conditional distribution ofr t-distribution

[fisher garrry]

I have not added the rest of Example 5c because it is not relevant for my problem. In the example they obtain the conditional distribution of T given a given y. They do that by adressing y as a constant and the fact that Z is a standard normal variable. The conditional distribution can also be noted: \(\displaystyle f_{T/Y}(t|y)=\frac{f(t,y)}{f(y)}\). My problem is that I dont see that \(\displaystyle f_{T/Y}(t|y)=\frac{f(t,y)}{f(y)}\) and that the conditional distribution of \(\displaystyle T=\sqrt{n}\frac{Z}{\sqrt{Y}}\) if Y=y is the same. I get that the mean would be 0 as for Z and that the variance would be determined: \(\displaystyle var(\frac{\sqrt{n}}{\sqrt{Y}}Z)=\frac{n}{Y}var(Z)\) and that you could use thoose in the normal distribution but why would that approach be the same as obtaining \(\displaystyle f_{T/Y}(t|y)=\frac{f(t,y)}{f(y)}\). Can someone prove the link between \(\displaystyle f_{T/Y}(t|y)=\frac{f(t,y)}{f(y)}\) and threating y as a constant and writing the normal distribution of \(\displaystyle T=\sqrt{n}\frac{Z}{\sqrt{Y}}\) mathematically?