1
$\begingroup$

I know that if $Z \sim \mathrm{N}(0,1)$, then $Z^2 \sim \chi^2_1$.

  1. Is the converse true: if $X\sim \chi^2_1$ is the square of a random variable $U$, then $U$ is necessarily standard normal? I think it is false, but it seems my professor thinks it is true. For example $U = |Z|$ is not standard normal, but $U^2 = |Z|^2 = Z^2= X$, seems (to me) a chi-square random variable. What is the truth?

  2. Are $|Z|$ and $Z$ the only (real) random variables which square to $Z^2$?

$\endgroup$

1 Answer 1

1
$\begingroup$

Your counterexample seems correct to me. You can even do $U=ZW$ where $Z \sim N(0,1)$ and $W$ is some random variable that takes values $1$ and $-1$.

$\endgroup$
1
  • $\begingroup$ though that generalisation is only interesting when there is a dependency between $Z$ and $W$. So the answer is to (2) is "No" $\endgroup$
    – Henry
    Commented Dec 15, 2017 at 22:22

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .