=1/\sqrt(\lambda)$ will give the standardized t distribution with center 0 and disperson 1. But not sure what $Z$ above will look like.