Product of two distribution functions.

Question

Let F and G be two distribution functions, does the product FG still a distribution function?

Answer 1

If, as the caps hint at, you mean that $F$ and $G$ are cumulative distribution functions, the answer is yes. We need to verify that the product has the required properties. So we want to show that $F(x)G(x)$ is continuous from the right, that $\lim_{x\to\infty}F(x)G(x)=1$, that $F(x)G(x)$ is non-decreasing, that $\lim_{x\to -\infty} F(x)G(x)=0$. The verifications are straightforward.