Question

Let X be a continuous random variable with a density function which is symmetric about 0. Show that E(X) = 0.

Answer 1

A probabilistic ( and therefore better ) way of doing the problem is to first let X=a +Y m Assume that E(X) exists ( it need not) . Then E(X) = a+ E (Y) . We need to show that

E (Y)=0.

Then density function g(y) of Y is symmetric about y=0 . It follows that the random variable Y has the same distribution as the random variable -Y. Thus E(Y) = E(-Y) =-E(Y) and therefore each is zero.