7. Let X be a continuous random variable with density function f . We say that X is symmetric about if for all x,

7. Let X be a continuous random variable with density function f . We say that X is symmetric about α if for all x, P (X ≥ α + x) = P (X ≤ α − x).

(a) Prove thatX is symmetric about α if and only if for all x, we have f (α−x) = f (α + x).

(b) Let X be a continuous random variable with probability density function f (x) = 1 √2π e−(x−3)2/2 , x ∈ R, and Y be a continuous random variable with probability density function g(x) = 1 π 4 1 + (x − 1)2 5, x ∈ R. Find the points about which X and Y are symmetric.