I recently learnt that r.v. $T=\frac{X_0}{\sqrt{\frac{1}{n}(Y_1^2+..+Y_n^2)}}$ follows t-distribution with parameter n, when $X_0,Y_1,..,Y_n \overset{i.i.d.}{\sim} \mathcal{N}(0,1)$

I also learnt that for a small sample size n, $T_0=\frac{\bar{X}-\mu}{S/\sqrt{n}}$ follows t-distribution when $X_1,..,X_n \overset{i.i.d.}{\sim} \mathcal{N}(\mu,\sigma^2)$. $\bar{X}$ and $S$ are sample mean and s.d. respectively.

However, when I try to derive $T$ from $T_0$, I am stuck.

I could derive that $\bar{X} \sim \mathcal{N}(\mu,\sigma^2/n)$. This means $X_0=\frac{\bar{X} - \mu}{\sigma^2/n}\ \sim \mathcal{N}(0,1)$

I could derive that $X_1^2+..+X_n^2 \sim \chi_n^2$

But can't mode further.

I'll appreciate any help here.

  • $\begingroup$ Does this answer your question? A normal divided by $\sqrt{\chi^2(s)/s}$ $\endgroup$
    – jcken
    Commented Jun 9, 2020 at 19:36
  • $\begingroup$ @jcken - Not really - until I found this [tutorial and slide #13][1]. I still need to complete the proof. So, do you still think an answer to this question will not help others! [1]: ocw.mit.edu/courses/mathematics/… $\endgroup$
    – KGhatak
    Commented Jun 11, 2020 at 18:16
  • $\begingroup$ You defined $X_0$ isn't a $N(0,1)$ RV. You need to divide by the standard error/deviation ($\sigma/ \sqrt (n) $) to get a standard normal $\endgroup$
    – jcken
    Commented Jun 12, 2020 at 18:48
  • $\begingroup$ Can anyone pls suggest few books that I can refer to for self-learning these proofs and concepts thoroughly. $\endgroup$
    – KGhatak
    Commented Jun 13, 2020 at 20:29