Let X be a continuous random variable with probability density function f. We say that X is symmetric about a if for all x,
P(X ≥ a+x)=P(X ≤ a-x).
(a) Prove that X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).
(b) Show that X is symmetric about a if and only if f(x) = f(2a - x) for all x.
(c) Let X be a continuous random variable with probability density function
f(x) = [1 / √(2phi)] e^-(x-3)²/2, x E R,
and Y be a continuous random variable with probability density function
g(x) = 1 / phi [1 + (x-1)²], x E R.
Find the points about which X and Y are symmetric.
