9. A random variable X is called symmetric about 0 if for all x ∈ R , P (X ≥ x) = P (X ≤ −x). Prove that if X is symmetric about 0, then for all t > 0 its distribution function F satisfies the following relations:

(a) P ! |X| ≤ t " = 2F (t) − 1.

(b) P ! |X| > t" = 2 4 1 − F (t)5 .

(c) P (X = t) = F (t) + F (−t) − 1.