Question

If a random variable u has t -distribution with n degree of freedom, find the 

distribution of u².

Answer 1

Solution:

Given, $u$ has t-distribution.

So, $u\sim \dfrac{Z}{\sqrt{X/n}}=\dfrac{N(0,1)}{\chi^2_n/n}$

Then, $u^2\sim \dfrac{Z^2}{{X/n}}=\dfrac{\chi^2_1/1}{\chi^2_n/n}=\dfrac{n\chi^2_1}{\chi^2_n}$

Thus, distribution of $u^2$ is $\dfrac{n\chi^2_1}{\chi^2_n}$