If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ "have the same distribution"?

Question

Q: If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ have the same distribution?

In a way, this seems correct: both $X$ and $-X$ have the same probability density functions. But, in other ways, it seems incorrect: (a) they're clearly not independent, and (b) $X \neq -X$, in general.

Is there a better way of phrasing this, both in this case, and in general (where $X$ and $f(X)$ have the same probability density functions)? Or is the original phrasing acceptable?

Answer 1

Hint: Two random variables $X,Y$ are equal in distribution if $$ \mathbb{P}(X\leq z)=\mathbb{P}(Y\leq z) \, \mbox{for all } z. $$

It doesn't have to do with independance either. Clearly $X$ and $X$ have the same distribution but$\ldots$

Answer 2

Since $X$ has a normal distribution, which is symmetric around its mean, then yes, $X$ and $-X$ have the same distribution.

For example, if $X \sim N(0,17)$, then $-X \sim N(0,17)$.

In addition, $X$ and $-X$ are negatively correlated, since $-X = -x$ when $X = x$. But two identically distributed variables can still be correlated, so that's OK.