0
$\begingroup$

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

$\endgroup$
1
  • 3
    $\begingroup$ Show us what you have done so far to try to solve your problem $\endgroup$ Dec 10, 2020 at 22:13

1 Answer 1

1
$\begingroup$

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, $X^2 \sim F_{1,n}$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.