If Y is a continuous variable with density function f(Y) that is symmetric about 0, that is, f(y) = f(y) for all y, and E(Y) exists.

Show that E(Y) = 0.

Hint:

E(Y) =

yf(y)dy=0yf(y)dy+0yf(y)dy

Make the change of variable x=-y in the first integral.