Expected Value and Statistics

Question

If X is a continuous random variable with density function f (x) that is an even function and E(X) exists, show that E(X) = 0

Answer 1

As E(X) exists and f(x) is continuous, E(X) = ∫_-∞^∞xf(x)dx = ∫_-∞⁰xf(x)dx + ∫₀^∞xf(x)dx

Let Y = ∫₀^∞xf(x)dx

Let z =-x

Then,

∫_-∞⁰xf(x)dx = -∫₀^∞zf(-z)dz =, as f(z) is even, - ∫₀^∞zf(z)dz = - Y

∫_-∞⁰xf(x)dx + ∫₀^∞xf(x)dx = -A + A = 0

E(X) = 0