Mathematical Statistics

Question

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

Answer 1

The question seems a little fishy for a couple reasons:

1. A probability density function should be non-negative. An odd function is necessarily negative for half of the inputs by definition: f(-x)=-f(x).
2. E(x)=∫xf(x)dx, integrating from minus infinity to infinity. Integrating an odd function would result in zero as long as the integral is convergent, but since f(x) is said to be odd, and x is an odd function, xf(x) is an even function. The only even function who's integral over the entire x axis equals zero, is the zero function, z(x)=0. Since g(x)=x is nonzero, xf(x)=0 implies that f(x)=0, but this is an even function, which is a contradiction.

So I believe the problem as stated is not solvable.