Derivation of t-distribution for small sample from Normal population [duplicate]

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.

• Does this answer your question? A normal divided by $\sqrt{\chi^2(s)/s}$ Commented Jun 9, 2020 at 19:36
• @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/… Commented Jun 11, 2020 at 18:16
• 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 Commented Jun 12, 2020 at 18:48
• Can anyone pls suggest few books that I can refer to for self-learning these proofs and concepts thoroughly. Commented Jun 13, 2020 at 20:29