The question states:
A random variable $X$ is called symmetric about 0 if for all $x \in \mathbb R$, $\mathbb P(X \geq x) = \mathbb P(X \leq -x)$.
Prove that if $X$ is symmetric about 0, then for all $t > 0$, its distribution function $F$ satisfies the following relations: (I'm only going to give one example so I can do the rest myself)
a) $\mathbb P(|X|\leq t) = 2F(t)-1$.
How do I prove this?
And also what does it mean that the random variable $X$ is symmetric about 0?