# What is the distribution of X? [closed]

Let a random variable X have a t distribution with n degrees of freedom. What is the distribution of X^2?

• Show us what you have done so far to try to solve your problem Dec 10, 2020 at 22:13

Suppose $$X \sim t_n$$. By definition, we have
$$X = \frac{Z}{\sqrt{U/n}}$$
where $$Z \sim \mathcal{N}(0,1)$$ and $$U \sim \chi^2_n$$, and $$Z$$ and $$U$$ are independent. Therefore,
$$X^2 = \frac{Z^2 / 1}{U/n}$$.
Notice that $$Z^2$$ and $$U$$ are still independent. Furthermore, $$Z^2 \sim \chi^2_1$$. $$X^2$$ is the ratio of two independent chi-squared random variables divided by their degrees of freedom. So, $$X^2 \sim F_{1,n}$$.